Mathematics

Dr. Sheldon Gordon

SUNY Distinguished Teaching Professor gordonsp@retiree,farmingdale.edu

If you are interested in receiving gratis any, or all, of these files for the DIGMath spreadsheets and the copies of the recent articles for your personal use or to provide to your students, please feel free to contact me at the e-mail address directly above this paragraph.

DIGMath Software Packages

Dynamic Investigatory Graphical Displays of Math: DIGMath Graphical Simulations for Statistics and Probability in Excel
Dynamic Investigatory Graphical Displays of Math: DIGMath Graphical Explorations for College Algebra and Precalculus in Excel
Dynamic Investigatory Graphical Displays of Math: DIGMath Graphical Explorations for Calculus in Excel
Dynamic Investigatory Graphical Displays of Math: DIGMath Graphical Explorations for Numerical Analysis in Excel
Dynamic Investigatory Graphical Displays of Math: DIGMath Graphical Explorations for Differential Equations in Excel

Note that most of these DIGMath modules require the use of macros to operate. To use these spreadsheets, Excel must be set to accept macros. To change the security setting on macros:

When you open the spreadsheet, a new bar appears near the top of the window that says something like: "Security Warning: Some active content has been disabled" followed by a button reading "Enable Content". The exact phrasing varies with the version of Excel you are using.
Click on "Enable Content" and then click OK.

Books

Functions, Data, and Models: An Applied Approach to College Algebra (co-author), MAA Textbooks, Mathematical Association of America, 2010.
A Fresh Start for Collegiate Mathematics: Rethinking the Courses Below Calculus, (co-editor), Nancy Baxter Hastings, et al., Eds., MAA Notes #69, Mathematical Association of America, 2006. I co-edited this highly influential collection of articles by leading mathematics educators at major universities across the country as part of the NSF’s major initiative to change the focus of the courses below calculus that was published by the MAA.
Functioning in the Real World: A PreCalculus Experience, (principal author), Math Modeling/ PreCalculus Reform Project, Addison-Wesley, 2004. This textbook was awarded the first prize in a national competition sponsored jointly by the Annenberg Foundation and the Corporation for Public Broadcasting as the best Innovative Project Using Technology in the algebra through precalculus category. The book was the first to emphasize conceptual understanding and realistic applications of the mathematics drawn from virtually every other quantitative field instead of focusing exclusively on the development of algebraic skills presumably needed in subsequent calculus courses that vanishingly few students ever take.
Contemporary Statistics: A Computer Approach, textbook and software package (co-author), McGraw- Hill, 1994. This was the first statistic book to integrate computer graphic software throughout the book to demonstrate visually and to simulate virtually every topic and method in statistics and probability to promote a significantly deeper level of conceptual understanding of the material. The book influenced all subsequent statistics texts to emphasize the concepts and computer usage instead of statistical computations. The book was translated into Chinese.
Calculus, The Harvard Core Calculus Consortium (co-author), Wiley, 1998. The calculus books developed by the Calculus Consortium based at Harvard, a group of eminent mathematics educators from institutions including Harvard, Stanford, Colgate, and about half a dozen others under major grants from the NSF to develop a completely fresh approach to calculus that focused on conceptual understanding and real-world applications, not just the manipulative skills that had previously been the basis for most calculus courses and textbooks. Our work had immense impact on calculus across the country and abroad and became the basis for the AP Calculus course.
Calculus: Single and Multivariable, The Harvard Core Calculus Consortium (co-author), Wiley, 1998.
Multivariable Calculus, The Harvard Core Calculus Consortium (co-author), Wiley, 1997.
Calculo, Spanish translation of Calculus, 1st Edition, by Hughes Hallett, Gleason, Gordon, et al., Mexico City, Mexico, 1996.
Calculo, Portuguese translation of Calculus, 1st Edition, by Hughes Hallett, Gleason, Gordon, et al., Sao Paulo, Brazil.
Calculo de Varias Variaveis, Portuguese translation of Multivariable Calculus, 1st Edition, by McCallum, Hughes Hallett, Gleason, Gordon, et al., Sao Paulo, Brazil.
Applied Calculus for Business, Economics and Biology, The Harvard Core Calculus Consortium (co-author), Wiley, 1996.
Brief Calculus for Business, Economics and Biology, The Harvard Core Calculus Consortium (co-author), Wiley, 1997.
Statistics for the Twenty First Century, MAA Notes #26, 1992. I co-edited this highly influential collection of articles on new and innovative ways to teach statistics courses written by several dozen of the most distinguished statistics educators in the U.S. and around the world. This volume was published by the MAA and, for over a decade, was used as the basis for every workshop run by both the MAA and the American Statistical Association to prepare thousands of collegiate and high school mathematics teachers to teach statistics courses.
Calculus: The Dynamics of Change, (co-editor), Wayne Roberts, et al., Eds., MAA Notes, 1996. I co-edited this highly influential collection of articles by leading mathematics educators at major universities across the country as part of the NSF’s major initiative to change the focus of calculus that was published by the MAA.
Calculus and the Computer, Prindle, Weber & Schmidt, 1986.

Articles (since 2000)

Visualizing and Understanding the Chain Rule in Calculus, (submitted).
Approximating Exponential and Logarithmic Functions with Newton Interpolation (with Yajun Yang), (submitted).
Exploring Logarithmic Functions with Fractional Bases (with Florence Gordon), The MathAMATYC Educator, ( to appear).
Buffonery in the Math Classroom (with Florence Gordon), (submitted).
Visualizing and Understanding Exponential and Logarithmic Identities Using Dynamic Software (with Florence Gordon), The MathAMATYC Educator, (to appear).
Using Moving Averages to Find Trends in Data (with Florence Gordon), The MathAMATYC Educator, 2023.
Introducing Tangentoidal Functions, The MathAMATYC Educator, 2023.
Dynamic Visualizations of Taylor Approximations to Enhance Student Understanding, Int'l J of Technology in Math Education, (to appear).
Visualizing and Understanding P-Values in Hypothesis Testing using Dynamic Software, The MathAMATYC Educator, 2023.
Variations on a Theme of Fibonacci, (with Michael Burns), Mathematics Teacher and Learner, 2023.
The 'Calculus' of Moving Averages (with Florence Gordon), PRIMUS, 2022.
Developing a Mathematics Curriculum for the Biosciences (with Sarah Gross, Jessica Seifert, Carla Martin, Yajun Yang, Jack Winn, and Matthew Bahamonde), PRIMUS, 2021.
Incorporating Statistics into College Algebra (with Florence Gordon), Int'l J of Technology in Math Education, 2021.
Doubling A Cube and Other Geometric Solids, (with Michael Burns), The MathAMATYC Educator, 2021.
Visualizing and Understanding Optimization Problems Using Dynamic Software, Int'l J. of Technology in Mathematics Education, 2020.
Visualizing and Understanding Sampling Distributions Using Dynamic Software, (with Florence Gordon), Int'l J. of Technology in Mathematics Education, 2019.
Using Secant Parabolas for Root Finding, (with Yajun Yang), The Mathematics Teacher, 2018.
Visualizing and Understanding Regression and Correlation using Dynamic Software, (with Florence Gordon), Int'l J. of Technology in Mathematics Education, 2018.
Approximating Sinusoidal Functions Using Lagrange Interpolation, (with Yajun Yang), The MathAMATYC Educator, 2018.
Visualizing and Understanding Hypothesis Testing Using Dynamic Software, (with Florence Gordon), PRIMUS, 2018.
Visualizing and Understanding l'Hopital's Rule, Int'l J of Math Education in Science and Technology, 2017.
Approximating Exponential and Logarithmic Functions with Lagrange Interpolating Polynomials, (with Yajun Yang), Int'l J of Math Education in Science and Technology, 2017).
Visualizing and Understanding Confidence Intervals Using Dynamic Software, (with Florence Gordon), Math and Computer Education, 2016.
Taking Your Chances on Excel.
Taking Control of Excel, Math and Computer Education, 2016.
Visualizing the Components of Lagrange and Newton Interpolation (with Yajun Yang), PRIMUS, 2015.
Curve Fitting and Lagrange Interpolation, (with Yajun Yang).
Deriving Simpson's Rule Using Newton Interpolation, (with Yajun Yang), Math and Computer Education, 2016.
Functioning in Excel, Math and Computer Education, 2016.
The Flavor of a Modeling Based College Algebra or Precalculus Course, UMAP Journal, 2016.
Finding the Complex Roots of Polynomials, The MathAMATYC Educator, 2016.
Visualizing Techniques of Integration, Math and Computer Education, 2015.
Numerical Integration - One Step at a Time, (with Yajun Yang), PRIMUS, 2016.
How Rational is the Tangent Function, The MathAMATYC Educator, 2015.
Using Quartile-Quartile Lines as Linear Models, PRIMUS, 2015.
Approximating Sinusoidal Functions with Interpolating Polynomials, (with Yajun Yang).
Finding Polynomial Patterns and Newton Interpolation, (with Yajun Yang), Math and Computer Education, 2014.
How Rational is a Logarithmic Function, Math and Computer Education, 2014.
Subdivisions for Definite Integrals with Uniform Spacing, Math and Computer Education, 2014.
Interpolation and Polynomial Curve Fitting (with Yajun Yang), The Mathematics Teacher, 2014.
The Geology of Mathematics, Guest Editorial, UMAP Journal, 2014.
Integrating Statistics into College Algebra to Meet the Needs of Biology Students, (with Florence Gordon), in Undergraduate Mathematics for the Life Sciences: Processes, Models, Assessment, and Directions, MAA Notes, 2013.
Investigating Convergence Patterns for Numerical Methods Using Data Analysis, PRIMUS 2013.
Adding A New Dimension to Mathematics, The Mathematics Teacher, 2013.
Mathematics Education Must Evolve, Math and Computer Education, 2013.
Fitting Tangent Functions to Data, Math and Computer Education, 2013.
Time, Tides, and Temps Wait for All Students The MathAMATYC Educator, 2013.
Some Surprising Errors in Numerical Differentiation, PRIMUS, 2012.
Algebra Must Evolve, Vision - Potential: College Algebra Newsletter, 2012.
Building on Errors in Numerical Integration, Math & Computer Education, 2011.
Introducing Functions of Several Variables in Lower Division Courses, MathAMATYC Educator, 2011.
Placement Testing: The Great Abyss, Vision -- Potential, 2011.
Errors in Mathematics are Not Always Bad, The Mathematics Teacher, 2011.
Finding the Best Quadratic Fit to a Function (with Yajun Yang), Int'l J. of Mathematics Education in Science and Technology, 2011.
Exposing the Mathematical Wizard: Approximating Trigonometric and Other Transcendental Functions, The Mathematics Teacher, 2011.
Intermediate Algebra in High School vs. College, Vision - Potential: College Algebra Newsletter, 2011.
The Average Distance in an Ellipse, Math and Computer Education, 2010.
The State of Mathematics Education Today: What Happens in the Classroom, J of Economics and Finance, 2010.
What the Partner Disciplines Want from College Algebra, Vision - Potential: College Algebra Newsletter, 2010.
The Status and Purpose of College Algebra, Vision - Potential: College Algebra Newsletter, 2010.
The Statistics of a Function (with Florence Gordon), International J. of Mathematics Education in Science and Technology, 2010.
Mathematics for the Laboratory Sciences, MathAMATYC Educator (premier issue), 2009.
Visualizing and Understanding Probability and Statistics: Graphical Simulations Using Excel, (with Florence Gordon), PRIMUS, 2009.
Doubling Time for Non-Exponential Families of Functions The Mathematics Teacher, 2009.
Modeling Population Growth and Extinction PRIMUS, 2009.
The Curriculum Foundations Project in Mathematics and Economics, (with Richard Vogel), MAA Focus, 2009.
What's Wrong with College Algebra?, PRIMUS, 2008.
A Tale of Two Students, PRIMUS, 2008.
Taylor Approximations and Definite Integrals, PRIMUS, 2007.
Using the Sum of the Squares, (with Florence Gordon), Mathematics and Computer Education, 2008.
Finding the Best Linear Approximation to a Function (with Yajun Yang), Mathematics and Computer Education, 2008.
Comparing the Continuous and Discrete Logistic Models, PRIMUS, 2008.
College Algebra: Today's Students, Tomorrow's Courses, MAA Focus, 2008.
Discovering the Fundamental Theorem of Calculus using Data Analysis, (with Florence Gordon), The Mathematics Teacher, 2007.
Placement: The Shaky Bridge between School and College Mathematics, (college version), Placement and Assessment Newsletter, AMATYC, 2006
Fitting Surge Functions to Data, PRIMUS, 2007.
When Does a Curve Bend Most Rapidly: The Curvature Function of a Function (with Loucas Chrysafi), Mathematics and Computer Education, 2006.
Placement: The Shaky Bridge between School and College Mathematics, The Mathematics Teacher, 2006.
What Does Conceptual Understanding Mean? The AMATYC Review, 2006.
Interconnected Quantitative Learning at Farmingdale State University, (with Jack Winn), in Models of Quantitative Literacy, Rick Gillman, ed, MAA Notes, 2006.
Approximating Functions with Exponential Functions, PRIMUS, 2006.
Rethinking Placement in Mathematics, AMATYC Committee on Articulation and Placement Newsletter, 2006.
Preparing Students for Calculus in the Twenty First Century, in A Fresh Start for Collegiate Mathematics: Rethinking the Road Toward Calculus, Nancy Baxter Hastings, Ed., MAA Notes, 2006.
The Need to Rethink Placement in Mathematics, in A Fresh Start for Collegiate Mathematics: Rethinking the Road Toward Calculus, Nancy Baxter Hastings, Ed., MAA Notes, 2006.
Where Do We Go From Here: Forging a National Initiative to Refocus the Courses Below Calculus, in A Fresh Start for Collegiate Mathematics: Rethinking the Road Toward Calculus, Nancy Baxter Hastings, Ed., MAA Notes, 2006.
The Functioning in the Real World Project, in A Fresh Start for Collegiate Mathematics: Rethinking the Preparation for Calculus, Nancy Baxter Hastings, Ed., MAA Notes, 2006.
Discovering the Chain Rule Graphically, Mathematics and Computer Education, 2005.
Developing a 2020 Vision for Mathematics Education, in Generating a Vision for Undergraduate Mathematics, Gary Krahn, Ed., West Point, 2005.
What’s Wrong with College Algebra? SUNY/UUP Working Papers Discussion series.
Deriving the Linear Regression Equations Using Algebra, Mathematics and Computer Education, 2004.
Deriving the Quadratic Regression Equations Using Algebra, Mathematics and Computer Education, 2004.
Taylor's Theorem and the Derivative Tests for Optima and Inflection Points, PRIMUS, 2004.
Algebra for the New Millennium, Int'l J. of Computer Algebra in Math Education, 2004.
An Individualized Term Project for Multivariate Calculus, PRIMUS, 2004.
Mathematics for the New Millennium, in Proc, International Conference on Mathematics Education, Copenhagen, 2004.
Grant Writing to Enhance Teaching and Learning, Teaching-Talk, 2002.
Using Data Analysis to Motivate Derivative Formulas, Mathematics and Computer Education, 2002.
Using Data Analysis to Discover the Fundamental Theorem of Calculus, PRIMUS, 2003.
Refocusing the Courses Below Calculus, in Proc 13th Intl Conf on Tech in Collegiate Math, Addison-Wesley, 2003.
Discovering New Trig Identities, The Mathematics Teacher, 2003.
Rethinking the Preparation for Calculus (co-author), MAA FOCUS, 2002.
The Implications of Hand-Held Computer Algebra System Calculators Throughout the Mathematics Curriculum (with Arlene Kleinstein), AMATYC Review, 2002.
A Spoonful of Medicine Makes the Mathematics Go Down, The AMATYC Review, 2001.
Renewing the Precursor Courses: New Challenges, Opportunities, and Connections, in Calculus Renewal: Issues for Undergraduate Mathematics Education in the Next Decade, Susan Ganter, Ed., Plenum Publishing, 2000.
Mathematical Articulation in New York State: Smoothing the Mathematical Transitions of Students, MetroMath, 2000.
Preparing Students Mathematically for the Twenty First Century, in Proc, Int' Conf. on Mathematics Education, Tokyo, 2000.
Mathematics Education at Farmingdale State College (with Arlene Kleinstein, Jack Winn). in Proc Int'l Conf on Mathematics Education, 2000.
Preparing Students Mathematically for the Twenty First Century, in Proc Int'l Conf on Mathematics Education, 2000.
What Does Being a Good Math Student Mean?, in Technology in Mathematics Education, AMATYC, 2000.
Revisiting Lanchester's Square Law for Military Stalemates, Math & Computer Education, 2000.

Dynamic Investigatory Graphical Displays of Math: DIGMath Graphical Simulations for Statistics and Probability using Excel

As mentioned above, Excel must be set to accept macros by selecting "Enable Content".

The following are the DIGMath explorations that are currently completed and ready for use. If you have any suggestions for improvements or for new topics, please pass them on also.

DIGMath Spreadsheets for Statistics and Probability
Fair Coin This DIGMath spreadsheet is a coin flipping simulation in which you have the choice of the number n of repetitions of flipping a fair coin. It shows both graphically and numerically the results of each fun of the simulation to let you visualize the shape of the distribution of outcomes and the associated likelihood (the probability) of each.
Dice Role A dice rolling simulation in which the user can choose the number of roles of a pair of fair dice.
Coin Flips A coin flipping simulation in which the user has the choice of the number of fair coins being flipped and the number of repetitions.
Binomial Distribution This DIGMath module lets you investigate the binomial distribution based on n trials with a probability p of success. You can select values of n and p using sliders and the program draws the histogram for the corresponding binomial distribution. It also shows the mean and standard deviation of the distribution.
Binomial Simulation This DIGMath spreadsheet is a simulation of binomial probabilities. You have the choice of the number of coins, the probability of success, and the number of repetitions.
Binomial Probabilities This DIGMath module lets you investigate the probabilities associated with a binomial distribution. The spreadsheet covers six different cases, covering virtually al the standard kinds of problems with binomial probabilities. It will calculate and display the binomial probabilities of getting (1) exactly x successes in a binomial process with n trials with a probability p of success; (2) between two given numbers of successes; (3) at least a given number of successes; (4) more than a given number of successes; (5) at most a given number of successes; and (6) less than a given number of successes. In each case, you select values of n, p, and x using sliders and the program draws the histogram to show the corresponding binomial probability.
Law of Large Numbers This DIGMath module lets you investigate the Law of Large Numbers -- what happens in the long run. You have the choice of the desired probability of success p and the number of repetitions n in a binomial experiment to see the pattern of successes over the long run.
Chaos of Small Numbers The user has the choice of the desired probability of success and the number of repetitions (up to 25) to see the actual (simulated) outcomes and the cumulative frequency of success to demonstrate the unpredictable nature of the outcomes in the short run.
The Effects of an Extra Point on the Mean and Standard Deviation This DIGMath module lets you investigate the effect that changing an additional point has on the values for the mean and the standard deviation (both graphically and numerically). The spreadsheet starts with a set of either 5, 10, or 20 points and you can change the value for an additional point to see how it affects the calculations and the extent of the changes depending on the number of points.
Effect of an Extra Point on Statistics This DIGMath module lets you investigate the effects of an extra point on the mean, the median, the standard deviation, and the InterQuartile Range from two perspectives: depending on how close to or far from the center an extra point lies and depending on the size of the dataset.
The Normal Distribution This DIGMath module lets you investigate the normal distribution with mean μ and standard deviation σ. You enter both parameters using sliders and the program draws the corresponding normal distribution curve. You can watch the effects of changing the parameter values on the resulting curve.
Normal Probabilities This DIGMath spreadsheet lets you investigate the probabilities associated with a normal distribution . You can enter, via sliders, values for the mean μ and standard deviation σ and then an interval of x-values from xL to xR to visualize the probability that x lies between these two values . The program raws the graph of the normal distribution, and highlights graphically the region under the normal curve between xLand xR; it also shows numerically the probability that x lies between these two values.
Simulating the Normal Distribution This DIGMath spreadsheet lets you investigate the normal distribution in terms of a random simulation. You can enter, via sliders, values for the mean μ and standard deviation σ and then an interval of x-values from x-Min to x-Max. You can also select the number of random points you want in this normal distribution and the program will generate those points, plot them along with the graph of the normal distribution, and display graphically, with different colors, those that fall under the designated portion of the normal distribution curve and those that do not. The results are also shown numerically and compared to the theoretical values for the area under the normal curve.
Normal Approximation to the Binomial Distribution This DIGMath program lets you investigate how well a normal distribution approximates the binomial distribution based on the parameters n and p. You enter the values for n and p via sliders and the program draws the histogram for the binomial distribution and the corresponding normal distribution curve using μ = np and σ = √np(1-p) to compare the two distributions.
The Poisson Distribution This DIGMath spreadsheet lets you investigate two different aspects of the Poisson distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event. (1) The first looks at the shape of the Poisson distribution depending on its two parameters, the expected number of outcomes of an event in a given time period and the number of occurrences. You use a slider to vary the first parameters and see the effects on the shape of the distribution. (2) The second aspect is based on the idea that the Poisson distribution can be used to approximate the binomial distribution with probability of success p and number of trials n. Using sliders to change n and p, you can observe which combinations make for a good approximation and which do not.
Central Limit Theorem Simulation The user can choose any of four underlying populations (normal, uniformly distributed, skewed, and bimodal), the sample size, and the number of random samples. The simulation randomly generates the samples and plots the means of each sample. From the graphical display and the associated numerical displays, it becomes apparent that (1) the distribution of sample means is centered very close to the mean of the underlying population, that (2) the spread in the sample means is a fraction of the standard deviation of the underlying population (about one-half as large when n = 4, about one-third as large when n = 9, about one-quarter as large when n = 16, etc.), so that students quickly conjecture that the formula for the standard deviation of the distribution of sample means is s/√n, and that (3) as the sample size increases, the sampling distribution looks more and more like a normal distribution.
Visualizing the Sample Mean and the Sample Standard Deviation This DIGMath program helps you visualize the sample means and the sample standard deviations drawn from each of the four underlyiing populations used in the Central Limit Theorem Simulation. You can select your choice of the population and select, using sliders, the sample size and the number of desired samples. The program calculates and displays graphically the average of the sample means and the sample standard deviations.
t-Distributions This DIGMath module lets you investigate the properties of the t-distribution based on various numbers of degrees of freedom from 2 up to 31. You enter the desired number of degrees of freedom and the program draws the corresponding t-distribution curve as well as the curves for d.f. = 1, d.f. = 11, and d.f. = 21 and the limiting normal distribution curve when d.f. = 31.
Distribution of Sample Proportions The user chooses the probability of success π, the sample size n, and the number of random samples. The simulation randomly generates the samples, displays the corresponding proportion of successes, and displays the summary statistics. From these displays, the students quickly conjecture that (1) the mean of the distribution of sample proportions is equal to the proportion of successes in the underlying population, that (2) the simulated results with different values of the sample size n agree with the formula for the standard deviation of this sampling distribution, and that (3) the sampling distribution becomes more and more normal in appearance as the sample size increases.
Sample Medians This DIGMath simulation is similar to the Central Limit Theorem Simulation, but with sample medians instead of sample means.
Sample Midranges This DIGMath simulation is similar to the Central Limit Theorem Simulation, but with sample midranges instead of sample means.
Sample Modes This DIGMath simulation is similar to the Central Limit Theorem Simulation, but with sample modes instead of sample means.
Distribution of Sample Variances This DIGMath simulation is similar to the Central Limit Theorem Simulation, but instead of simulating sample means from a population, the program now generates and displays the sample variances.
Distribution of Sample Standard Deviations This DIGMath simulation is similar to the Central Limit Theorem Simulation, but instead of simulating sample means from a population, the program now generates and displays the sample standard deviations.
Sample IQR's This DIGMath simulation is similar to the Central Limit Theorem Simulation, but instead of simulating sample means from a population, the program now generates and displays the sample InterQuartile Ranges. The IQR is the difference between the first and third quartiles of a set of data and so represents a measure of the spread in the data.
Distribution of Sample Skewnesses This is similar to the Central Limit Theorem Simulation, but instead of simulating sample means from a population, the program now generates and displays the sample skewnesses -- a measure of how far a set of values deviates from a symmetric distribution.
Standard Deviations of Sample Proportions This DIGMath program simulates the distribution of the standard deviation of sample proportions p. You enter the probability of success p, the sample size and the number of random samples and the program randomly generates, calculates, and displays the standard deviation of those sample proportions.
Simulating Confidence Intervals The user has the choice of the same four underlying populations as in the Central Limit Theorem simulation (to see that the population does not affect the results) and the confidence level (90%, 95%, 98%, 99%), as well as the sample size n (from 10 to 50). The simulation generates random samples from the selected population, calculates and plots the corresponding confidence interval, and summarizes the number and percentage of confidence intervals that actually contain the mean of the underlying population. Students see that the actual (simulated) percentage is typically close to the selected value for the confidence level. They also see that typically the higher the confidence level, the longer the lines are that represent the actual confidence interval. They also see that typically those confidence intervals that do not contain the population mean are near-misses.
Constructing Confidence Intervals for Means This DIGMath spreadsheet assists you in constructing a confidence interval for the mean of a population. You enter the sample data -- the sample size n, the sample mean, and the sample standard deviation, and select the level of confidence (90%, 95%, 98%, or 99%) you want. The spreadsheet constructs the corresponding confidence interval and displays it, as well as compares it in size, to the confidence intervals with other levels of confidence.
Simulating Confidence Intervals for Proportions This DIGMath module simulates the notion of confidence intervals for the proportion of successes in a population. The user controls the choice of the population proportion for the underlying population, the confidence level (90%, 95%, 98%, 99%), and the sample size n. The simulation generates a fixed number of random samples from that population, calculates and plots the corresponding confidence interval, and summarizes the number and percentage of confidence intervals that actually contain the proportion of the underlying population. Students see that the actual (simulated) percentage is typically close to the selected value for the confidence level. They also see that typically the higher the confidence level, the longer the lines are that represent the actual confidence interval. They also see that, as the sample size increases, the lengths of the sample confidence intervals decrease and as the sample size decreases, the lengths of the confidence intervals increase. They also see that typically those confidence intervals that do not contain the population proportion are near-misses.
Constructing Confidence Intervals for Proportions This DIGMath spreadsheet assists you in constructing a confidence interval for the proportion of a population. You enter the sample data -- the sample size n and the number of "successes" in that sample -- and select the level of confidence (90%, 95%, 98%, or 99%) you want. The spreadsheet constructs the corresponding confidence interval and displays it, as well as compares it in size, to the confidence intervals with other levels of confidence.
Visualizing Confidence Intervals for the Mean This DIGMath spreadsheet helps you visualize what confidence intervals are for the mean of a population. You enter the sample data -- the sample size n, the sample mean, and the sample standard deviation, and select the level of confidence (90%, 95%, 98%, or 99%) you want. The spreadsheet constructs all four corresponding confidence interval and displays each of them so that you can compare their sizes, their centers, and their lengths to one another as you change the parameters using sliders.
Visualizing Confidence Intervals for the Proportion This DIGMath spreadsheet helps you visualize what confidence intervals are for the proportion of a population. You enter the sample data -- the sample size n and the proportion p of successes, and select the level of confidence (90%, 95%, 98%, or 99%) you want. The spreadsheet constructs all four corresponding confidence interval and displays each of them so that you can compare their sizes, their centers, and their lengths to one another as you change the parameters using sliders.
Confidence Intervals for the Difference of Means This DIGMath module helps in constructing a confidence interval for the difference of mean based on summary sample data from two samples: the size of the samples, the sample means, and the sample standard deviations. The user can choose the confidence level desired -- 90%, 95%, 98%, or 99% and the resulting intervals are shown graphically and numerically.
Confidence Intervals for Difference of Proportions This DIGMath module helps in constructing a confidence interval for the difference in population proportion based on summary sample data from two samples: the size of the samples and the number of "successes" in each sample. The user can choose the confidence level desired -- 90%, 95%, 98%, or 99% and the resulting intervals are shown graphically and numerically.
Simulating Hypothesis Testing This DIGMath module provides a visual simulation of the process of testing a hypothesis regarding the population mean π. You have the choice of the same four underlying populations (again, to see that the population does not affect the results) and the level of significance (10%, 5%, 2%, 1%). The simulation generates a fixed number of samples from the selected population, plots the mean of each sample with a vertical line at the appropriate location, and summarizes the number and percentage of sample means that fall into this region. The height of each line is equal to the standard deviation of that sample. Students see that the simulated percentage of sample means that fall in the rejection region is typically close to the selected level of significance. They also see that most of the sample means that fall into the rejection region tend to be quite close to the critical values. They also see that the lines representing the samples whose means are close to the population mean tend to be very tightly clustered compared to those that are near the extreme ends, which are sparsely distributed.
Visualizing Hypothesis Testing for the Mean This DIGMath spreadsheet helps you visualize the fundamental ideas related to testing a hypothesis for the mean of a population. You enter the sample data -- the sample size n, the sample mean, and the sample standard deviation, the choice of a two-tailed test or a one-tailed test with the tail on the right or the left, and select the significance level α you want highlighted. The spreadsheet displays all four corresponding critical values for a one-tailed test or all eight critical values for a two-tailed test and shows the position of the sample mean for the data. It also displays the associated z- or t-value and the conclusion of whether you can reject or fail to reject the null hypothesis as the selected significance level.
Hypothesis Tests for the Mean This DIGMath spreadsheet assists you in testing a hypothesis for the mean of a population. You enter the null hypothesis for the supposed value of µ and select the test you want -- either two tailed or one tail with either tail. You then enter the sample data -- the sample size n, the sample mean m and the sample standard deviation s -- and the level of significance α. The spreadsheet displays the corresponding normal or t-distribution, the location of the critical value(s), and the location of the sample mean. It also shows the associated z- or t-value, as well as the corresponding P-value, and the conclusion as to whether you Reject or Fail to Reject the null hypothesis.
Hypothesis Tests for the Proportion This DIGMath spreadsheet assists you in testing a hypothesis for the proportion of a population. You enter the null hypothesis for the supposed value of π and select the test you want -- either two tailed or one tail with either tail. You then enter the sample data -- the sample size n and the number of successes x in that sample -- and the level of significance α. The spreadsheet displays the corresponding normal distribution (when appropriate), the location of the critical value(s), and the location of the sample proportion p. It also shows the associated z-value, as well as the corresponding P-value, and the conclusion as to whether you Reject or Fail to Reject the null hypothesis.
Simulating the P-Values for Hypothesis Tests on the Mean This DIGMath module lets you visualize the P-values associated with sample means when conducting a hypothesis test on the population mean. The program randomly generates 100 samples of size n = 50 from your choice of the usual four underpopulations, calculates the P-value associated with each sample mean, and draws the P-values as a series of vertical lines. You can choose the significance level ∝ for a two-tailed test and the program also shows the vertical line associated with the P-value for that critical value. You can then see the number, and percentage, of sample means whose P-values are sufficiently unlikely that would indicate that you should reject the null hypothesis. The spreadsheet also displays the scatterplot of the 100 sample P-values plotted against the values of the 100 sample means to demonstrate that the pattern in the points typically looks like a normal distribution pattern. A horizontal line is also included at the height corresponding to the P-value associated with the critical value for the hypothesis test on the population mean at the selected level ∝ of significance.
Simulating the P-Values for Hypothesis Tests on the Proportion This DIGMath module lets you visualize the P-values associated with sample proportions when conducting a hypothesis test on the population proportion π. The program randomly generates 100 samples of size 100, calculates the P-value associated with each sample proportion, and draws the P-values as a series of vertical lines. You can choose the significance level a for a two-tailed test and the program also shows the vertical line associated with the P-value for that critical value. You can then see the number, and percentage, of the sample proportions whose P-values are sufficiently unlikely that would indicate that you should Reject the null hypothesis. The spreadsheet also displays the scatterplot of the 100 sample P-values plotted against the values of the 100 sample proportions to demonstrate that the pattern in the points typically looks like a normal distribution pattern. A horizontal line is also included at the height corresponding to the P-value associated with the critical value for the hypothesis test on the population proportion at the selected level ∝ of significance.
Hypothesis Test on the Difference of Means This DIGMath spreadsheet assists you in testing a hypothesis for the difference in means of two populations. The null hypothesis is that the two means are equal, and you have to select the alternate hypothesis test you want -- either two tailed or one tail with either tail. You then enter the sample data for the two samples -- the sample size n, the sample mean m and the sample standard deviation s -- and the level of significance α. The spreadsheet displays the corresponding normal or t-distribution for the distribution of differences of sample means, the location of the critical value(s), and the location of the difference in the two sample means. It also shows the associated z- or t-value, as well as the corresponding P-value, and the conclusion as to whether you Reject or Fail to Reject the null hypothesis.
Hypothesis Test on the Difference of Proportions This DIGMath spreadsheet assists you in testing a hypothesis for the difference in proportions of two populations. The null hypothesis is that the two proportions are equal, and you have to select the alternate hypothesis test you want -- either two tailed or one tail with either tail. You then enter the sample data for the two samples -- the sample size n and the number of successes x in each sample -- and the level of significance α. The spreadsheet displays the corresponding normal distribution (if appropriate) for the distribution of differences of sample proportions, the location of the critical value(s), and the location of the difference in the two sample proportions. It also shows the associated z-value, as well as the corresponding P-value, and the conclusion as to whether you Reject or Fail to Reject the null hypothesis.
The Distribution of the Difference of Means This DIGMath module lets you investigate the distribution of the difference of means based on summary sample data from two samples drawn from the choice of four underlying populations (to see that the population does not affect the results). The user can choose the sample size (from n = 2 to n = 50) from each sample and the number (from 50 to 250) of samples. The simulation generates that number of samples from the selected populations, plots the difference in the sample means of each sample, and displays the mean and standard deviation of the differences in the sample means compared to the theoretical predictions based on the population of differences of means of all possible samples.
The Distribution of the Difference of Sample Proportions This DIGMath module lets you investigate the distribution of the difference of proportions based on summary sample data from two samples drawn from two binomial populations. For each population, you can choose probability p of success and the sample size (from n = 2 to n = 100) from each sample, as well as the number of random samples (between 50 and 300). The simulation generates that number of samples from the two populations, plots the difference in the sample proportions of each set of samples, and displays the mean and standard deviation of the differences in the sample proportions compared to the theoretical predictions based on the population of differences of proportions of all possible samples.
Linear Regression: Fitting a Line to Data This DIGMath module performs a linear regression analysis on any set of up to 50 (x, y) data points. It shows graphically the points and the associated regression line and also displays the equation of the regression line, the value for the correlation coefficient r, and the value for the Sum of the Squares that measures how close the line comes to all the data points.
Sum of the SquaresThis spreadsheet allows you to investigate dynamically how the sum of the squares measures how well a line fits a set of data. You can enter a set of data and select the number of data points you want to use. You also enter the values you want for the slope and the vertical intercept of a line. The display shows the data points with the line based on those parameters and also shows the value for the sum of the squares associated with that linear fit.
Regression Simulation This DIGMath spreadsheet performs a graphical simulation on the regression lines based on a variety of samples drawn from an underlying population. You have the choice of the sample size (n > 2) and the number of samples. The simulation generates the random samples, calculates the equation of and plots the corresponding sample regression line, and also draws the population regression line. It also displays the mean of the slopes and of the vertical intercepts, so that users can compare these values to the slope and intercept of the underlying population. The students quickly see that, with small sample sizes, the likelihood of the sample regression line being close to the population regression line may be very small with widely varying slopes for many of the sample lines. As the sample size increases, the sample regression lines become ever more closely matched to the population line.
Simulating the Correlation Coefficient This DIGMath spreadsheet lets you investigate the sample distribution for the correlation coefficient r based on repeated random samples drawn from a bivariate population. You can choose between n = 3 and n = 50 random points for each sample and between 50 and 250 such samples from the underlying population. For each sample, it then calculates the correlation coefficient and displays a histogram showing the values of r from the samples. It also calculates and displays the mean of the sample correlation coefficients and compares it to the correlation coefficient for the underlying bivariate population.
Simulating the Regression Coefficients This DIGMath module lets you investigate the sample distributions for the two regression coefficient a and b in the regression equation y = ax + b based on repeated random samples drawn from a bivariate population. You can choose between n = 3 and n = 40 random points for each sample and between 50 and 250 such samples from the underlying population. For each sample, the program calculates the regression equation and displays the various regression lines along with the regression line for the underlying bivariate population. It then draws two histograms -- one showing the distribution of the values of the slope a from the random samples and the other showing the distribution of the values of the vertical intercepts b from those samples. It also calculates and displays the mean of each of the sample regression coefficients and compares it to the regression coefficients for the underlying bivariate population.
The Effects of an Extra Point on the Regression Line and the Correlation Coefficient This DIGMath module lets you investigate the effect that changing an additional point has on the regression line (both graphically and numerically) and on the correlation coefficient. You have the choice of 5, 10, or 20 fixed points and can move an additional point using sliders to see how it affects the calculations and the extent of the changes depending on the number of points.
Fitting a Median-Median Line to Data This DIGMath module fits a median-median line to any set of up to 50 (x, y) data points. It shows graphically the points and the associated median-median line and also displays the equation of the median-median line and the value for the Sum of the Squares that measures how close the line comes to all the data points.
Simulating the Quartile-Quartile Line This DIGMath module lets you investigate the quartile-quartile line that fits a set of data via a simulation. The quartile-quartile line is based on finding the 1st and 3rd quartiles for both the x and the y values in a set of data and then creating the line that passes through those two points. As such, it is a viable alternative to the usual least-squares regression line that is conceptually and computationally simpler. You have a choice of the sample size and the number of samples that will be drawn from an underlying population. The spreadsheet generates the random samples and draws all the corresponding quartile-quartile lines to help you see the effect of sample size on the consistency of the lines.
DataFit: Fitting Linear, Exponential, and Power Functions to Data This DIGMath spreadsheet is provided as a visual and computational tool for investigating the issue of fitting linear, exponential, and power functions to data and the underlying transformations used to create the nonlinear functions. You can enter a set of data and the spreadsheet displays six graphs: (1) For a linear fit: the regression line superimposed over the original (x, y) data; (2) For an exponential fit: the regression line superimposed over the transformed (x, log y) data values; (3) The exponential function superimposed over the original (x, y) data; (4) For a power fit: the regression line superimposed over the transformed (log x, log y) data values; (5) The power function superimposed over the original (x, y) data; (6) All three functions superimposed over the original (x, y) data. The spreadsheet also shows the values for the correlation coefficients associated with all three linear fits and the values for the sums of the squares associated with each of the three fits to the original data. On a separate page, the spreadsheet also shows the residual plots associated with each of the three function fits.
Multivariate Linear Regression This DIGMath spreadsheet lets you perform multivariate linear regression when the dependent variable Y is a function of two independent variables X1 and X2 or a function of three independent variables X1, X2, and X3. You enter the number of data points (up to a maximum of 50) and then the values for the dependent and independent variables in the appropriate columns. The spreadsheet responds with the equation of the associated linear regression equation, the value for the sum of the squares, and the value for the coefficient of determination, R²; note that this value tells you the percentage of the variation that is explained by the linear function.
Comparing Moving Averages This DIGMath program lets you compare different moving averages to one another as well as to the underlying set of data. Moving averages are used widely in many different fields to identify patterns and trends in data where there are wild fluctuations on a day-to-day basis. This spreadsheet uses some actual data on the spread of the COVID pandemic to illustrate the concept by plotting the underlying data, the 3-day moving average, the 20-day moving average, and a third moving average that you can choose based on either 4-days, 5-days, up through 19-days.
Fitting Functions to Moving Averages This DIGMath spreadsheet lets you investigate how moving averages can be used to identify the pattern in a set of data that has extreme daily fluctuations. The program creates charts for the 5-day moving average, the 10-day moving average, the 15-day moving average, and the 20-day moving average. It also allows you to select the type of function you would like to use -- linear, exponential, power, quadratic, or cubic -- using a slider and displays the resulting functions that fit both the underlying data and the corresponding moving average. It also displays the resulting equations of the functions and the values of the corresponding correlation coefficients, r, (for the linear, exponential, and power fits) and the coefficient of multiple correlation, R, (for the polynomial fits) to assess how well each function fits the data.
The Kolmogorov-Smirnov Test for Normality This program is a test to determine whether or not a set of data might be normally distributed via an hypothesis test when you can safely assume that the underlying population is normally distributed. You enter a relatively small (up to 30 numbers) in ascending numerical order and the spreadsheet performs the test at the 5% significance level while providing a graphical interpretation of the procedure. It displays the associated test-statistic and reports whether you can reject the claim that the data is normally distributed or whether you fail to reject the claim.
The Lillifors Test for NormalityThis program is a test to determine whether or not a set of data might be normally distributed via an hypothesis test. It is a special case of the Kolmogorov-Smirnov Test when you don't know if you can assume that the underlying population is itself normally distributed. You enter a relatively small (up to 30 numbers) in ascending numerical order and the spreadsheet performs the test at the 5% significance level while providing a graphical interpretation of the procedure. It displays the associated test-statistic and reports whether you can reject the claim that the data is normally distributed or whether you fail to reject the claim.
The Birthday Problem This DIGMath module lets you investigate the Birthday Problem, which asks for the probability that two people in a group of n people will have the same birthday. The program lets you decide on the number of people in a group (between 1 and 100) and displays the graph of the probability of a match versus the number of people in the group.
Number of Boys vs. Girls Born in a Family This DIGMath spreadsheet lets you investigate the number of Boys and Girls born into a family based on the fact that 51.2% of all live births are Boys. You can choose the number of children (1-10) in a family and the number of such families. The program simulates this and displays the outcomes graphically in a histogram and numerically with a table of outcomes and the mean and standard deviation of the results.
Simulating the Birthday Problem This DIGMath module lets you investigate the Birthday Problem from the point of view of a random simulation. You have the choice of the number of people in a group (from 2 to 50). The program then generates a random sample of birth-dates for each of the people and displays the list, including highlighting those that match. It presents the results, including the theoretical probability of a match and the number of matches.
The Drunkard's (or Random) Walk Simulation This DIGMath module lets you investigate the notion of a random walk in the plane. You have the choice of the number of random steps (between 1 and 1000) and the length of each step. The program then generates a random collection of steps and displays the results graphically, as well as some numerical analysis on the actual distance covered from the starting point compared to the theoretical predictions.
Buffon's Needle Problem This DIGMath module lets you experiment with a graphical simulation of Buffon's Needle Problem -- the probability that a needle of length L lands on the seam between parallel strips of flooring of width W when it falls to the floor. You can select the number of random "needles" that fall, the width of the strip of flooring, and the length of the needles and see the results graphically and numerically.
Buffon's Needle Problem on Square Tiles This DIGMath module lets you experiment with a graphical simulation of the Laplace/Buffon's Needle Problem -- the probability that a needle of length L lands on the seam between square tiles of width W when it falls to the floor. You can select the number of random "needles" that fall, the width of the tile, and the length of the needles and see the results graphically and numerically.
Buffon's Disks on a Long Plank Problem This DIGMath spreadsheet lets you experiment with a graphical simulation of a variation of Buffon's Needle Problem -- the probability that a circular disk of radius r lands on the seam between parallel strips of flooring of width W when it falls to the floor. You can select the number of random "disks" that fall, the width of the strip of flooring, and the length of the needles and see the results graphically and numerically.
Buffon's Disks on a Square Tile Problem This DIGMath spreadsheet lets you experiment with a graphical simulation of a variation of Buffon's Needle Problem -- the probability that a circular disk of radius r lands on the seam between square tiles of width W when it falls to the floor. You can select the number of random "disks" that fall, the width of the tile, and the length of the needles and see the results graphically and numerically.
Buffon's Disks in a Circle Problem This DIGMath module lets you experiment with a graphical simulation of a variation of Buffon's Needle Problem -- the probability that a circular disk of radius r lands entirely within a larger circle of radius R when it falls to the floor or if it crosses the boundary circle. You can select the number of random "disks" that fall, the radius r of the disks, and the radius R of the circle, and see the results graphically and numerically.
Buffon's Needle Problem in Concentric Circles This DIGMath spreadsheet lets you experiment with a graphical simulation of a variation of Buffon's Needle Problem -- the probability that a circular disk of radius r lands on the seam between a group of concentric circles when it falls to the floor. You can select the number of random "disks" that fall, the fixed difference in the radii of the concentric circles on the floor, and the length of the needles and see the results graphically and numerically.
Buffon Problem for Square Coins on a Square Tile This DIGMath spreadsheet lets you experiment with a graphical simulation of a variation on Buffon's Needle Problem -- the probability that a square coin of length L lands on the edge of a square tile of width W . You can select the number of random "coins" that fall, the length of each square coin, and the width of the flooring tile and see the results graphically and numerically.
Simulation of Gambler's Ruin This DIGMath simulation lets you investigate the notion of Gambler's Ruin in which a person enters a game of chance with a fixed amount to bet (the stake) and repeatedly bets a fixed amount on one particular outcome until he or she runs out of money or reaches a certain amount of winnings. You can enter the stake, the amount of the bet, the amount won on each successful bet, the fixed probability of winning on each bet, and the number of bets (up to 1000) that will be displayed using sliders. In most realistic situations, the "house" sets the payoff amount low enough to assure that the gambler will eventually run out of money -- that is, lose his or her shirt. This is why it is known as Gambler's Ruin.
Simulation of Dart Throwing This DIGMath module lets you investigate the process of throwing random darts at a dartboard. You can select between 100 and 1000 random darts and the spreadsheet shows the position that each dart lands and displays the breakdown of how many, and what percentage, of the darts all into each of the rings in the dartboard.
Difference of the Faces of Two Dice This DIGMath simulation lets you investigate the differences of the faces on a pair of fair dice. You can choose up to 720 rolls of the two dice and the spreadsheet shows the distribution of the outcomes and a list of the number and percentage of each possible outcome compared to the theoretical predictions.
Product of the Faces of Two Dice This DIGMath simulation lets you investigate the product of the faces on a pair of dice. You can choose up to 720 rolls of the two dice and the spreadsheet shows the distribution of the outcomes and a list of the number and percentage of each possible outcome compared to the theoretical predictions.
Simulation of Rolling 4-Sided Dice This DIGMath spreadsheet lets you investigate the probability experiment of rolling a pair of fair 4-sided dice, instead of the usual 6-sided dice, so that the possible sums are now 2, 3, ..., 8. The simulation shows the results of repeated trials both graphically in a histogram and numerically in terms of the number of times each of the possible outcomes arises.
Simulation of Rolling 8-Sided Dice This DIGMath spreadsheet lets you investigate the probability experiment of rolling a pair of fair 8-sided dice, instead of the usual 6-sided dice, so that the possible sums are now 2, 3, ..., 16. The simulation shows the results of repeated trials both graphically in a histogram and numerically in terms of the number of times each of the possible outcomes arises.
Yahtzee: Rolling Five Dice Simulation The game of Yahtzee^TM involves rolling a set of five fair dice. This DIGMath module lets you investigate this experiment by simulating repeated random rolls (up to 720 times) of five dice. It displays the results in a histogram as well as a table showing the simulated outcomes and the expected theoretical outcomes.
Visualizing Conditional Probability This DIGMath spreadsheet helps you visualize the idea of conditional probability where the usual sample space for a probability experiment is reduced by knowing some other detail of the event. The spreadsheet looks at the sum of the faces of two dice and allows you to investigate what happens if either (1) you know the result of the second die or (2) you know the product of the two faces. The spreadsheet displays the corresponding histogram and the associated numerical outcomes.
Hypergeometric Probabilities This DIGMath module helps you visualize the probabilities associated with a hypergeometric distribution, which is based on selecting a sample of size n from a population having N elements of which K are considered successes. The standard probability problem asks what is the probability of having exactly x successes in that sample? You can enter the three values N, K, and n and the desired number of successes x in that sample. The spreadsheet draws the corresponding hypergeometric distribution and highlights the desired outcome, as well as the numerical results.
Waiting Time Simulation This DIGMath module lets you investigate the length of time a car will wait at a red light. You can select the total length of the cycle and the length of time that the light is red. The results -- the number of times that the wait is 0, 1, 2, ..., seconds -- are shown graphically in a histogram and in a table listing the outcomes. The average wait over all repetitions is also shown.
Chi-Square Analysis This DIGMath spreadsheet is designed to perform a complete chi-square analysis on many different sized contingency tables, including 2 by 2, 2 by 3, 2 by 4, 3 by 2, 3 by 3, and 3 by 4. On the Set-Up screen, the user first enters the number of rows and the number of columns of the desired contingency table and is then instructed to click on an appropriate tab to go to the corresponding input screen. On that screen, you then enters the observed values into the various positions in the contingency table. The spreadsheet displays the resulting table of expected frequencies, the number of degrees of freedom, and the value of the chi-square statistic based on the values in the table. It also draws the graph of the corresponding chi-square distribution and indicates the location of the critical value, based on the desired level of significance, separating the rejection region from the region where one cannot reject the null hypothesis. In addition, the program indicates the location of the chi-square statistic corresponding to the data in the contingency table. Finally, the program indicates whether or not one can reject the null hypothesis at that significance level.
The Chi-Square Distributions This DIGMath module lets you explore the behavior of various chi-square distributions, which depend on the number n of degrees of freedom. You can enter any desired number of degrees of freedom from 2 to 31 and the program draws the graphs of that chi-square distribution as well as those with 3, 7, 11, 15, ..., 27 degrees of freedom. Because the chi-square distributions become more normal in shape as n increases, the program also draws the standard normal distribution with mean μ = 0 and standard deviation σ = 1 for comparison.
Chi-Square Simulation This DIGMath spreadsheet lets you investigate the variation in the values that can arise for the chi-square statistic via a random simulation based on a two by three contingency table. You define the table by entering the column and row totals and the number of random samples drawn from that population and the spreadsheet generates and graphs the corresponding values of the chi-square statistic.
One-Way Analysis of Variance (ANOVA) This DIGMath spreadsheet is designed to perform a complete one-way analysis of variance (ANOVA) to test whether the means of two or more (up to 5 sample means) may come from populations with the same mean (the null hypothesis) or from populations with different means (the alternate hypothesis). Each sample can contain up to 10 entries. The spreadsheet displays the resulting ANOVA table, including the value of the F-statistic. It also draws the graph of the corresponding F-distribution and indicates the location of the critical value, based on the 5% level of significance, separating the rejection region from the region where one cannot reject the null hypothesis. In addition, the program indicates the location of the F-statistic corresponding to the data in the table. Finally, the program indicates whether or not one can reject the null hypothesis at the 5% significance level.
Simulating the Runs Test This DIGMath spreadsheet lets you investigate the Runs Test both graphically and numerically. When there is a collection of outcomes consisting of A's and B's, the object is to see the number of runs that occur. The module lets you select the total number of A's and B's, the number of A's, and the number of random samples. The distribution of the number of runs is drawn as well as numerical measures for the mean and standard deviation.

Dynamic Investigatory Graphical Displays of Math: DIGMath Graphical EXPLORATIONS for College Algebra and Precalculus using Excel

As mentioned above, Excel must be set to accept macros by selecting "Enable Content".

The following are the DIGMath explorations that are currently completed and ready for use in Excel . (Many others are under development.) If you have any suggestions for improvements or for new topics, please pass them on also.

DIGMath Spreadsheets for College Algebra and Precalculus

Linear Functions This DIGMath spreadsheet allows you to investigate visually three different aspects of linear functions. (1) You can enter the slope and vertical intercept and watch the effects of changing either of them via a slider on the resulting graph. (2) You can enter a point and the slope and watch the effects of changing either of them on the graph via the point-slope formula. (3) You can enter two points and change either of them to see the effects.

Linear Regression: Fitting a Linear Function to Data This DIGMath module performs a linear regression analysis on any set of up to 50 (x, y) data points. It shows graphically the points and the associated regression line and also displays the equation of the regression line, the value for the correlation coefficient r, and the value for the Sum of the Squares that measures how close the line comes to all the data points.

Regression Simulation The user has the choice of the sample size (n> 2) and the number of samples. The simulation generates repeated random samples, calculates the equation of and plots the corresponding sample regression line, and also draws the population regression line. The students quickly see that, with small sample sizes, the likelihood of the sample regression line being close to the population regression line may be very small with widely varying slopes for many of the sample lines. As the sample size increases, the sample regression lines become ever more closely matched to the population regression line.

Fitting a Median-Median Line to Data This DIGMath module fits a median-median line to any set of up to 50 (x, y) data points. It shows graphically the points and the associated median-median line and also displays the equation of the median-median line and the value for the Sum of the Squares that measures how close the line comes to all the data points.

Median-Median Line Simulation The user has the choice of the sample size (n > 2) and the number of samples. The simulation generates repeated random samples, calculates the equation of and plots the corresponding sample median-median line, and also draws the population median-median line. The students quickly see that, with small sample sizes, the likelihood of the sample median-median line being close to the population median-median line may be very small with widely varying slopes for many of the sample lines. As the sample size increases, the sample median-median lines slowly become more closely matched to the population median-median line.

Comparing Lines that Fit Data This DIGMath program lets you compare how well the least-squares line, the median-median line (that is built into many calculators), and the quartile-quartile line fit sets of data. You can choose the number of random data points from an underlying population and the spreadsheet generates a random sample and displays the three lines, along with the data points, so that you can compare how well the three lines fit the data and how they compare to one another, particularly as the sample size increases.

Moving Along Lines This DIGMath spreadsheet lets you visualize dynamically the notion of moving along the Supply and Demand lines, which is a fundamental concept in microeconomics. You can vary the vertical intercepts of the two lines and see the effects on the graphs of the two lines and on the point of intersection as each line seems to move along the other line.

Exponential Functions This DIGMath spreadsheet allows you to investigate visually two different aspects of exponential functions. (1) You can enter the growth/decay factor b in y = b^xand watch the effects on the resulting graph of changing it via a slider. (2) You can enter two points and change either of them to see the effects.

Comparing Exponential Growth Functions This DIGMath spreadsheet is designed to give you a deeper understanding of the behavior of exponential growth functions y = b^x, with b > 1. It lets you select a value for the base b between 0.15 and 0.85 using a slider and draws the graphs of the three exponential growth functions y = 1.10^x, y = b^x with your choice of b, and y = 2^x. You can also trace all three curves using a slider and see the effects on the graphs and on the printed values, all of which are color coordinated to help you see what is happening.

Comparing Exponential Decay Functions This DIGMath spreadsheet is designed to give you a deeper understanding of the behavior of exponential decay functions y = b^x, with 0 < b < 1. It lets you select a value for the base b between 0.15 and 0.85 using a slider and draws the graphs of the three exponential decay functions y = 0.10^x, y = b^xwith your choice of b, and y = 0.90 ^x. You can also trace all three curves using a slider and see the effects on the graphs and on the printed values, all of which are color coordinated to help you see what is happening.

Exponential Regression: Fitting an Exponential Function to Data This DIGMath module performs an exponential regression analysis on any set of up to 50 (x, y) data points. It shows graphically the points and the associated exponential regression function and also displays the equation of the exponential regression function, the value for the associated correlation coefficient r based on the transformed (x, log y) data, and the value for the Sum of the Squares that measures how close the exponential function comes to all the data points.

Drug Level Models This DIGMath module lets you investigate two different models for the level of a medication in the bloodstream. The first model is for the drug level after a single dose of the medication; the second model is for the repeated (say daily) dose of a medication. In both cases, the key parameters are the amount of the medication that is taken (either once or repeatedly) and the percentage of the drug that is washed out of the blood each time period.

Doubling Time and Half-Life This DIGMath spreadsheet is intended to let you investigate visually two important applications of exponential functions. First, you can explore the relationship between the growth factor b and the doubling time of an exponential growth process. Second, you can investigate the relationship between the decay factor b and the half-life of an exponential decay process.

Amortization: Financing a Home or a Car This DIGMath module is intended to let you investigate the mathematics involved in the amortization process -- taking out a loan to finance a home or a car. You have a choice of financing a home or a car on different pages of the spreadsheet. For each, you enter the amount of the loan, the length of the loan, and the annual interest rate. The spreadsheet displays the graph of the monthly balance of the loan and two different views of the breakdown of how the monthly payment is split between paying off the principal and paying for the interest on the loan. It also lets you trace across the graphs to see the amounts in any month.

Effects of Compound Interest This DIGMath spreadsheet lets you see the effects of compounding with different frequencies -- yearly, quarterly, monthly, daily, by the hour, and by the minute -- on the balance in an account. You can enter your choice of interest rate and initial balance and see both graphically and numerically the growth in the balance over time with the different compounding periods.

Exploring the Sum Identity for Exponents This DIGMath module lets you investigate graphically and numerically the sum identity for exponents: e^a+b = e^a e^b for any a and b. You enter the values for the parameters using sliders and the spreadsheet draws the graph of the exponential function and highlights the values of a and b as horizontal distances and shows the corresponding heights along with a visualization of the identity.

Exploring the Difference Identity for Exponents This DIGMath module lets you investigate graphically and numerically the difference identity for exponents: e^a-b = e^a /e^b for any a and b. You enter the values for the parameters using sliders and the spreadsheet draws the graph of the exponential function and highlights the heights for e^a, e^b, and e^a-b. The spreadsheet focuses on the fact that e^a appears to be a multiple of e^a-b = e^a /e^b and that the multiple seems to be equal to e^b, both graphically and numerically. That realization is just a short step to the discovery of the identity.

Haldane Functions in Genetics This module draws the graph of the Haldane function that is used in genetics to relate the recombination fraction C_AB between loci A and B in a gene and the distance X_AB between the loci. A slider lets you change the value for the distance x_AB between loci and see the effect on the corresponding point on the graph of the Haldane function that involves an exponential function.

Power Functions This DIGMath spreadsheet allows you to investigate visually the behavior of power functions. You can enter the power p and watch the effect on the graph of changing it via a slider.

Comparing Power Functions This DIGMath program lets you investigate the behavior of the three different kinds of power functions -- those with p > 1, 0 < p < 1, and p < 0. In each case (on a different page of the spreadsheet), the curves for two "extreme" power functions are drawn and a third graph, based on your choice of the power p between the extremes is also drawn. As you change the value of p, you can see how the graph compares to the two extreme curves.

Power Regression: Fitting a Power Function to Data This DIGMath module performs a power regression analysis on any set of up to 50 (x, y) data points. It shows graphically the points and the associated power regression function and also displays the equation of the power regression function, the value for the associated correlation coefficient r based on the transformed (log x, log y) data, and the value for the Sum of the Squares that measures how close the power function comes to all the data points.

Sum of the Squares This DIGMath module allow you to investigate dynamically how the sum of the squares measures how well a line fits a set of data. You can enter a set of data and select the number of data points you want to use. You also enter the values you want for the slope and the vertical intercept of a line. The display shows the data points with the line based on those parameters and also shows the value for the sum of the squares associated with that linear fit.

DataFit: Fitting Linear, Exponential, and Power Functions to Data This DIGMath spreadsheet is provided as a visual and computational tool for investigating the issue of fitting linear, exponential, and power functions to data and the underlying transformations used to create the nonlinear functions. You can enter a set of data and the spreadsheet displays six graphs:
(1) For a linear fit: the regression line superimposed over the original (x, y) data;
(2) For an exponential fit: the regression line superimposed over the transformed (x, log y) data values;
(3) The exponential function superimposed over the original (x, y) data;
(4) For a power fit: the regression line superimposed over the transformed (log x, log y) data values;
(5) The power function superimposed over the original (x, y) data;
(6) All three functions superimposed over the original (x, y) data.
The spreadsheet also shows the values for the correlation coefficients associated with all three linear fits and the values for the sums of the squares associated with each of the three fits to the original data. On a separate page, the spreadsheet also shows the residual plots associated with each of the three function fits.

Logarithmic Growth This DIGMath spreadsheet lets you investigate how fast (or actually how slow) logarithmic growth is. In particular, it lets you see, both graphically and numerically, what the cost is in terms of how much the variable t must increase in order for the logarithmic function f(t) = log t to increase by 1 unit for different values of t.

Comparing Logarithmic Functions This DIGMath spreadsheet is designed to give you a deeper understanding of the behavior of logarithmic functions y = log_bx, with b > 1. It lets you select a value for the base b between 0.15 and 0.85 using a slider and draws the graphs of the three exponential growth functions y = log₂ x, log_b x with your choice of b, and y = log₁₀ x . You can also trace all three curves using a slider and see the effects on the graphs and on the printed values, all of which are color coordinated to help you see what is happening.

Logarithmic Functions with Fractional Bases Typically, logarithmic functions are only considered in which the base b > 1. This DIGMath program lets you investigate what happens if fractional bases with 0 < b < 1 are introduced. The resulting behavior patterns are very different from what you would expect based on the usual logarithmic functions.

The Logistic Model This DIGMath module allows you to investigate visually two different aspects of the discrete logistic model based on the logistic difference equation P_n+₁= aP_n- bP_n². (1) You can enter, via sliders, values for the two parameters a and b, as well as the initial population value P₀and watch dynamically the effects on the resulting graph of the population, and also see the effects of changing any of these values. (2) You can also investigate visually the effects on a population of changes in the initial growth rate a and the maximum sustainable population (the limit to growth), along with the initial population value P₀, using sliders, and watching the dynamic effects on the graph of changing any of them.

Doubling Times of Various Functions This DIGMath module lets you investigate the notion of the doubling time associated with a variety of different functions, including linear, power and logarithmic. For an exponential function f(t) = Ab^t , the doubling time depends only on the growth rate b, but not on the point t or the value of the coefficient A. The module lets you see what happens with the other families of function to see which, if any, of the parameters or values of the independent variable contribute to the value for the doubling time.

The Surge Function This DIGMath module lets you investigate the properties of the surge function, which has the equation y = A t ^p b^t, where p > 0 and b < 1. It is used to model phenomena where there is an initial growth spurt in a quantity followed by a slow decay toward zero. You can experiment with different values for the parameters, using sliders, to see the effects on the behavior of the function.

Exploring the Product Identity for Logarithms This DIGMath module lets you investigate graphically and numerically the product identity for logarithms: log (a* b) = log (a) + log (b) for any a and b. You enter the values for the parameters using sliders and the spreadsheet draws the graph of the logarithmic function and highlights the values of a and b as horizontal distances and shows the corresponding heights along with a visualization of the identity.

Exploring the Logarithm of a Reciprocal This DIGMath module lets you investigate graphically and numerically the rule for the logarithm of a reciprocal that log (1/x) = - log (x), for any base b > 0 and any x > 0. You enter the values for the base b using a slider and the spreadsheet draws the graph of y = log (1/x) and shows the numerical values for both y = log (1/x) and y = log (x). One page deals with the case with b > 1 and another with 0 < b < 1.

Exploring the Quotient Identity for Exponents This DIGMath module lets you investigate graphically and numerically the quotient identity for logarithms: log (a / b) = log (a) - log (b) for any a and b. You enter the values for the parameters using sliders and the spreadsheet draws the graph of the exponential function and highlights the values of a and b as horizontal distances and shows the corresponding heights along with a visualization of the identity.

Exploring the Log of a Power Identity for Logarithms This DIGMath spreadsheet lets you investigate graphically and numerically the identity: log (x^p) =p log (x) for logarithms for any power p and any x > 0 . You enter the value for p (p = 2, 3, 4, and 5) using a slider and the spreadsheet draws the graphs of y = log (x) and y = log (x^p) and demonstrates graphically and numerically how log (x^p) is always equal to log (x) as x changes as a dynamic visualization of the identity.

The Log of a Power Identity This DIGMath spreadsheet lets you investigate visually, numerically, and dynamically the fundamental identity log _b(b^x) = x, for any base b > 1 and any value x. It draws the graph of the exponential term y = b^xand the expression y = log_b(b^x) and allows the user to trace along both curves as it displays the algebraic steps and numerical results to show that the result is always equal to x. Simultaneously, the second graph looks linear and in fact it is the line y = x.

The Power of a Log Identity This DIGMath spreadsheet lets you investigate visually, numerically, and dynamically the fundamental identity b^log_b( x) = x, for any base b > 1 and any value x. It draws the graph of the logarithmic term y = log_b x and the expression y = b^{log b( x)}and allows the user to trace along both curves as it displays the algebraic steps and numerical results to show that the result is always equal to x. Simultaneously, the second graph looks linear and is, in fact, the line y = x.

Quadratic Functions This DIGMath spreadsheet allows you to investigate visually two different aspects of quadratic functions. (1) You can enter, via sliders, values for the three coefficients in a quadratic function and watch dynamically the effects on the resulting graph of changing any of them via sliders. (2) You can also investigate visually the fact that a quadratic is determined by three points by entering the coordinates of three points, using sliders, and watching the dynamic effects on the graph of changing any of them.

Cubic Functions This DIGMath spreadsheet allows you to investigate visually two different aspects of cubic functions. (1) You can enter, via sliders, values for the four coefficients in a cubic function and watch dynamically the effects on the resulting graph of changing the values of any of them via sliders. (2) You can also investigate visually the fact that a cubic is determined by four points by entering the coordinates of four points, using sliders, and watching the dynamic effects on the graph of changing any of them.

Quartic Functions This DIGMath spreadsheet allows you to investigate visually two different aspects of quartic functions. (1) You can enter, via sliders, values for the five coefficients in a cubic function and watch dynamically the effects on the resulting graph of changing the values of any of them via sliders. (2) You can also investigate visually the fact that a quartic is determined by five points by entering the coordinates of five points, using sliders, and watching the dynamic effects on the graph of changing any of them.

Polynomial Graphs This DIGMath spreadsheet lets you investigate the graph of any polynomial up to eighth degree by entering the values for the coefficients and the interval over which you want to see the graph. You can also control a point on the graph by means of a slider to see the coordinates of that point and so locate real roots, turning points, and inflection points.

End Behavior: A Polynomial vs. Its Power Function This DIGMath module lets you investigate the end behavior of any polynomial up to eighth degree. You must enter the values for the coefficients. The spreadsheet then displays the graph of the polynomial as well as the graph of the power function corresponding to the leading term of the polynomial. A slider lets you expand the interval for the display, so that you can see how different the two graphs are when the interval is small and the turning points and inflection points of the polynomial are clearly in view. As the interval expands, the polynomial looks more and more like the power function.

Polynomial Regression: Fitting a Polynomial to Data This DIGMath module performs a polynomial regression analysis (linear, quadratic, ..., up to sixth degree) on any set of up to 50 (x, y) data points, which goes beyond what is possible with graphing calculators. It shows graphically the points and the associated polynomial regression function and also displays the equation of the regression polynomial, the value for the associated coefficient of multiple determination R² based on multivariate linear regression of y on x, x², x³, ..., x⁶ (depending on the choice of degree) and its significance in terms of what percentage of the variation in the data is explained by the regression polynomial, and the value for the Sum of the Squares that measures how close the power function comes to all the data points.

Squaring a Binomial Term This DIGMath module lets you practice squaring a binomial term (x + a)² both algebraically and graphically. The spreadsheet randomly generates the binomial term and allows you to select the coefficients b and c in the quadratic product x² + bx + c using sliders. It displays the graphs of the squared term and the quadratic to show how you eventually get exact overlap when you get the correct result.

Product of Binomial Terms This DIGMath module lets you practice multiplying two randomly generated binomial terms (x + p) and (x + q) both algebraically and graphically. The spreadsheet randomly generates the two binomial terms and allows you to select the coefficients b and c in the quadratic product x² + bx + c using sliders. It displays the graphs of the product of the two binomial terms and the quadratic based on b and c to show how you eventually get exact overlap when you get the correct result.

Factoring Quadratics This DIGMath module lets you practice factoring randomly generated quadratic polynomials of the form x² + bx + c by selecting the numbers p and q for two binomial terms (x + p) and (x + q). The spreadsheet displays the graphs of the target quadratic and the quadratic based on your choice of p and q to show how you eventually get exact overlap when you get the correct result.

Estimating the Complex Zeros of Cubic Polynomials This DIGMath spreadsheet lets you locate all three zeros, particularly any pairs of complex conjugate zeros, of cubic polynomials using a graphical exploration. The spreadsheet also displays the factorization and the values of the three zeros, either real or complex.

Estimating the Complex Zeros of Quartic Polynomials This DIGMath module lets you approximate all four zeros, particularly any pairs of complex conjugate zeros, of quartic polynomials of the form y = x⁴ + ax³ + bx² + cx + d using a graphical exploration. As you vary the coefficients in an approximation to a possible quadratic factor, you see how well the resulting approximate factorization fits the original quartic both graphically and in terms of the Sum of the Squares. The spreadsheet also displays the factorization and the values of the four zeros, either real or complex.

Estimating the Complex Zeros of a Polynomial This DIGMath spreadsheet lets you approximate the complex zeros of any polynomial up to degree 8. You enter the desired coefficients and then enter six points that lie along the curve for the polynomial. The program draws the graph and then applies quadratic regression to find the best-fitting quadratic approximation, draws its graph, and displays its zeros, whether real or complex, as approximations to two of the zeros of the original polynomial.

Bairstow's Method for Complex Roots This DIGMath program lets you investigate Bairstow's Method for approximating the complex roots of a polynomial by creating a sequence of quadratic functions that converges to an irreducible quadratic factor of the polynomial. Once you know a quadratic factor, you can find the complex roots by applying the quadratic formula. The spreadsheet lets you use any polynomial up to 8^th degree and any choice of initial estimate for the quadratic factor. The spreadsheet draws the first five quadratic approximations to illustrate the convergence both graphically and numerically.

Components of the Lagrange Interpolating Polynomial This DIGMath spreadsheet lets you investigate the Lagrange Interpolating Polynomial and how it is formed by a set of components. You can work with a linear function based on two interpolating points, a quadratic function based on three interpolating points, or a cubic function based on four interpolating points of your choice. The spreadsheet displays the resulting Lagrange interpolating polynomial, as well as the components of that polynomial -- the two linear functions that combine to form the linear Lagrange polynomial, the three quadratic functions that combine to form the quadratic Lagrange polynomial, and the four cubic functions that combine to form the cubic Lagrange polynomial.

Difference Tables and the Newton Interpolating Polynomial This DIGMath spreadsheet lets you investigate the Newton Interpolating Polynomial and how it is based on a table of successive differences formed from a set of data. You can enter your data values with equally spaced x-values and the spreadsheet displays the corresponding table of differences, as well as the graph of the resulting Newton Interpolating Polynomial and the formula for that polynomial.

A Bouncing Ball This DIGMath module lets you investigate the mathematics behind a bouncing ball. You can work in either the English or the metric system. You input the initial height from which a ball is dropped and the percentage of the velocity that is lost on each bounce. The spreadsheet draws the graphs of the height of the ball as a function of time and the velocity of the ball as a function of time. The investigations also include the fact that the shape of each portion of the height function is a polynomial.

Graph of a Function This DIGMath spreadsheet allows you to investigate the graph of any desired function of the form y = f(x) on any desired interval a to b (or equivalently, xMin to xMax).

Shifting and Stretching This DIGMath spreadsheet allows you to investigate visually the four different aspects of shifting and stretching/squeezing a function. The function used in the dynamic presentation is a zig-zag function (basically a saw-tooth wave that serves as a precursor to the sine function). (1) The first investigation involves experimenting with the effects of changing the parameters a and c in the zig-zag function y - c = zig (x - a). You can enter, via sliders, values for these parameters and watch dynamically the effects on the resulting graph of changing their values via sliders to see the horizontal and vertical shifts. (2) The second investigation involves experimenting with the effects of changing the parameters k and m in the zig-zag function k * y = zig (m * x). You can enter, via sliders, values for these parameters and watch dynamically the effects on the resulting graph of changing their values via sliders to see the horizontal and vertical stretches and squeezes that occur.

Operations on Functions This DIGMath module lets you visualize the operations of adding or subtracting two functions, using a zig-zag function.

Symmetry of Inverse Functions This DIGMath program lets you investigate the symmetric nature of a function y = f(x) and its inverse function y =g(x) . It uses the exponential function y = b^x compared to its inverse, the logarithmic function y = log_b(x), where you use a slider to select the base b between 0 and 2. When b > 1, the exponential function is increasing and concave up and the logarithmic function is increasing and concave down. When b < 1, the exponential function is decreasing and concave up and the logarithmic function with fractional base is decreasing and concave up. The program demonstrates graphically and numerically how the two functions, for any b > 0, is symmetric about the diagonal line y = x, which is true for any function and its inverse.

The Bisection Method This DIGMath module illustrates the convergence of the Bisection Method for finding the real zeros of any function y = f(x). To use it, you enter the desired function and an initial interval from a to b over which the function has at least one zero and the program performs successive iterations, or repetitions, of the method, displaying the results both graphically and in a table to illustrate how the successive approximations converge to a real zero.

Correlation Coefficient and the Sum of the Squares This DIGMath spreadsheet lets you investigate how the correlation coefficient and the sum of the squares capture the trend in a set of data. You use a slider to vary a parameter that represents by how much a set of data is "squeezed" or "stretched" vertically to see the effects on the correlation coefficient and on the sum of the squares.

Newton's Laws of Heating and Cooling This DIGMath module lets you explore both Newton 's Law of Heating and Newton 's Law of Cooling. Using sliders, you can enter the temperature of the medium, the heating or cooling constant (essentially, the rate at which the object heats up or cools), and the initial temperature of the object. The program draws the graph of the temperature function and allows the user to trace along the curve to see the temperature value at different times.

Normal Distribution Function This DIGMath module allows you to investigate the behavior of the normal distribution function based on its two parameters: the mean m (which produces horizontal shifts) and the standard deviation s (which primarily produces vertical stretches and squeezes). You can change either of them using sliders to see the effect on the normal distribution curve.

Multivariate Linear Regression This DIGMath spreadsheet lets you perform multivariate linear regression when the dependent variable Y is a function of two independent variables X1 and X2 or a function of three independent variables X1, X2, and X3. You enter the number of data points (up to a maximum of 50) and then the values for the dependent and independent variables in the appropriate columns. The spreadsheet responds with the equation of the associated linear regression equation, the value for the sum of the squares, and the value for the coefficient of determination, R²; note that this value tells you the percentage of the variation that is explained by the linear function.

Comparing Moving Averages This DIGMath program lets you compare different moving averages to one another as well as to the underlying set of data. Moving averages are used widely in many different fields to identify patterns and trends in data where there are wild fluctuations on a day-to-day basis. This spreadsheet uses some actual data on the spread of the COVID pandemic to illustrate the concept by plotting the underlying data, the 3-day moving average, the 20-day moving average, and a third moving average that you can choose based on either 4-days, 5-days, up through 19-days.

Fitting Functions to Moving AveragesThis DIGMath spreadsheet lets you investigate how moving averages can be used to identify the pattern in a set of data that has extreme daily fluctuations. The program creates charts for the 5-day moving average, the 10-day moving average, the 15-day moving average, and the 20-day moving average. It also allows you to select the type of function you would like to use -- linear, exponential, power, quadratic, or cubic -- using a slider and displays the resulting functions that fit both the underlying data and the corresponding moving average. It also displays the resulting equations of the functions and the values of the corresponding correlation coefficients, r, (for the linear, exponential, and power fits) and the coefficient of multiple correlation, R, (for the polynomial fits) to assess how well each function fits the data.

Visualizing Sine and Cosine This DIGMath spreadsheet is intended to introduce, visually and dynamically, the graphs of the cosine and sine functions based on the movement of the minute hand of a clock over a 60 minute period. (1) The cosine function is introduced as the vertical distance, at time t, of the end of the minute hand above/below the horizontal axis. Two graphs are shown, one being that of the clock as time passes and the other being that of the associated vertical distances as time passes. You control the time t via a slider to see how the curve generated is related to the time on the clock. (2) Similarly, the sine function is introduced as the horizontal distance, at time t, from the vertical axis to the end of the minute hand.

Parameters of the Sine Function This DIGMath module lets you explore the effects of each of the four parameters A, B, C and D (each on a different page) of the sine function f(x) = A + B sin (C(x - D)). In each case, three curves are shown corresponding to specific values of the parameter and the fourth curve corresponds to the value of the parameter you select via a slider.

Parameters of the Cosine Function This DIGMath module lets you explore the effects of each of the four parameters A, B, C and D (each on a different page) of the cosine function f(x) = A + B cos (C(x - D)). In each case, three curves are shown corresponding to specific values of the parameter and the fourth curve corresponds to the value of the parameter you select via a slider.

Sinusoidal Functions This DIGMath spreadsheet allows you to investigate dynamically the effects of the four parameters A, B, C, and D on a sinusoidal function. (1) For the sine curve f(x) = A + B sin (C(x - D)), you can vary the values of the midline, the amplitude, the frequency, and the phase shift via sliders and see the effects on the corresponding graph. (2) You can conduct the same kind of experiments on the cosine function f(x) = A + B cos (C(x - D)).

Fitting Sinusoidal Functions to Data This DIGMath spreadsheet lets you investigate dynamically the problem of fitting a sinusoidal function to a set of data. You can opt to use either a sine or a cosine function. You enter the desired set of periodic data values and then enter values for the four parameters -- the midline, the amplitude, the period, and the phase shift. The spreadsheet displays the corresponding sinusoidal function superimposed over the data, so you can visually assess how well the function fits the data. It also gives the value for the sum of the squares associated with the function, so you can assess numerically how well the function fits the data. You can then adjust any of the four parameters that are reasonable to see if you can improve on the fit.

Level of the Tides Model This DIGMath spreadsheet lets you explore a simple sinusoidal model for the level of the tides. You enter the time (hour and minute) of the projected high tide, the following low tide, and the heights of those two tides. The program generates the graph of the sinusoidal model over a 24 hour day and you can trace along the curve to see the corresponding level of the tide at each time and the associated rate at this the tide is rising or falling at that time. The graph of the rate of rise/fall of the tide is also shown.

Visualizing Trig Identities This DIGMath module lets you investigate dynamically, graphically, and numerically the notion of a trig identity. You have the choice of sin 2x, cos 2x, sin 3x, ..., cos 5x. The spreadsheet draws the graphs of the function and the identity function to show that they agree exactly at every point on the interval [0, 2π]. As you trace along the curves, the spreadsheet also displays the values of y along both curves associated with the tracing point to illustrate that the agreement is identical numerically at each point. Finally, in most of the cases, the spreadsheet also draws the graphs of the components of the identity function and shows the difference in heights between the two as well as the graph of the function to demonstrate how the identity function is created.

Verifying Trig Identities This DIGMath spreadsheet lets you verify graphically whether a supposed trig identity is actually true for all values of x on any desired interval. You enter the expressions for the function on the left-hand side and the right-hand side of the supposed identity. The spreadsheet then draws the two graphs and displays the coordinates of a tracing point. If it is truly an identity, the two graphs should completely overlap across the entire interval and the values associated with the tracing point should be the same; otherwise, it is not an identity.

Approximating Sinusoidal Functions This DIGMath spreadsheet lets you investigate dynamically the idea of approximating a sinusoidal function with a polynomial. You can choose to work with either the sine or the cosine function and can use an approximating polynomial up to sixth degree in the form y = a + (1/b)x + (1/c)x²+ (1/d)x³ + (1/e)x⁴ + (1/f)x⁵ + (1/g)x⁶, where the seven parameters are all integers. You use sliders to change each of the parameter values and the associated graph shows how well the corresponding polynomial fits the sinusoidal curve. The spreadsheet also shows the value for the sum of the squares to provide a numerical measure for the goodness of the fit. This investigation is intended as a precursor to the notion of Taylor polynomial approximations.

Approximating the Exponential Function This DIGMath spreadsheet lets you investigate dynamically the idea of approximating the exponential function f(x) = b^x= e^xwith base e = 2.71828... with a polynomial. You can use an approximating polynomial up to fifth degree in the form y = a + (1/b)x +(1/c) x²+ (1/d) x³+ (1/e) x⁴+ (1/f) x⁵, where the six parameters a, b, c, d, e, and f are all integers. You use sliders to change each of the parameter values and the associated graph shows how well the corresponding polynomial fits the exponential curve. The spreadsheet also shows the value for the sum of the squares to provide a numerical measure for the goodness of the fit. This investigation is intended as a precursor to the notion of Taylor polynomial approximations.

Approximating the Natural Logarithmic Function This DIGMath spreadsheet lets you investigate dynamically the idea of approximating the natural logarithmic function f(x) = ln x with base e = 2.71828... with a polynomial. You can use an approximating polynomial up to fifth degree in the form y = a + (1/b)(x - 1) + (1/c) (x-1)²+ (1/d) (x-1)³+ (1/e) (x-1)⁴+ (1/f) (x-1)⁵, where the six parameters a, b, c, d, e, and f are all integers. You use sliders to change each of the parameter values and the associated graph shows how well the corresponding polynomial fits the exponential curve. The spreadsheet also shows the value for the sum of the squares to provide a numerical measure for the goodness of the fit. This investigation is intended as a precursor to the notion of Taylor polynomial approximations.

Approximating the Sine Function with Lagrange Interpolation This DIGMath module lets you investigate dynamically the way that the Lagrange Interpolating Polynomial is used to approximate values of the sine function on the interval [0, π/4] using either a quadratic polynomial (based on three interpolating points) or a cubic polynomial (based on four interpolating points).

Approximating the Cosine Function with Lagrange Interpolation This DIGMath module lets you investigate dynamically the way that the Lagrange Interpolating Polynomial is used to approximate values of the cosine function on the interval [0, π/4] using either a quadratic polynomial (based on three interpolating points) or a cubic polynomial (based on four interpolating points).

Approximating the Exponential Function with Lagrange Interpolation This DIGMath spreadsheet lets you investigate dynamically the way that the Lagrange Interpolating Polynomial is used to approximate values of the exponential function. One page lets you work with a quadratic polynomial based on three points on the exponential curve and another page lets you use a cubic polynomial based on four points.

Approximating the Sine Function with Newton Interpolation This DIGMath module lets you investigate dynamically the way that the Newton Interpolating Polynomial is used to approximate values of the sine function on the interval [0, π/4]using either a quadratic polynomial (based on three uniformly spaced interpolating points) or a cubic polynomial (based on four uniformly spaced interpolating points).

Approximating the Cosine Function with Newton Interpolation This DIGMath module lets you investigate dynamically the way that the Newton Interpolating Polynomial is used to approximate values of the cosine function on the interval [0, π/4]using either a quadratic polynomial (based on three uniformly spaced interpolating points) or a cubic polynomial (based on four uniformly spaced interpolating points).

Approximating the Exponential Function with Newton Interpolation This DIGMath spreadsheet lets you investigate dynamically the way that the Newton Interpolating Polynomial is used to approximate values of the exponential function. One page lets you work with a quadratic polynomial based on three uniformly spaced points on the exponential curve and another page lets you use a cubic polynomial based on four uniformly spaced points. The spreadsheet calculates and draws the graph of the interpolating function as it passes through the interpolating points and seeks to match the exponential curve. It also draws the graph of the error function -- the difference between the polynomial and the exponential function.

Approximating the Natural Logarithm with Newton Interpolation This DIGMath module lets you investigate dynamically the way that the Newton Interpolating Polynomial is used to approximate values of the natural logarithm function. The first page lets you compare the effectiveness of the quadratic, the cubic, the quartic, and the quintic Newton polynomials based on uniformly spaced points on the curve. The spreadsheet calculates and draws the graphs of the four interpolating polynomials as they pass through the interpolating points on the natural logarithm curve. It also lets you select a particular degree for the polynomial and draws the graph of the associated error function -- the different between the polynomial and the logarithmic function. A second page shows all four interpolating polynomials separately and all four error functions separately.

Chebyshev Nodes This DIGMath module lets you investigate the so-called Chebyshev nodes that are often the best points to use to construct an interpolating polynomial based on a set of points. The Chebyshev nodes have the property that they provide the minimum value of the maximum error in the approximation at the interpolating points. The spreadsheet lets you enter your choice of either three or four interpolating points on the interval [-1, 1] and compares, both numerically and graphically, the maximum error to the corresponding maximum error using the Chebyshev nodes.

The Tangent Function This DIGMath module lets you investigate the definition of the tangent function as the ratio of the sine and the cosine. You can enter any desired interval in radians (or in degrees on a separate sheet of the spreadsheet). The spreadsheet draws two charts, one for the graphs of the sine and cosine functions, and the other for the graph of the tangent function. It also provides numerical results, showing the values of the sine, the cosine, and the tangent at any desired tracing point, to demonstrate that the tangent function is indeed the ratio of the sine and cosine at each point.

Vertical Asymptotes for Rational Functions This DIGMath spreadsheet lets you investigate the notion of the vertical asymptotes associated with a rational function of the form y = (x - a)(x - b)/(x - c)/(x - d) for your choice of the four parameters a, b, c, and d. You can select the values for the four parameters and the spreadsheet displays the corresponding graph showing the usual two vertical asymptotes (there may be only 1 or none, in some very special combinations of the parameters) and watch how the locations of those asymptotes change as you change the parameter values.

DeMoivre's Theorem This DIGMath module lets you explore DeMoivre's Theorem. You enter the values for any desired complex number and use a slider to select the power to which z = a + bi is to be raised. You can investigate the results both graphically and numerically in three situations: using the trigonometric form for the complex number and its powers, using the rectangular form for both, and using non-integer powers.

Tangentoidal Functions This DIGMath module lets you investigate the behavior of the so-called tangentoidal functions that are defined as f(x) = sin x /[a + cos x]. You can enter any desired value for the parameter a and any desired interval in radians. The spreadsheet draws the graph of the tangentoidal function. It also raises questions about the behavior of these functions in terms of the location and existence of vertical asymptotes and how that is related to the value of a.

Systems of Linear Equations This DIGMath modules provides a tool for solving systems of linear equations using matrix methods. You have the choice of a 2 x 2 system, a 3 x 3 system, or a 4 x 4 system. In each case, you enter the components of the matrix of coefficients A and the vector (matrix) of constants B in AX = B and the program responds with the corresponding solution vector X.

Homogeneous Linear Systems This DIGMath spreadsheet lets you investigate the solutions of homogeneous systems of linear equations AX = 0 from a graphical perspective. You can use sliders to change the coefficients and watch the effects on the corresponding lines to see the conditions under which the lines have a unique point of intersection, are parallel so there is no solution, or overlap completely so that there are infinitely many solutions.

Matrix Powers This DIGMath module lets you investigate the successive powers of a 2 by 2 matrix A applied to a vector X₀. The results are displayed both graphically and numerically.

Rotation Matrices This DIGMath module lets you investigate the effects of successively applying a rotation matrix A to any desired initial vector X₀ using any desired angle. The results are displayed both graphically and numerically.

Markov Chains This DIGMath module lets you investigate the effects of successive terms in a Markov chain by applying a transition matrix A to any desired initial vector X₀. The results are displayed both graphically and numerically.

Finding Eigenvectors Graphically This DIGMath program lets you investigate graphically the notion of the eigenvectors and eigenvalues of the 2 x 2 matrix A. You can enter the four entries in the matrix and the components of an initial vector X using sliders. The spreadsheet draws the graph of the vector X and the resultant vector AX. You can estimate the eigenvectors by adjusting the components of X until X and AX are essentially pointing in the same direction. The program also displays the coordinates and shows the actual eigenvalues, either real or complex.

Eigenvalues and Eigenvectors This DIGMath modules finds the eigenvalues and eigenvectors of any two by two matrix that you enter, whether there are two distinct real eigenvalues, a double real eigenvalue, or a pair of complex conjugate eigenvalues. It also displays the characteristic equation, the value of the discriminant, and graphs of the real eigenvectors.

The Ellipse This DIGMath spreadsheet lets you investigate the properties of the ellipse. One sheet is based on the equation of the ellipse and you can select and vary the values for the center x = c of the ellipse and the parameters a and b in the ellipse and see the effects on the resulting graphs instantly. The second sheet is based on the definition of the ellipse as the set of points such that the sum of the distances to two fixed points, the foci, is a constant. You can enter and vary the locations of the two foci that lie on the x-axis and the value for the sum of the two distances and see the effects on the ellipse so determined, as well as its equation.

The Hyperbola This DIGMath spreadsheet lets you investigate the properties of the hyperbola. One sheet is based on the equation of the hyperbola where the foci are located on the horizontal axis and you can select and vary the values for the center x = x₀ of the hyperbola and the parameter a in its equation and see the effects on the resulting graph instantly. A second sheet is based on equation of the hyperbola where the foci are located on the vertical axis and you can select and vary the values for the center y = y₀ of the hyperbola and the parameter a in its equation and see the effects on the resulting graph instantly. The third sheet is based on the definition of the hyperbola as the set of points such that the difference in the distances to two fixed points, the foci, is a constant. You can enter and vary the location of the foci that lie on the x-axis symmetrically and the value for the difference of the two distances and see the effects on the hyperbola so determined, as well as its equation.

The Parabola This DIGMath spreadsheet lets you investigate the geometric definition of the parabola. One sheet is based on the equation of the parabola where the focus is located on the vertical axis and you can select and vary the value for the focus c and the width of the display window and see the effects on the resulting graph instantly. It also shows the directrix and displays the distances from a tracing point both to the focus and to the directrix. A second sheet is based on equation of the parabola where the focus lies on the horizontal axis and you can select and vary the values for the value for the focus c and the height of the display window and see the effects on the resulting graph instantly. It also shows the distances from a moving point to the focus and the directrix.

Rational Functions This DIGMath spreadsheet lets you investigate the properties of rational functions. On different sheets, you have the choice of a variety of different forms for a rational function, including (1) the product of two linear terms divided by a quadratic term, (2) the product of three linear terms divided by a quadratic term, and (3) the product of one linear and one quadratic term divided by a quadratic term. In each case, you enter the values of the parameters in each factor and the spreadsheet displays the corresponding graph instantly, as well as the equation of that rational function.

Exploring Vertical Asymptotes of Rational Functions This DIGMath module lets you explore the vertical asymptotes of rational functions of the form f(x) = (x - a)(x - b)/(x - c)(x - d). You can enter the values for the four parameters and the spreadsheet produces the graph of the function, highlighting both the real roots and the vertical asymptotes. As you vary the parameters c and d, you can see how the vertical asymptotes change accordingly to give a dynamic feel to the location of the asymptotes and their correspondence to the roots of the denominator.

The Fibonacci Sequence and the Golden Ratio This DIGMath module lets you explore the Fibonacci Sequence {1, 1, 2, 3, 5, 8, 13, ...} in which each term is the sum of the previous two terms. It draws the graph of the points in the sequence along with a visual depiction of how each term is the sum of the preceding two. It also draws the values of the ratios of the successive terms to illustrate how they converge to the Golden Ratio and the graph of the differences between each pair of successive ratios and the value of the Golden Ratio to highlight how the convergence is oscillatory in manner.

The Lucas Sequence and the Golden Ratio This DIGMath spreadsheet lets you explore the Lucas Sequence, which is an extension of the Fibonacci Sequence in which each entry is likewise the sum of the previous two, but the first two terms can be selected rather than being 1 and 1. The spreadsheet draws the graph of the points in the sequence along with a visual depiction of how each term is the sum of the preceding two. It also draws the ratios of the successive terms in the sequence to illustrate how they also converge to the Golden Ratio and the graph of the differences between each pair of successive ratios and the value of the Golden Ratio to highlight how the convergence is oscillatory in manner.

Variations on the Fibonacci Sequence This DIGMath module lets you investigate some variations on the Fibonacci Sequence in which the underlying rule is G_n+2 = aG_n₊₁ + bG_n, where a and b are two constant multiples and the two starting values can also be selected. The spreadsheet displays a chart showing the points in the resulting sequence and a second chart showing the successive ratios of the terms in the sequence. For many values of a and b, the terms of the sequence increase rapidly in an exponential-type pattern and the successive ratios approach a limiting value. For other values of a and b, the terms of the sequence oscillate in a periodic pattern, as do the successive ratios. Associated with the relationship among the G's is a quadratic polynomial r² - ar - b = 0, known as the characteristic polynomial. When this quadratic has two real roots (depending on the sign of the discriminant), the sequence of the successive ratios approach a limiting value and when there are a pair of complex conjugate roots, the sequence of values oscillate periodically and the successive ratios also oscillate periodically. The spreadsheet also has a chart of the graph of this polynomial and it is reasonably evident that the larger of the two real roots is equal to the limiting value for the successive ratios.

The Tribonacci Sequence This DIGMath module lets you explore the so-called Tribonacci Sequence as an extension of the Fibonacci Sequence. Instead of each term being the sum of the previous two terms, each term is now the sum of the previous three terms. For instance, if the first three terms are 1, 1, and 3, then the following term is 5 and the one after that is 9. The spreadsheet draws the graph of the points in the sequence along with a visual depiction of how each term is the sum of the preceding three. It also draws the values of the ratios of the successive terms to illustrate how they converge to a fixed limit (not the Golden Ratio) and the graph of the differences between each pair of successive ratios and the value of the limiting value to highlight how the convergence is oscillatory in manner.

The Fibonacci-like Sequence with Differences instead of Sums This DIGMath spreadsheet lets you explore a variation on the Fibonacci Sequence in which each entry is the difference of the previous two terms. You can select the first two terms using sliders to determine the specific sequence based on them. The spreadsheet draws the graph of the points in the sequence along with a visual depiction of how each term is the difference of the preceding two. Surprisingly, the terms in this sequence are periodic with period 6.

The Fibonacci-like Sequence with Products instead of Sums This DIGMath spreadsheet lets you investigate a variation on the Fibonacci Sequence in which each entry is the product of the previous two terms. You can select the first two terms using sliders to determine the specific sequence based on them. The spreadsheet draws the graph of the points in the sequence. The exploration of this sequence suggests that you write out the terms in the solution as powers of the two initial values and look for a very surprisingly pattern in those powers.

Dynamic Investigatory Graphical Displays of Math: DIGMath Graphical EXPLORATIONS for Calculus using Excel

As mentioned above, Excel must be set to accept macros by selecting "Enable Content".

DIGMath Spreadsheets for Calculus
Graph of a Function This DIGMath spreadsheet allows you to investigate the graph of any desired function of the form y = f(x) on any desired interval a to b (or equivalently, xMin to xMax).
Graphs of Two Functions This DIGMath utility allows you to investigate the graphs of any two functions of the form y = f(x) and y = g(x) on any desired interval a to b. It is probably most useful as a tool for estimating the point(s) of intersection of the two functions by numerically zooming in on the graphs.
Delta-Epsilon Definition of the Limit This DIGMath module allows you to explore the delta-epsilon definition of the limit of a function at a point. For your choice of any point x = a on any function's graph, you can select the value of e and see graphically the notion that you want to construct a box centered at the presumed limit point that contains the portion of the curve from x = a - d to x = a + d, for some d > 0.
The Newton Difference-Quotient This DIGMath module lets you investigate the Newton Difference-Quotient, which is the basis for the definition of the derivative of a function at a point. You can enter any desired function on any interval, adjust the size of the step-size using a slider, and select the point on the curve. As the step-size decreases toward zero, you can see, both visually and numerically, how the difference-quotient approaches the value for the derivative of the function at the point, which is equivalent to the slope of the tangent line at that point.
Tangent Line to a Curve This DIGMath spreadsheet allows you to investigate the tangent line to the graph of any desired function of the form y = f(x) on any desired interval a to b. You control the point of tangency using a slider and watch the effects on the resulting tangent line to the curve as the point changes. The length of the tangent line also changes to reflect the size of the slope or, equivalently, the value of the derivative of the function at the point.
The Angle of Inclination of the Tangent Line This DIGMath module lets you investigate the angle of inclination of the tangent line to a curve as a function of the point x as you use a slider to move along the curve.
Tangent Parabola to a Curve This DIGMath spreadsheet allows you to investigate the idea that, at each point on a smooth curve, there is a parabola (the second order Taylor polynomial approximation, actually) that is tangent to the graph of any desired function of the form y = f(x) and has the same curvature as the graph on any desired interval a to b. You control the point of tangency using a slider and watch the effects on the resulting tangent parabola as the point changes.
Secant Lines This DIGMath module lets you investigate the convergence, as h approaches 0, of a sequence of secant lines to any desired curve at any desired point. Graphically, it is clear that the successive secant lines converge to the tangent line and numerically, the slopes of the successive secant lines converge to the slope of the tangent line at the point, which is the value of the derivative at that point.
The Bisection Method This DIGMath module illustrates the convergence of the Bisection Method for finding the real zeros of any function y = f(x). To use it, you enter the desired function and an initial interval from a to b over which the function has at least one zero and the program performs successive iterations of the method, displaying the results both graphically and in a table.
The Secant Method This DIGMath module illustrates the convergence of the Secant Method for finding the real zeros of any function y = f(x). To use it, you enter the desired function and an initial estimate x₀ of any real zero and the program performs successive iterations of the method, displaying the results both graphically and in a table.
The Regula Falsi (or False Position) Method This DIGMath module illustrates the convergence of the Regula Falsi Method for finding the real zeros of any function y = f(x). To use it, you enter the desired function and an initial interval that brackets a real zero x₀ and the program performs successive iterations of the method, displaying the results both graphically and in a table.
The Secant Parabola Method for Root Finding This DIGMath module illustrates the convergence of the Secant Parabola Method for finding the real zeros of any function y = f(x). It is based on the idea of using a parabola determined by three points to find the next approximation to a real zero x₀ of the function. The next approximation is determined by one of the two real roots of the quadratic polynomial based on the quadratic formula. The program performs successive iterations of the method, displaying the results both graphically and in a table.
A Function and its Derivatives This DIGMath spreadsheet allows you to investigate the behavior of any desired function of the form y = f(x) and its first and second derivatives on any desired interval a to b. Using a slider, you can control the position of a point on the curve of the function and see the tangent line to the curve at that point. A vertical line is also drawn through that point and connects to the other two curves, so you can see the corresponding slope of the tangent line on the graph of the derivative and the corresponding point on the graph of the second derivative.
Mean Value Theorem This DIGMath spreadsheet allows you to investigate the Mean Value Theorem of any desired smooth function of the form y = f(x) on any desired interval a to b. There are two components to the investigation. First, you can control the point of tangency using a slider and watch the effects on the resulting tangent line both visually and numerically as the point changes in order to locate all the points c where the tangent line is parallel to the secant line connecting the endpoints of the curve on the desired interval. Second, using a slider, you can slide a line that is parallel to the secant line until it is tangent to the curve and then find the point of tangency.
Derivative of the Exponential Function This DIGMath spreadsheet allows you to investigate the derivative of exponential functions of the form y = b^xand discover the base e. The spreadsheet shows the graph of the exponential function and its derivative. For some values of b, the derivative is below the function; for other values of b, the derivative is above the function. The challenge is to find the value for b as accurately as possible for which the derivative is exactly the same as the function itself. You control the value of b using a slider to produce the dynamic effects
Derivative of the Natural Logarithm This DIGMath spreadsheet allows you to investigate the derivative of the natural logarithm function y = ln x. There are two components to this investigation. First, you can control the location of four points on the curve of the logarithmic function using sliders and see the corresponding four tangent lines. An associated chart shows the slopes of those four tangent lines, which fall into the pattern of a decaying power function y = x^p with p < 0. The spreadsheet uses Excel’s power function fit routine to calculate, graph, and display the equation of that power function. The second component generates sets of 20 random points on the natural logarithm curve, calculates the slope of the tangent line at each of these points, displays the results, and fits a power function to the 20 points.
Exponential Rate of Change This DIGMath module lets you investigate the rate of change of exponential functions based on the base b. You can compare the rate of change of three exponential growth functions both graphically and numerically by looking at the slope of the tangent line to the three curves at different points of tangency as you use a slider.
Quadratic Rate of Change This DIGMath module lets you investigate the rate of change of quadratic functions y = ax + bx + c based on the parameters a, b, and c. You can compare the rate of change of three quadratic functions both graphically and numerically by looking at the slope of the tangent line to the three curves at different points of tangency as you use a slider. There are three cases, one where you can select and change the values of a and b using sliders while c takes on three fixed values, a second where you can select and changes the values of a and c while b takes on three fixed values, and a third where you can select and change the values of b and c while a takes on three fixed values.
Derivative of the Sine and Cosine This DIGMath spreadsheet allows you to investigate the derivative of both the sine and the cosine functions . The spreadsheet shows the graph of the either function and its derivative based on the slope of the tangent line. A slider allows the user to control a moving point along both curves drawn to see the way that the slope of the tangent line in the derivative plot relates to points on the graph of the original function.
Discovering the Chain Rule This DIGMath module allows the user to discover the chain rule by examining the graphs of y = sin x, y = sin 2x, y = sin 3x, and y = sin x², along with the associated graph of its derivative. The graphs clearly indicate that the derivative of each sine function has the form of a cosine function, but with different amplitudes; for the last of the four functions, the amplitude is not constant, but rather increases in a linear pattern that is enveloped by the function 2x.
Newton's Method This DIGMath module lets you investigate Newton's Method for finding the roots of a function f, both numerically and graphically. For any desired function, any desired starting value, and any desired number of iterations, you can see the set of iterated approximations in a table and the graph of the process, either in a fixed window of your choice (although the sequence of approximations may leave the window) or in a variable window that follows the sequence of iterations.
The Differential This DIGMath spreadsheet lets you investigate the differential dy associated with a change dx in the independent variable for any desired function y = f(x). In particular, you can compare, both graphically and numerically, the change in the function along the tangent line to the curve at any given point and the actual change along the curve for any value of Dx at any desired point.
Visualizing the Product Rule This DIGMath module helps you understand the product rule through a visual image. You can enter any two functions f(x) and g(x). The spreadsheet draws the graph of the two and the graph of the product and, as you trace along the curves, it shows the various values, including the slope of the point on the product curve. It also shows the graph of the product of the two derivatives, y = f(x) g(x), as well as the graph of the product rule function, y = f(x) g'(x) + f'(x) g(x).
Visualizing the Quotient Rule This DIGMath spreadsheet helps you understand the quotient rule for the derivative of the quotient of two functions through a visual image. You can enter any two functions f(x) and g(x). The spreadsheet draws the graph of the two and the graph of the quotient f(x) / g(x); as you trace along the curves, it shows the various values, including the slope of the point on the quotient curve. It also shows the graph of the quotient of the two derivatives, y = f '(x)/ g'(x), as well as the graph of the quotient rule function, y = [f(x) g'(x) - f'(x) g(x)]/g²(x).
Projectile Motion This DIGMath spreadsheet allows you to investigate the path of a projectile launched from ground level with initial velocity and initial angle of inclination a . You control the values for the initial velocity and the angle via sliders and the spreadsheet draws the path of the projectile, allows you to trace along the path via another slider, and displays the time t, the coordinates of the tracing point, and the vertical velocity at each point. One page uses the English system of measurements in feet and seconds and another pages does the comparable displays in the metric system with centimeters and seconds. Among the suggested investigations is one involving finding the angle a for which the range of the projectile is maximum.
The Farmer Brown Fencing Problem This DIGMath program lets you investigate graphically the standard optimization problem of finding the dimensions of the largest rectangular pen(s) that a farmer can construct with a given amount of fencing (the perimeter). There are several scenarios: a single rectangle, a single rectangle using an existing wall or fence or river for one side, two rectangular pens, and three rectangular pens. Using a slider, you can see the effect on the total area of the pen(s) based on the perimeter and compare the solution observed graphically and numerically with the analytic solution.
The Optimal Sum or Product of Numbers Problem This DIGMath module lets you investigate graphically the standard optimization problems of finding either two numbers with a given sum whose product is maximum or two numbers with a given product whose sum is minimal. Using a slider in each case, you can see the effect both graphically and numerically on the quantity being optimized and compare the solution observed with the analytic solution.
The Optimal Sum of Squares of Two Numbers Problem This DIGMath module lets you investigate graphically the standard optimization problems of finding two numbers such that the sum of their squares is either a maximum or a minimum. Using a slider, you can see the effect both graphically and numerically on the sum of the squares of the two numbers with a sum that you select and you can then compare the solution observed with the analytic solution.
The Largest Rectangle that Fits in a Circle Problem This DIGMath module lets you investigate graphically the standard optimization problems of finding the largest rectangle that fits into the unit circle. It draws the graph of the area function for the rectangle as a function of its horizontal length. Using a slider, you can see the effect both graphically and numerically on the area of the rectangle and you can then compare the solution observed with the analytic solution.
The Wire Into a Square Plus a Circle Problem This DIGMath program lets you investigate graphically the standard calculus problem of cutting a length of wire into two pieces to form a square and a circle that encompass the greatest area. Using a slider, you can see the effect on the total area based on the length of wire used to form the square and compare the solution observed graphically and numerically with the analytic solution.
The Wire Into a Square Plus an Equilateral Triangle Problem This DIGMath program lets you investigate graphically the standard calculus problem of cutting a length of wire into two pieces to form a square and a an equilateral triangle that encompass either the minimum area or the maximum area. Using sliders, you can see the effect on the total area based on the length of wire used to form the square and the triangle and compare the solution you observe graphically and numerically with the analytic solution.
The Distance from a Point to a Parabola Problem This DIGMath spreadsheet lets you investigate graphically the standard calculus problem of finding the point on the parabola y = x² that is closest to a given point P(a, b). You use sliders to enter the coordinates of P, and the spreadsheet shows the graph of the situation along with the graph of the distance function as a function of x = a. Using a slider, you can see the effect on both the overall situation and the distance function as the point changes and then you can compare the solution observed graphically and numerically with the analytic solution.
The Distance from a Point to a Circle Problem This DIGMath spreadsheet lets you investigate graphically the standard calculus problem of finding the point on the circle x² + y² = r² that is closest to a given point P(a, b). You use sliders to enter the coordinates of P and the value of the radius r, and the spreadsheet shows the graph of the situation along with the graph of the distance function as a function of x as well as the graph of the derivative function. Using a slider, you can see the effect on both the overall situation, the distance function and the derivative as the point changes and then you can compare the solution observed graphically and numerically with the analytic solution.
The Run and Swim Problem This DIGMath module lets you investigate graphically the standard optimization problem of finding the optimal path for a person to run along a shore and then swim out to a particular point. Using a slider, you can see the effect on the total time based on the point where the person takes to the water and compare the solution observed graphically with the analytic solution.
The Two Poles Staked to the Ground Problem This DIGMath spreadsheet lets you investigate the standard optimization problem in which there are two vertical poles and a guy wire that stakes both of them to the ground at some point between the two poles. The problem is to find the point so that the length of wire used is a minimum, based on the height of the two poles and the distance between them. You enter these three values using sliders and then trace both the graph of the total length of wire function and the derivative function, using a slider, to determine where the minimum occurs. You can then compare the solution you observe graphically and vertically with the analytic solution
The Ladder Around a Corner Problem This DIGMath module lets you investigate graphically the standard optimization problem of finding the longest ladder that can be carried horizontally around a corner from one corridor to another corridor. Using a slider, you can see the effect on the length of the ladder based on the widths of the two corridors and compare the solution observed graphically and numerically with the analytic solution.
The Printed Page Problem This DIGMath module lets you investigate graphically the standard optimization problem of finding the dimensions of the smallest sheet of paper that will contain a given area of printed material. Using a slider, you can see the effect on the total area of the page based on the side and top/bottom margins and the area of the printed material and compare the solution observed graphically and numerically with the analytic solution.
The Minimum Land Needed for a Building Problem This DIGMath spreadsheet lets you investigate graphically the standard optimization problem of finding the dimensions of the smallest plot of land on which a building of a given area can be constructed given building code requirements about the free space needed on each of the four sides. Using a slider, you can see the effect on the total area of the plot based on the front, back, and side margins and compare the solution observed graphically and numerically with the analytic solution.
The Norman Window Problem This DIGMath module lets you investigate graphically the standard optimization problem of finding the dimensions of the largest Norman Window (a rectangle surmounted by a semicircle) that can be constructed with a given perimeter. Using a slider, you can see the effect on the total area of the window based on the perimeter and the radius of the semicircle and compare the solution observed graphically and numerically with the analytic solution.
The Open Box Problem This DIGMath program lets you investigate graphically the standard optimization problem of finding the dimensions of the largest (meaning greatest volume) open box that can be constructed by snipping off the four corners of a sheet of cardboard. The program has two pages: the first is the usual problem where the cardboard sheet is square and the second is the more sophisticated problem when the cardboard is rectangular. Using sliders, you can see the effect on the total volume of the box based on the lengths of the sides of the cardboard and the size of the corner being snipped away. You can also compare the solution observed graphically and numerically with the analytic solution.
The Open Box with a Given Amount of Material Problem This DIGMath program lets you investigate graphically the standard optimization problem of finding the dimensions of the largest (meaning greatest volume) open box with a square base that can be constructed using a given amount of cardboard for the base and four sides. Using sliders, you can see the effect on the total volume of the box and the associated derivative function based on the lengths of the sides and the height. You can also compare the solution observed graphically and numerically with the analytic solution.
The Cheapest Tin Can Problem This DIGMath module lets you investigate graphically the standard optimization problem of finding the dimensions of the cheapest (meaning, least surface area) open cylindrical tin can that can be constructed having a given volume. Using a slider, you can see the effect on the surface area of the tin can based on the volume and the radius of the tin can and the associated derivative function. You can compare the solution observed graphically and numerically with the analytic solution.
The Cost of a Tin Can Problem This DIGMath module lets you investigate graphically the standard optimization problem of finding the dimensions of the cheapest cylindrical tin can that can be constructed having a given volume where there are costs associated with the metal used for the sides and for the top and bottom. Using a slider, you can see the effect on the total cost of the tin can based on the volume and the radius of the tin can and compare the solution observed graphically and numerically with the analytic solution.
The Cylinder Inscribed in a Cone Problem This DIGMath spreadsheet lets you investigate graphically the standard optimization problem of finding the dimensions of the largest (meaning greatest volume) cylinder that can be inscribed in a right circular cone. The program lets you select the radius and height of the cone, using sliders. Then as you use a slider to select the radius of the cylinder, you can see the effect on the volume of the cylinder that is inscribed in the cone, as well as the value of the height of the cylinder. You can also compare the solution observed graphically and numerically with the analytic solution.
The Maximum Viewing Angle Problem This DIGMath spreadsheet lets you investigate graphically the standard optimization problem of finding the distance from a wall to stand to have the maximum viewing angle with which to view a painting hanging on the wall. Using sliders, you can select the height of the painting as well as the height from eye level to the bottom of the painting. You can then see the effect on the viewing angle a as you change the distance x from the wall. You can also compare the solution observed graphically and numerically with the analytic solution.
The Blowing Up a Balloon Related Rate Problem This DIGMath module lets you investigate graphically the standard related rate problem of finding the rate at which the radius of a balloon changes, at a particular instant, when air is blown into the balloon at a fixed rate. Using sliders, you can see how the volume of the balloon depends on the radius, how the volume changes over time, and the rate at which the radius changes as air is blown into the balloon, and you can also compare the solution observed graphically and numerically with the analytic solution.
The Two Cars Approaching an Intersection Related Rate Problem This DIGMath spreadsheet lets you investigate graphically the standard related rate problem of finding the rate at which the distance between two cars approaching an intersection at right angles at different speeds, at a particular instant. Using sliders, you can see how the distances of the two cars from the intersection changes with respect to time, how the distance between the two cars changes over time, and how the rate of change of the distance changes over time. You can also compare the solution observed graphically and numerically with the analytic solution.
The Plane vs. the Radar Station Related Rates Problem This DIGMath module lets you investigate graphically the standard related rate problem of finding the rate at which the distance from a plane to a radar station changes at a particular instant when the plane flies at a fixed altitude with a fixed speed. Using sliders, you can see how the distance from the plane to the radar dish changes with respect to time, how the distance changes with respect to the horizontal distance, and how the rate of change of the distance changes over time. You can also compare the solution observed graphically and numerically with the analytic solution.
The Slipping Ladder Related Rate Problem This DIGMath module lets you investigate graphically the standard related rate problem of finding the rate at which the height of a ladder leaning against a wall decreases, at a particular instant, when the base of the ladder slips away from the wall at a given rate. Using sliders, you can see how the height of the ladder changes as a function of the distance the base is from the wall, how the height of the ladder changes with respect to time, and how the rate of change of the height changes over time. You can also compare the solution observed graphically and numerically with the analytic solution.
The Conical Pile of Sand Related Rate Problem This DIGMath program lets you investigate graphically the standard related rate problem of finding the rate at which the height of a conical pile of sand grows, at a particular instant, as additional sand is added to the pile. Using sliders, you can see the rate at which the height changes as the sand is added to the pile and compare the solution observed graphically and numerically with the analytic solution.
The Filling a Conical Cup Related Rate Problem This DIGMath module lets you investigate graphically the standard related rate problem of finding the rate at which the height of liquid in a cone-shaped cup increase, at a particular instant, when the liquid is poured into the cup at a given rate. Using sliders, you can see how the height of the liquid changes as a function of the radius of the liquid at the surface, how the height of the liquid changes with respect to time, and how the rate of change of the height changes over time. You can also compare the solution observed graphically and numerically with the analytic solution.
The Length of Shadow Related Rate Problem This DIGMath program lets you investigate graphically the standard related rate problem of finding the rate at which the shadow of a person walking away from a lamppost changes, at a particular instant. There are two scenarios. One is where the quantity of interest is the length of the shadow; the other is the rate at which the tip of the shadow is moving away from the light. Using sliders, you can see the rate at which the length of the shadow changes or the rate at which the tip of the shadow moves as the person's distance from the lamppost changes based on the person's height, the height of the light, and the rate at which the person walks. You can then compare the solution observed graphically and numerically with the analytic solution.
The Water into a Trapezoidal Trough Related Rate Problem This DIGMath module lets you investigate graphically the standard related rate problem of finding the rate at which the height of the volume of water in a watering trough with a trapezoidal cross-section changes, at a particular instant, where the water is being poured into the trough at a fixed rate. Using sliders, you can enter the rate at which the water comes in, the height of the trough, the length of the trough, and the bottom and top bases of the trapezoidal face of the trough. The spreadsheet displays the graph of the volume V of the water versus the height y of the water, the graph of the volume V versus time t, and the graph of the rate of change of the height of water, dy/dt versus t. You can then compare the solution observed graphically and numerically with the analytic solution.
Tangentoidal Functions This DIGMath module lets you investigate the behavior of the so-called tangentoidal functions that are defined as f(x) = sin x /[a + cos x]. You can enter any desired value for the parameter a and any desired interval in radians. The spreadsheet draws the graph of the associated tangentoidal function. It also raises questions about the behavior of these functions in terms of the location and existence of vertical asymptotes and how that is related to the value of a.
The Third Derivative This DIGMath module lets you investigate the properties of the third derivative of any function and how it relates to the function, to the first derivative, and to the second derivative.
Inverse Functions This DIGMath module lets you explore graphically the inverse of a function f. For any choice of a function that is strictly increasing or decreasing on an interval [a, b[, the program draws the graph of both the function and the inverse to demonstrate the symmetric relationship between the two.
Cubic Splines This DIGMath spreadsheet lets you investigate the notion of cubic splines, a way to construct a smooth curve determined by a set of points in such a way that the curve is made up of a series of smoothly connected cubic curves. The spreadsheet has two components. In the first, the data points are grouped three at a time subject to the condition that the slope at the third point must be equal to the slope of the following cubic at the first point. In the second, the points are grouped two at a time subject to the two conditions that the slope and the value of the second derivative must agree at each of the overlapping points.
Antiderivatives of a Function This DIGMath spreadsheet allows you to investigate two different aspects of the antiderivative of a function. First, you can enter any function on any interval and the minimum and maximum "starting" values for the antiderivative of the function. The spreadsheet draws three graphs; two correspond to the minimum and maximum starting values and the third is controlled by a slider that lets you vary the "starting" value, so that you can see a spectrum of different antiderivative functions. The second aspect of the antiderivative on a separate page draws the graph of the function along with one antiderivative and, with the use of a slider, allows you to see the correspondence of points on the two curves.
The Second Fundamental Theorem This DIGMath module lets you investigate the Second Fundamental Theorem of Calculus, which says that the derivative of a definite integral with a variable limit of integration is equal to the function evaluated at that upper limit of integration. You can enter any desired function of x on any interval from a to b and the spreadsheet shows the graph of the function. You can then select any point between a and b with a slider and the spreadsheet sweeps out the area under the curve in one chart and also the graph of the area function in a second chart.
Numerical Integration This DIGMath spreadsheet allows you to investigate four different methods to approximate the value of a definite integral -- using left and right-hand Riemann Sums, using the Trapezoid Rule, the MidPoint Rule, and Simpson's Rule for any function of the form y = f(x) on any desired interval [a, b]. You control the number of subdivisions for each method using a slider and the spreadsheet draws the graph of the function, draws the approximating subdivisions, and displays the associated approximation to the definite integral.
Monte Carlo Method for Definite Integrals This DIGMath spreadsheet allows you to investigate visually and numerically the use of Monte Carlo simulations for estimating the value of the definite integral of any function of the form y = f(x) that is non-negative on any desired interval [a, b]. You control the number of random points, between 500 and 2500, via a slider and the spreadsheet draws the graph of the function, plots the random points, and displays the number and percentage of them that fall under the curve, and uses that percentage to estimate the area of the region.
Monte Carlo Methods for Graphing a Function This DIGMath unit uses Monte Carlo simulation methods to produce the graph of a function on any desired interval. You can select the number of random points (between 10 and 50) on the function curve to see how the sample may be adequate to create the curve. You can also request that new samples of the same size be generated to observe how the pattern of points generated varies from one sample to another.
Mean Value Theorem for Integrals This DIGMath spreadsheet allows you to investigate the Mean Value Theorem for Integrals of any desired smooth function of the form y = f(x) on any desired interval a to b. First, you slide a horizontal line up and down until the area of the rectangle roughly matches the area of the region under the curve. Second, using a slider, you can slide a point along the curve to find the coordinates of the points where the horizontal line crosses the curve and so determine the values of c for which the theorem holds.
Integrating the Acceleration Function This DIGMath module allows you to investigate visually the process of starting with the function representing the acceleration of a body as a function of time and then integrating the acceleration once to produce the velocity function and then integrating the velocity to produce the position function.
Arc Length This DIGMath spreadsheet allows you to investigate the arc length of any curve y = f(x) on any desired interval a to b. You have the choice of the desired number of subdivisions, n = 4, 8, 16, ..., 128 and the program draws all of the associated piece-wise linear approximations to the arc length to illustrate the convergence graphically to the curve. It also displays the corresponding numerical values in a table to illustrate the convergence numerically.
The Logistic Model This DIGMath module allows you to investigate visually two different aspects of the continuous logistic model based on the logistic differential equation P' = aP - bP². (1) You can enter, via sliders, values for the two parameters a and b, as well as the initial population value P₀ and watch dynamically the effects on the resulting graph of the population, and also see the effects of changing any of these values. (2) You can also investigate visually the effects on the population of changes in the initial growth rate a and the maximum sustainable population (the limit to growth) L, along with the initial population value P₀, using sliders, and watching the dynamic effects on the graph of changing any of them.
Comparing the Discrete and Continuous Logistic Growth Models This DIGMath spreadsheet lets you investigate the differences between the solutions of the discrete and the continuous logistic growth models. The discrete logistic model based on the logistic difference equation P_n+1 = aP_n - bP_n² and the continuous model is based on the differential equation P' = aP - bP². You can enter, via sliders, values for the two parameters a and b, as well as the initial population value P₀and watch dynamically the effects on the resulting graph of the two population models, and also see the effects of changing any of these values. The spreadsheet also shows the graph of the difference between the two model functions, which gives a different, and often more insightful, view of how the two models compare.
The Slope, or Tangent, Field of a Differential Equation This DIGMath module allows you to investigate the slope field (also called the tangent field) associated with a differential equation of the form y' = f(x, y). You can enter your choice of function, the window with x from xMin to xMax and y from yMin to yMax over which the tangent lines are to extend. The program then draws the associated slope field and, as you vary the coordinates of the initial point (x₀, y₀), it also draws the graph of the solution, which you can see following the path determined by the tangent line segments.
*Euler's Method for Numerical Solutions to Differential Equations y' = f(x, y)* This DIGMath spreadsheet lets you investigate Euler's Method for generating numerical approximations to the solution of the differential equation y' = f(x, y), for any desired function of x and y, with any desired initial condition. The spreadsheet calculates and displays the approximation solutions corresponding to n = 4, 8, 16, ..., 128 steps across any desired interval, so you can observe the convergence of the successive approximations toward a smooth curve.
*Euler's Method for Numerical Solutions to Differential Equations y' = f(x)* This DIGMath spreadsheet lets you investigate Euler's Method for generating numerical approximations to the solution of the differential equation y' = f(x), for any desired function of x (but not y) with any desired initial condition. The spreadsheet calculates and displays the approximation solutions corresponding to n = 4, 8, 16, ..., 128 steps across any desired interval, so you can observe the convergence of the successive approximations toward a smooth curve.
Integration via Trig Substitutions This DIGMath spreadsheets lets you investigate the process involved in integration via trig substitutions. You can consider either substitutions of the form x = a/b sin θ or x = a/b tan θ. In either case, you can enter the values of the parameters a and b corresponding, respectively, to the coefficients of a² - b²x² or a² + b²x². The program draws the graph of the original function on any desired interval of x-values, the graph of the area function on the same interval, and the graph of the transformed function in terms of the angle q on the equivalent interval of q -values. You can trace along all three curves simultaneously to see that the area under the transformed graph is always precisely the same as the area under the original graph.
Integration by Parts This DIGMath spreadsheets lets you investigate graphically the process involved in integration by parts. You can consider three different forms for the integrand: x^p e ^cx , x^p sin (cx), and x^p cos (cx). In each case, you can enter the values of the parameters p and c. The program draws (1) the graph of the original function on any desired interval of x-values, (2) the graph of the area function on the same interval, (3) the graph of the function y = uv, (4) the graph of the integral of v du, and (5) the graph of the difference between uv and the integral of v du. You can trace along all five curves simultaneously to see that the area under the final graph is always precisely the same as the area under the original graph.
Integration via the z-Substitution This DIGMath module lets you investigate graphically the process involved in integration by using the z-substitution z = tan (x/2), which is used to integrate rational functions of sine and cosine. For any choice of the three parameters a, b and c in the function 1/(a + b sin x + c cos x), the spreadsheet shows the result of the substitution and displays the graphs of the original function with the associated area highlighted on any desired interval, the graph of the area function, and the graph of the transformed function in terms of z with the area highlighted on the resulting transformed interval. In this way, it is evident that, as you trace along the various curves, the area swept out under the original and the transformed curves are identical.
Partial Fraction Decompositions This DIGMath spreadsheet lets you investigate graphically the partial fraction decomposition of a rational function. There are three cases considered: (a) rational functions where the denominator consists of the product of two different linear terms; (2) rational functions where the denominator consists of the product of a linear function and an irreducible quadratic term; and (3) rational functions where the denominator consists of the product of a repeated (double) linear factor and a different linear factor.
Universal Law of Gravitation This DIGMath spreadsheet lets you investigate the Universal Law of Gravitation that says that the gravitational force on an object is proportional to the product of the masses and inversely proportional to the square of the distance between them. You can select the relative masses of the two objects -- say, planets -- and select the proportion of the distance between them for a spacecraft travelling from one to the other.
Modeling a Spring This DIGMath module lets you investigate the behavior of a bob attached to a vertical spring. There are two options -- no damping where the motion depends only on the mass of the bob, the initial displacement, and the spring constants or damping where the motion also depends on the viscous resistance coefficient. You can experiment with the effects of the coefficients in the case of simple harmonic motion (no damping) or the special cases of underdamping and overdamping when the resistance force is included.
Modeling a Pendulum: Simple Harmonic Motion This DIGMath module lets you investigate the behavior of pendulum, which consists of a bob attached to a relatively long string. Typically, it assumed that there are no forces to slow down the movement of the bob (called no damping) when the bob is released from some initial displacement, so theoretically it continues to oscillate back and forth indefinitely (known as Simple Harmonic Motion). The spreadsheet allows you to enter the length of the string and the initial vertical displacement of the bob and shows both the movement and the path of the bob over time.
Series vs. Sequences This DIGMath module lets you investigate the meaning of a sequence compared to that of a series. You an enter the expression for any desired sequence, a_k, in terms of k. You can select the number of points you want displayed. The spreadsheet then draws that number of the points in one chart and simultaneously draws the associated graph showing the sum of the values of those terms from the sequence.
A Bouncing Ball This DIGMath module lets you investigate the mathematics behind a bouncing ball. You can work in either the English or the metric system. You input the initial height from which a ball is dropped and the percentage of the velocity that is lost on each bounce. The spreadsheet draws the graphs of the height of the ball as a function of time, the velocity of the ball as a function of time, and the function giving the total distance traversed by the ball from the instant it is dropped to any time thereafter.
Visualizing l'Hopital's Rule: 0/0 at x = aThis DIGMath module lets you investigate l'Hopital's Rule both graphically and numerically for the limit of the ratio of two functions that leads to the indeterminate form 0/0. You can provide any two functions f and g you want that are both zero at a point x = a. The spreadsheet creates the graphs of both f/g and f'/g' and allows you to trace along both curves. It also provides the numerical values as you trace, particularly as you approach the limiting point a. It also shows the graphs of the two functions f and g together, as well as the graphs of the two derivative functions f' and g'.
Visualizing l'Hopital's Rule: ∞/∞ at x = aThis DIGMath module lets you investigate l'Hopital's Rule both graphically and numerically for the limit of the ratio of two functions that leads to the indeterminate form ∞/∞ as x approaches a finite point x = a. You can provide any two functions f and g you want that both become infinite as x approaches a point x = a. The spreadsheet creates the graphs of both f/g and f'/g' and allows you to trace along both curves. It also provides the numerical values as you trace, particularly as you approach the limiting point a. It also shows the graphs of the two functions f and g together, as well as the graphs of the two derivative functions f' and g'.
Visualizing l'Hopital's Rule: 0/0 as x approaches ∞ This DIGMath module lets you investigate l'Hopital's Rule both graphically and numerically for the limit of the ratio of two functions that leads to the indeterminate form 0/0 as x approaches infinity. You can provide any two functions f and g you want that both approach zero as x becomes infinite. The spreadsheet creates the graphs of both f/g and f'/g' and allows you to trace along both curves. It also provides the numerical values as you trace, particularly as x increases toward infinity. It also shows the graphs of the two functions f and g together, as well as the graphs of the two derivative functions f' and g'.
Visualizing l'Hopital's Rule: 0/0 as x approaches ∞ This DIGMath module lets you investigate l'Hopital's Rule both graphically and numerically for the limit of the ratio of two functions that leads to the indeterminate form 0/0 as x approaches infinity. You can provide any two functions f and g you want that both approach zero as x becomes infinite. The spreadsheet creates the graphs of both f/g and f'/g' and allows you to trace along both curves. It also provides the numerical values as you trace, particularly as x increases toward infinity. It also shows the graphs of the two functions f and g together, as well as the graphs of the two derivative functions f' and g'.
Taylor Approximations to the Exponential Function This DIGMath spreadsheet lets you investigate ideas on building polynomial approximations to the exponential function on any desired interval. Individual pages let you build linear, quadratic, cubic, quartic, and quintic polynomials by entering values for the coefficients via sliders and judging how well the resulting function fits the exponential curve graphically and numerically by the values of the greatest deviation and the sum of the squares of the deviations.
Taylor Approximations to the Sine Function This DIGMath spreadsheet lets you investigate ideas on building polynomial approximations to the sine function on any desired interval. Individual pages let you build linear, quadratic, cubic, quartic, and quintic polynomials by entering values for the coefficients via sliders and judging how well the resulting function fits the sine curve graphically and numerically by the values of the greatest deviation and the sum of the squares of the deviations.
Taylor Approximations to the Cosine Function This DIGMath spreadsheet lets you investigate ideas on building polynomial approximations to the cosine function on any desired interval. Individual pages let you build linear, quadratic, cubic, quartic, quintic, and 6th degree polynomials by entering values for the coefficients via sliders and judging how well the resulting function fits the cosine curve graphically and numerically by the values of the greatest deviation and the sum of the squares of the deviations.
Taylor Approximations to the Natural Logarithm Function This DIGMath spreadsheet lets you investigate ideas on building polynomial approximations to the natural logarithm function on any desired interval within (0, 2). Individual pages let you build linear, quadratic, cubic, quartic, and quintic polynomials by entering values for the coefficients via sliders and judging how well the resulting function fits the natural logarithm curve graphically and numerically by the values of the greatest deviation and the sum of the squares of the deviations.
Taylor Polynomial Approximations This DIGMath spreadsheet allows you to investigate the Taylor polynomial approximations to the four most common transcendental functions: the exponential function, the sine function, the cosine function, and the natural logarithm function. In each case, you can enter any desired interval and select which polynomial approximations you want to see displayed along with the function. For instance, with the exponential function, you can select any or all of the linear through the fifth degree polynomials; with the sine function, you can select any or all of the polynomials of odd degree up to the seventh degree.
Taylor Polynomials for Any Function This DIGMath spreadsheet allows you to investigate the Taylor polynomial approximations to any desired functions. You need to enter the formula for the function, the center point for the polynomials, and the desired interval. You can select which polynomial approximations (linear, quadratic, ..., sixth degree) you want to see displayed along with the function. You can also trace along the curves and see the various numerical approximations for each of the active curves.
Taylor Error for Exponential Functions This DIGMath program lets you investigate the error function corresponding to the Taylor polynomial approximation to the exponential function. You can enter any desired interval of values for x and choose to examine the error associated with the linear, quadratic, cubic, ..., quintic fits both graphically and numerically.
Creating Polynomial Approximations to the Exponential Function This DIGMath spreadsheet allows you try to create the best possible polynomial approximations to the exponential function centered at x = 0 on any desired interval. You have the choice of linear, quadratic, cubic, quartic, and quintic polynomials. For each, you can enter and vary the various coefficients using sliders and see the results of that polynomial versus the exponential curve graphically, as well as numerical measures such as the largest deviation and the sum of the squares of the deviations on the interval selected, so that you can decide on which coefficients produce the best fit and then continue the investigation with the next degree polynomial.
Creating Polynomial Approximations to the Logarithmic Function This DIGMath spreadsheet allows you try to create the best possible polynomial approximations to the logarithmic function about x = 1 on any desired interval. You have the choice of linear, quadratic, cubic, quartic, and quintic polynomials. For each, you can enter and vary the various coefficients using sliders and see the results of that polynomial versus the logarithmic curve graphically, as well as numerical measures such as the largest deviation and the sum of the squares of the deviations on the interval selected, so that you can decide on which coefficients produce the best fit and then continue the investigation with the next degree polynomial.
Creating Polynomial Approximations to the Sine Function This DIGMath spreadsheet allows you try to create the best possible polynomial approximations to the sine function centered at x = 0 on any desired interval. You have the choice of linear, quadratic, cubic, quartic, and quintic polynomials. For each, you can enter and vary the various coefficients using sliders and see the results of that polynomial versus the sine curve graphically, as well as numerical measures such as the largest deviation and the sum of the squares of the deviations on the interval selected, so that you can decide on which coefficients produce the best fit and then continue the investigation with the next degree polynomial.
Creating Polynomial Approximations to the Cosine Function This DIGMath spreadsheet allows you try to create the best possible polynomial approximations to the cosine function centered at x = 0 on any desired interval. You have the choice of linear, quadratic, cubic, ... sixth degree polynomials. For each, you can enter and vary the various coefficients using sliders and see the results of that polynomial versus the cosine curve graphically, as well as numerical measures such as the largest deviation and the sum of the squares of the deviations on the interval selected, so that you can decide on which coefficients produce the best fit and then continue the investigation with the next degree polynomial.
Taylor Polynomial Errors for the Exponential Function This DIGMath spreadsheet allows you try to investigate the best possible polynomial approximations to the exponential function centered at x = 0 on any desired interval. You have the choice of linear, quadratic, cubic, quartic, and quintic polynomials. For each, you can enter and vary the various coefficients using sliders and see the results of that polynomial versus the exponential curve graphically as well as the Error Function -- the difference between the function and the approximating polynomial. The spreadsheet also displays numerical measures such as the largest deviation and the sum of the squares of the deviations on the interval selected, so that you can decide on which coefficients produce smallest error and hence is the best fit and then continue the investigation with the next degree polynomial.
Taylor Polynomials vs. the Center Point This DIGMath module lets you investigate the effects of changing the center point x₀ at which Taylor polynomial approximations are based on the quadratic polynomial so created. The spreadsheet lets you explore the effects on both the cosine function (when the center is other than x = 0) or the natural logarithm function (when the center is other than x = 1).
The Indeterminate Form 0/0 and Taylor Approximations This DIGMath module lets you investigate the indeterminate form 0/0 that arises in the limit as x approaches a of the ratio f(x)/g(x). l'Hopital's Rule is a way to find the value of this limit, but to understand where the limiting value comes from, it is better to look at the ratio of the corresponding Taylor approximations to f and g. You enter the two functions, the limit point a, the two Taylor polynomials and the spreadsheet produces a number of graphs, most importantly that of f/g and the ratio of the two Taylor approximations.
Power Series Approximations This DIGMath spreadsheet allows you to investigate the successive polynomial approximations to a power series Σ a_k (x - c)_^k based on the coefficients of the series and the center point. You can enter any desired interval and select which polynomial approximations you want to see displayed from the constant up through the quintic (fifth degree).
Hyperbolic Functions This DIGMath module lets you investigate the hyperbolic functions y = sinh x and y = cosh x graphically and numerically based on their definitions in terms of the exponential functions y = e^xand y = e^-x.
Fourier Series Approximations This DIGMath spreadsheet allows you to investigate the successive Fourier approximations to three common periodic functions -- the square wave, the triangle wave, and the sawtooth wave. In each case, you can select which of the first four Fourier approximations you want to see displayed and can turn them on or off via sliders to observe how the successive approximations relate to one another and how they begin to converge to the shape of the desired target wave.
Curvature Function This DIGMath module lets you investigate the curvature function associated with a function of the form y = f(x) on any desired interval. You can trace along the curves of the function and the curvature function to see the coordinates of the point and the value of the curvature at that point.
Osculating Circle and the Radius of Curvature This DIGMath spreadsheet lets you explore the notion of the osculating circle -- the circle that is tangent to a function y = f(x) at any given point and whose radius is equal to the radius of curvature of the function at that point. It draws the graph of the function and the associated osculating circle as you trace along the curve in the chart on the left. Simultaneously, it shows the graph of the function representing the radius of curvature as a function of x and the corresponding tracing point in the chart on the right.
Graphs in Polar Coordinates This DIGMath spreadsheet draws the graph of any function r = f(Q) in polar coordinates on any interval of angles in radians from Q = a to Q = b. You can use a slider to trace out a moving point along the curve.
Intersection of Polar Curves This DIGMath spreadsheet lets you locate the points of intersection of two curves in polar coordinates. You enter both desired functions in terms of Q and an interval of Q-values from a to b. You can select a moving point along each curve using a slider to find the points where the curves apparently intersect and see whether or not the two curves have the same pairs of coordinates.
Slopes of Polar Curves This DIGMath spreadsheet lets you experiment with the slope of a curve in polar coordinates at any point along the curve. You enter any desired polar function in terms of Q on any desired interval and use a slider to move a tracing point along the curve. The program calculates and draws the associated tangent line.
Secant Lines to a Polar Curve This DIGMath module lets you investigate the convergence of a series of secant lines to the tangent line to a polar curve in terms of Q. You enter the desired polar function in terms of Q and the desired interval. The spreadsheet draws the curve, along with four secant lines and the tangent line and displays the slopes of the five lines. A slider allows you to trace along the curve and watch the way that the secant lines and the tangent line move accordingly.
Investigating Rose Curves in Polar Coordinates This DIGMath module lets you explore the so-called rose curves in polar coordinates given by r = a sin (nQ) or r = a cos (nQ) (on separate pages of the spreadsheet). You can select the values of a and n using sliders and the program draws the associated curve and allows you to trace along the curve. When n is an odd integer, the number of petals is clearly equal to n and when n is even, the number of petals is 2n. When n is not an integer, it is easy to watch how one configuration morphs into the other.
Investigating Cardioids in Polar Coordinates This DIGMath module lets you explore the cardioid and related curves in polar coordinates given by r = a sin (Q) + b or r = a cos (Q) + b (on separate pages of the spreadsheet). You select the values of a and b using sliders and the program draws the associated curve and allows you to trace along it. The values allowed provide you the opportunity to see what happens when a and b do not form a cardioid, so you have the opportunity to watch how one shape morphs into another as you change either a or b.
Investigating Limacons in Polar Coordinates This DIGMath module lets you explore the limacon and related curves in polar coordinates given by r = a sin (Q) + b or r = a cos (Q) + b (on separate pages of the spreadsheet). You select the values of a and b using sliders and the program draws the associated curve (either a limacon without a loop or a limacon with a loop, as well as other related shapes) and allows you to trace along it. The values allowed provide you the opportunity to see what happens when a and b do not form a limacon, so you have the opportunity to watch how one shape morphs into another as you change either a or b.
Investigating Lemniscates in Polar Coordinates This DIGMath module lets you explore the lemniscate curve given by r² = a² sin (2Q) or r² = a² cos (2Q), where a≠0 (on separate pages of the spreadsheet). You select the value of a using a slider and the program draws the associated curve, which is a figure-8 shape.
Taylor Approximations to Polar Curves This DIGMath spreadsheet lets you investigate how well Taylor polynomial approximations in terms of the variable Q approximate a polar curve r = f(Q), also in terms of Q.
Curvature of Polar Curves This DIGMath module lets you investigate the curvature function associated with any polar curve in terms of Q. You enter the desired polar function in terms ofQand the desired interval. The spreadsheet draws the curve and the graph of the curvature function. A slider allows you to trace along the curve and look for points where the curvature is maximal or minimal.
Approximating Polar Curves with Newton Interpolation This DIGMath module lets you investigate how any polar coordinate curve r = f (θ ) on any interval [α, β] can be approximated using Newton’s forward interpolation polynomials. You enter your choice for the function in terms of the variable Q (instead of θ ) and the desired interval using a slider. You also have the choice of the degree n, between 1 and 6. The spreadsheet draws the polar curve on the desired interval and superimposes the associated Newton interpolating polynomial. You can see how well the corresponding polynomial attempts to match the curve. As you change the degree, the smaller the interval, the better the fit, usually. You can also trace along both curves and see the corresponding coordinates of both displayed.
Approximating Polar Curves with Lagrange Interpolation This DIGMath module lets you investigate how any polar coordinate curve r = f (θ ) on any interval [α, β] can be approximated using the Lagrange interpolation polynomials. You enter your choice for the function in terms of the variable Q (instead of θ ) and the desired interval using a slider. You also have the choice of the degree n, between 1 and 6. The spreadsheet draws the polar curve on the desired interval and superimposes the associated Lagrange interpolating polynomial. You can see how well the corresponding polynomial attempts to match the curve. As you change the degree, you can see the effect on the approximation – the higher the degree, usually the better the fit. Also, the smaller the interval, the better the fit, usually. You can also trace along both curves and see the corresponding coordinates of both displayed.
Graphs of Parametric Functions This DIGMath module lets you explore the graphs of parametric functions of the form x = f(t) and y = g(t) on any desired interval. You can trace along the curve using a moving point and see the coordinates of that point.
Slope of a Parametric Curve This DIGMath spreadsheet lets you investigate the slope of the tangent line at any point along a parametric curve of the form x = f(t), y = g(t).
Tangent and Normal Vectors to a Parametric Curve This DIGMath spreadsheet lets you investigate the unit tangent and normal vectors at any point along a parametric curve of the form x = f(t), y = g(t).
Length of the Tangent Vector to a Parametric Curve This DIGMath spreadsheet lets you investigate the length of the tangent vector at any point along a parametric curve of the form x = f(t), y = g(t).
Hypocycloids A hypocycloid is the curve traced out when a fixed point on a small circle of radius r rolls around the inside rim of a larger circle of radius R. The path traced out by that point is called the hypocycloid and is represented by a pair of parametric functions. This DIGMath module graphs the hypocycloid based on your choice of the two radii r and R.
Epicycloids A epicycloid is the curve traced out when a fixed point on a small circle of radius r rolls around the outside rim of a larger circle of radius R. The path traced out by that point is called the epicycloid and is represented by a pair of parametric functions. This DIGMath module graphs the epicycloid based on your choice of the two radii r and R.
Taylor Polynomial Approximations to Parametric Functions This DIGMath spreadsheet allows you to investigate Taylor polynomial approximations to a function given in parametric form: x = f(t) and y = g(t) on any desired interval. You have to enter the two functions f and g in terms of the parameter t, as well as the expressions for the desired Taylor approximations x = F(t) and y = G(t) of any degree you like to each. The spreadsheet then draws the graphs of the two curves, so you can compare how well the approximation matches and use a slider to trace around the original parametric curve. The spreadsheet also displays the coordinates of the points on both curves as you trace around.
Curvature of Parametric Functions This DIGMath module lets you investigate the curvature function associated with any pair of parametric functions x = f(t) and y = g(t). You enter the desired parametric functions in terms of Q and the desired interval. The spreadsheet draws the curve and the graph of the curvature function. A slider allows you to trace along the curve and look for points where the curvature is maximal or minimal.
Approximating Parametric Functions with Newton InterpolationThis DIGMath module lets you investigate how any curve given as a pair of parametric functions x = f (t ), y = g(t) on any interval [a, b] can be approximated using Newton’s forward interpolation polynomials. You enter your choice for the two functions in terms of the variable t and the desired interval using a slider. You also have the choice of the degree n, between 1 and 6, for the polynomial. The spreadsheet draws the parametric curve on the desired interval and superimposes the associated Newton interpolating polynomial. You can see how well the corresponding polynomial attempts to match the curve. As you change the degree, you can see the effect on the approximation – the higher the degree, usually the better the fit. Also, the smaller the interval, the better the fit, usually. You can also trace along both curves and see the corresponding coordinates of both displayed.
Approximating Parametric Functions with Lagrange InterpolationThis DIGMath module lets you investigate how any curve given as a pair of parametric functions x = f (t), y = g(t) on any interval [a, b] can be approximated using Lagrange interpolation polynomials. You enter your choice for the two functions in terms of the variable t and the desired interval using a slider. You also have the choice of the degree n, between 1 and 6, for the polynomial. The spreadsheet draws the parametric curve on the desired interval and superimposes the associated Lagrange interpolating polynomial. You can see how well the corresponding polynomial attempts to match the curve As you change the degree, you can see the effect on the approximation – the higher the degree, usually the better the fit. Also, the smaller the interval, the better the fit, usually. You can also trace along both curves and see the corresponding coordinates of both displayed
Linear Functions in Bi-angular Coordinates This DIGMath spreadsheet lets you investigate the graphs of linear functions in bi-angular coordinates, which are based on locating points in the plane in terms of two angles, θ and φ, at two points, the poles. The linear function takes the form φ =mθ + b and some very surprising shapes result, particularly as you use the sliders to vary the parameters.
Functions in Bi-angular Coordinates This DIGMath spreadsheet lets you investigate the graphs of any function φ = f(θ) in bi-angular coordinates, which are based on locating points in the plane in terms of two angles, θ and φ, at two points, the poles.
The Curvature Function This DIGMath module lets you investigate the curvature function associated with a function of the form y = f(x) on any desired interval. You can trace along the curves of the function and the curvature function to see the coordinates of the point and the value of the curvature at that point.
The Osculating Circle and the Radius of Curvature This DIGMath spreadsheet lets you This DIGMath spreadsheet lets you explore the notion of the osculating circle -- the circle that is tangent to a function y = f(x) at any given point and whose radius is equal to the radius of curvature of the function at that point. It draws the graph of the function and the associated osculating circle as you trace along the curve in the chart on the left. Simultaneously, it shows the graph of the function representing the radius of curvature as a function of x and the corresponding tracing point in the chart on the right.
Surface Plot This DIGMath module lets you produce the graph (a surface plot) of a function of two variables, z = f(x, y) defined over any rectangular domain, x between xMin and xMax and y between yMin and yMax. You are able to rotate and make other changes to the view from within Excel.
Contour Plots This DIGMath module produces the contour plot of a function of two variables, z = f(x, y) defined over any rectangular domain with x between xMin and xMax and y between yMin and yMax. You are able to rotate and make other changes to the view from within Excel.
Contour Plot of the Area Function for a Rectangle This DIGMath module produces the contour plot of the area function A = x y for a rectangle, which is a function of two variables. It draws three contours automatically and lets you select a fourth contour value via a slider, so you can see the effects of changing that value, as well as tracing along all four contours.
Contour Plot of the Area Function for an Ellipse This DIGMath module produces the contour plot of the area function A = π a b for an ellipse, which is a function of two variables -- the semi-major and the semi-minor axes a and b. It draws three contours automatically and lets you select a fourth contour value for A via a slider, so you can see the effects of changing that value, as well as tracing along all four contours.
Contour Plot of the Volume Function for a Right-Circular Cylinder This DIGMath spreadsheet produces the contour plot of the volume function V = π r² h for a right-circular cylinder of radius r and height h, which is a function of two variables. It draws three contours automatically and lets you select a fourth contour value for V via a slider, so you can see the effects of changing that value, as well as tracing along all four contours. You can investigate either the case where r is in terms of h or h is in terms of r.
Contour Plot of the Volume Function for a Right-Circular Cone This DIGMath spreadsheet produces the contour plot of the volume function V = 1/3 π r² h for a right-circular cone having base radius r and height h, which is a function of two variables. It draws three contours automatically and lets you select a fourth contour value for V via a slider, so you can see the effects of changing that value, as well as tracing along all four contours. You can investigate either the case where r is in terms of h or h is in terms of r.
Curves in Space This DIGMath spreadsheet creates a representation of a curve in space based on the three parametric equations x = f(t), y = g(t), and z = h(t) on any desired interval for t from t = a to t = b.
Curves in Space with Tangent and Normal Vectors This DIGMath spreadsheet creates a representation of a curve in space based on the three parametric equations x = f(t), y = g(t), and z = h(t) on any desired interval for t from t = a to t = b. It also shows, both graphically and numerically, the unit tangent vector and the unit normal vector to the curve at any desired point as you trace along the curve.

Dynamic Investigatory Graphical Displays of Math: DIGMath Graphical EXPLORATIONS for Numerical Analysis using Excel

As mentioned above, Excel must be set to accept macros by selecting "Enable Content".

The following are the DIGMath explorations that are currently completed and ready for use. (Many others are under development.) If you have any suggestions for improvements or for new topics, please pass them on also.

DIGMath Spreadsheets for Numerical Analysis	Category	Filter
The Bisection Method This DIGMath module illustrates the convergence of the Bisection Method for finding the real zeros of any function y = f(x). To use it, you enter the desired function and an initial interval from a to b over which the function has at least one zero and the program performs successive iterations of the method, displaying the results both graphically and in a table.	Root Finding Methods	Root Finding Methods
The Secant Method This DIGMath module illustrates the convergence of the Secant Method for finding the real zeros of any function y = f(x). To use it, you enter the desired function and an initial estimate x0 of any real zero and the program performs successive iterations of the method, displaying the results both graphically and in a table.	Root Finding Methods	Root Finding Methods
Newton's Method This DIGMath module lets you investigate Newton's Method for finding the roots of a function f, both numerically and graphically. For any desired function, any desired starting value, and any desired number of iterations, you can see the set of iterated approximations in a table and the graph of the process, either in a fixed window of your choice (although the sequence of approximations may leave the window) or in a variable window that follows the sequence of iterations.	Root Finding Methods	Root Finding Methods
Bairstow's Method This DIGMath spreadsheet lets you investigate Bairstow's Method for approximating the complex roots of a polynomial by creating a sequence of quadratic functions that converges to an irreducible quadratic factor of the polynomial. You can use any polynomial up to 8th degree and any choice of initial estimate for the quadratic factor. The spreadsheet draws the first five quadratic approximations to illustrate the convergence both graphically and numerically.	Root Finding Methods	Root Finding Methods
Taylor Polynomial Approximations This DIGMath spreadsheet allows you to investigate the Taylor polynomial approximations to the four most common transcendental function: the exponential function, the sine function, the cosine function, and the natural logarithm function. In each case, you can enter any desired interval and select which polynomial approximations you want to see displayed along with the function. For instance, with the exponential function, you can select any or all of the linear through the fifth degree polynomials; with the sine function, you can select any or all of the polynomials of odd degree up to the seventh degree.	Approximating a Function	Approximating a Function
Power Series Approximations This DIGMath spreadsheet allows you to investigate the successive polynomial approximations to a power series Σ a_k (x - c)^k based on the coefficients of the series and the center point. You can enter any desired interval and select which polynomial approximations you want to see displayed from the constant up through the quintic (fifth degree).	Approximating a Function	Approximating a Function
Fourier Series Approximations This DIGMath spreadsheet allows you to investigate the successive Fourier approximations to three common periodic functions -- the square wave, the triangle wave, and the sawtooth wave. In each case, you can select which of the first four Fourier approximations you want to see displayed and can turn them on or off via sliders to observe how the successive approximations relate to one another and how they begin to converge to the shape of the desired target wave.	Approximating a Function	Approximating a Function
Lagrange Interpolation Formula This DIGMath module lets you investigate the Lagrange Interpolating Polynomial based on any set of up to 8 data points. The spreadsheet draws the graph of the Lagrange polynomial as it passes through the interpolating points. It also displays the equation of the polynomial using the Lagrange formula for the linear, quadratic, and cubic cases.	Approximating a Function	Approximating a Function
Building Lagrange Interpolation Polynomials This DIGMath spreadsheet lets you investigate the way that the Lagrange Interpolating Polynomial is created as a combination of the individual component polynomials. You can look at the way that the linear, the quadratic, and the cubic Lagrange polynomials are built based on the sum of two linear, three quadratic, or four cubic polynomials, respectively. In each case, the individual components are drawn along with the Lagrange polynomial based on the data points entered and you can then examine the relationships between the roots of the polynomial components, the interpolating points, and the associated function values.	Approximating a Function	Approximating a Function
Newton Interpolation Formula This DIGMath module lets you investigate the Newton Forward Interpolating Polynomial based on any set of up to 8 uniformly spaced data points. The spreadsheet draws the graph of each of the successive linear, quadratic, ... Newton polynomials as they pass through the various interpolating points.	Approximating a Function	Approximating a Function
Approximating the Sine using Lagrange Interpolation This DIGMath spreadsheet lets you investigate how the sine function can be approximated on the interval [0, π/4] using Lagrange Interpolating Polynomials based on either three or four interpolating points given either in radians or degrees. The spreadsheet draws the graph of sine and the interpolating polynomial as well as the graph of the error function P(x) - sin x so you can see how well or poorly the polynomial fits the function as you change the interpolating points.	Approximating a Function	Approximating a Function
Approximating the Sine using Newton Interpolation This DIGMath spreadsheet lets you investigate how the sine function can be approximated on the interval [0, π/4] using Newton Interpolating Polynomials based on either three or four uniformly spaced interpolating points given either in radians or degrees. The spreadsheet draws the graph of sine and the interpolating polynomial as well as the graph of the error function P(x) - sin x so you can see how well or poorly the polynomial fits the function as you change the interpolating points.	Approximating a Function	Approximating a Function
Approximating the Cosine using Lagrange Interpolation This DIGMath spreadsheet lets you investigate how the cosine function can be approximated on the interval [0, π/4] using Lagrange Interpolating Polynomials based on either three or four interpolating points given either in radians or degrees. The spreadsheet draws the graph of cosine and the interpolating polynomial as well as the graph of the error function P(x) - cos x so you can see how well or poorly the polynomial fits the function as you change the interpolating points.	Approximating a Function	Approximating a Function
Approximating the Cosine using Newton Interpolation This DIGMath spreadsheet lets you investigate how the cosine function can be approximated on the interval [0, π/4] using Newton Interpolating Polynomials based on either three or four uniformly spaced interpolating points given either in radians or degrees. The spreadsheet draws the graph of sine and the interpolating polynomial as well as the graph of the error function P(x) - cos x so you can see how well or poorly the polynomial fits the function as you change the interpolating points.	Approximating a Function	Approximating a Function
Approximating the Exponential Function using Lagrange Interpolation This DIGMath spreadsheet lets you investigate how the exponential function y = e^x can be approximated on the interval [0, 1] using Lagrange Interpolating Polynomials based on either three or four interpolating points. The spreadsheet draws the graph of exponential function and the interpolating polynomial as well as the graph of the error function P(x) - e^x so you can see how well or poorly the polynomial fits the function as you change the interpolating points.	Approximating a Function	Approximating a Function
Chebyshev Nodes This DIGMath module lets you investigate the so-called Chebyshev nodes that are often the best points to use to construct an interpolating polynomial based on a set of points. The Chebyshev nodes have the property that they provide the minimum value of the maximum error in the approximation at the interpolating points. The spreadsheet lets you enter your choice of either three or four interpolating points on the interval [-1, 1] and compares, both numerically and graphically, the maximum error to the corresponding maximum error using the Chebyshev nodes.	Approximating a Function	Approximating a Function
Cubic Splines This DIGMath spreadsheet lets you investigate the notion of cubic splines, a way to construct a smooth curve determined by a set of points in such a way that the curve is made up of a series of smoothly connected cubic curves. The spreadsheet has two components. In the first, the data points are grouped three at a time subject to the condition that the slope at the third point must be equal to the slope of the following cubic at the first point. In the second, the points are grouped two at a time subject to the two conditions that the slope and the value of the second derivative must agree at each of the overlapping points.	Approximating a Function	Approximating a Function
Tangent Line to a Curve This DIGMath spreadsheet allows you to investigate how the value of the derivative of a function at a point can be approximated by using the slope of the tangent line to the graph of any desired function of the form y = f(x) on any desired interval a to b. You control the point of tangency using a slider and watch the effects on the resulting tangent line to the curve, and therefore the derivative, as the point changes. The length of the tangent line also changes to reflect the size of the slope or, equivalently, the value of the derivative of the function at the point.	Approximating the Derivative	Approximating the Derivative
Newton Difference Quotients This DIGMath module lets you investigate the notion of the Newton difference quotient as an approximation to the slope of the tangent line to a curve at a point. You can use any function, any interval, any desired step-size, and any point of tangency. The spreadsheet displays graphically the graph, the associated secant line and the triangle based on the point of tangency. If also displays numerically slope of the secant line compared to the slope of the tangent line, so you can see what happens to the agreement as the step-size is decreased or as the point of tangency moves along the curve.	Approximating the Derivative	Approximating the Derivative
Numerical Integration Methods This DIGMath spreadsheet allows you to investigate four different methods to approximate the value of a definite integral -- using left and right-hand Riemann Sums, using the Trapezoid Rule, the MidPoint Rule, and Simpson's Rule for any function of the form y = f(x) on any desired interval [a, b]. You control the number of subdivisions for each method using a slider and the spreadsheet draws the graph of the function, draws the approximating subdivisions, and displays the associated approximation to the definite integral.	Numerical Integration	Numerical Integration
Monte Carlo Method for Definite Integrals This DIGMath spreadsheet allows you to investigate visually and numerically the use of Monte Carlo simulations for estimating the value of the definite integral of any function of the form y = f(x) that is non-negative on any desired interval [a, b]. You control the number of random points, between 500 and 2500, via a slider and the spreadsheet draws the graph of the function, plots the random points, and displays the number and percentage of them that fall under the curve, and uses that percentage to estimate the area of the region.	Numerical Integration	Numerical Integration
*Euler's Method for Numerical Solutions to Differential Equations y' = f(x, y)* This DIGMath spreadsheet lets you investigate Euler's Method for generating numerical approximations to the solution of the differential equation y' = f(x, y), for any desired function of x and y, with any desired initial condition. The spreadsheet calculates and displays the approximation solutions corresponding to n = 4, 8, 16, ..., 128 steps across any desired interval, so you can observe the convergence of the successive approximations toward a smooth curve.	Approximating Solutions of Differential Equations	Approximating Solutions of Differential Equations
*Euler's Method for Numerical Solutions to Differential Equations y' = f(x)* This DIGMath spreadsheet lets you investigate Euler's Method for generating numerical approximations to the solution of the differential equation y' = f(x), for any desired function of x (but not y) with any desired initial condition. The spreadsheet calculates and displays the approximation solutions corresponding to n = 4, 8, 16, ..., 128 steps across any desired interval, so you can observe the convergence of the successive approximations toward a smooth curve.	Approximating Solutions of Differential Equations	Approximating Solutions of Differential Equations
Picard’s Method for Numerical Solutions to the Differential Equation y’ = f(x, y) This DIGMath Module lets you investigate Picard’s Method for the differential equation y' = f(x, y) with any choice of f and any initial condition (x₀, y₀) and your choice of how long the interval over which the display is shown. The spreadsheet approximates the first six successive approximations using numerical integration methods and plots your choice of which of the six curves you want to see. This allows you to see how the successive approximations converge toward a single curve, the unique solution of the differential equation through the given initial point.	Approximating Solutions of Differential Equations	Approximating Solutions of Differential Equations
Second Order Homogeneous Differential Equations with Constant Coefficients This DIGMath module lets you investigate the behavior of the solutions of the second order homogeneous differential equation ay" + by' + cy = 0 for your choice of the coefficients a, b, and c and initial conditions y₀ and y'₀.	Approximating Solutions of Differential Equations	Approximating Solutions of Differential Equations

Dynamic Investigatory Graphical Displays of Math: DIGMath Graphical Investigations for DIFFERENTIAL EQUATIONS using Excel

As mentioned above, Excel must be set to accept macros by selecting "Enable Content".

DIGMath Graphical Investigations for DIFFERENTIAL EQUATIONS using Excel

The Slope, or Tangent, Field of a Differential Equation This DIGMath module allows you to investigate the slope field (also called the tangent field) associated with a differential equation of the form y' = f(x, y). You can enter your choice of function, and the window with x from xMin to xMax and y from yMin to yMax over which the tangent lines are to extend. The program then draws the associated slope field. Also, as you vary the coordinates of the initial point (x₀, y₀), it draws the graph of the corresponding solution, which you can see following the path determined by the tangent line segments.

Euler's Method for Numerical Solutions to Differential Equations y' = f(x, y) This DIGMath spreadsheet lets you investigate Euler's Method for generating numerical approximations to the solution of the differential equation y' = f(x, y), for any desired function of x and y, with any desired initial condition x₀ and y₀. The spreadsheet calculates and displays the approximation solutions corresponding to n = 4, 8, 16, ..., 128 steps across any desired interval, so you can observe the convergence of the successive approximations toward a smooth curve.

Picard’s Method for Numerical Solutions to Differential Equation y’ = f(x, y) This DIGMath Module lets you investigate the successive approximations to the solution of the differential equation y’ = f(x, y) subject to the initial conditions x₀ and y₀ generated by Picard’s iterative method. You can select, by checking the appropriate boxes, which of the successive approximations (n = 1, 2, …, 6) you want displayed to better let you compare the convergence of the approximations. You can also select how far from x₀ you want the approximations to extend.

First Order Homogeneous Linear Differential Equations This DIGMath module lets you investigate the behavior of the solutions of the first order linear homogeneous differential equation y' - ay = 0 with your choice of the coefficient a, and the initial condition y₀ at time t = 0. The spreadsheet allows you to explore the solution, both graphically and in closed form.

First Order Nonhomogeneous Differential Equations This DIGMath spreadsheet lets you investigate the behavior of the solutions of the first order linear non-homogeneous differential equation y' - ay = f(t), for a variety of basic families of functions -- exponential, sine, cosine, linear, and quadratic. Using sliders, you can select the coefficient a in the differential equation and the parameters in the function you chose, as well as the initial condition y₀ at time t = 0. The spreadsheet displays the closed form expression for the solution, as well as the graph of the solution.

The Logistic Model This DIGMath module allows you to investigate visually two different aspects of the continuous logistic model based on the logistic differential equation P' = aP - bP². (1) You can enter, via sliders, values for the two parameters a and b, as well as the initial population value P₀ and watch dynamically the effects on the resulting graph of the population, and also see the effects of changing any of these values. (2) You can also investigate visually the effects on the population of changes in the initial growth rate a and the maximum sustainable population (the limit to growth) L, along with the initial population value P₀, using sliders, and watch the dynamic effects on the graph of changing any of them.

The Predator-Prey Model This DIGMath spreadsheet lets you explore the Predator-Prey model that describes the interaction between two populations, one the predator (say, wolves) and the other the prey (say, rabbits). The model involves two interrelated differential equations, one for R ' and the other for W '. You can vary each of the four coefficients in the differential equations, as well as the initial values for the two populations to see the patterns over time. The spreadsheet also displays the phase plane portrait , which shows how one population changes with respect to the other population.

Projectile Motion This DIGMath spreadsheet allows you to investigate the path of a projectile launched from ground level with initial velocity and initial angle of inclination a . You control the values for the initial velocity and the angle via sliders and the spreadsheet draws the path of the projectile, allows you to trace along the path via another slider, and displays the time t, the coordinates of the tracing point, and the vertical velocity at each point. One page uses the English system of measurements in feet and seconds and another page does the comparable displays in the metric system with centimeters and seconds. Among the suggested investigations is one involving finding the angle α for which the range of the projectile is maximum.

A Falling Object with Air Resistance This DIGMath spreadsheet lets you investigate the situation where a falling object is subjected to air resistance, so the standard formulas for the height and velocity of the object as functions of time do not apply. Instead, you can investigate the two most widely used models with air resistance, one where the resistive force is proportional to the velocity v of the object and the other where the resistive force is proportional to the square of the velocity, v² . The spreadsheet lets you vary the initial height, the mass, and the resistive constant using sliders. It then displays the graph of the height as a function of time and the velocity of the object as a function of time.

Boundary Value Problems This DIGMath spreadsheet lets you explore the notion of a boundary value problem for the second order differential equation y" + ky = 0, whose solution is a sine curve. You can select either one page where you have the choice of an interval from t = 0 to an integer value or a second page where the interval extends from t = 0 to t = 2Π . In each case, you have the choice of the coefficient k and the boundary values. The program displays the graph of the solution and the formula for the closed-form particular solution.

Second Order Homogeneous Differential Equations with Constant Coefficients This DIGMath module lets you investigate the behavior of the solutions of the second order homogeneous differential equation ay" + by' + cy = 0 for your choice of the coefficients a, b, and c and initial conditions x₀, y₀, and y'₀.

Modeling a Spring This DIGMath module lets you investigate the behavior of a bob attached to a vertical spring. There are two options -- no damping where the motion depends only on the mass of the bob, the initial displacement, and the spring constants or damping where the motion also depends on the viscous resistance coefficient. You can experiment with the effects of the coefficients in the case of simple harmonic motion (no damping) or the special cases of underdamping and overdamping when the resistive force is included.

The RLC Circuit Model This DIGMath programs lets you investigate an RLC electric circuit, the most fundamental type of electrical circuit. It consists of three components, a resistor, whose resistance R is measured in ohms, an inductor, whose inductance L is measured in henries, and a capacitor, whose capacitance C is measured in farads. You can select the values for the inductance, the resistance, and the capacitance of the circuit, as well as the initial current. The spreadsheet then displays the graph of the current over time, which is the solution of the second order, linear homogeneous differential equation with constant coefficients L I" + R I' + (1/C) I = 0.

The Pendulum Model This DIGMath module provides a mathematical model for a pendulum – the motion of a bob hanging at the end of a string that oscillates back and forth repeatedly under the assumption that there is no air resistance that slows, or retards, the motion. You can select both the initial displacement angle q₀ and the length L of the string, as well as the length of time that you want the process continued. The spreadsheet displays an animation of the motion and a display of the height of the bob as a function of time.

Orthogonal Trajectories for the Differential Equation y' = f(x) This DIGMath spreadsheet lets you investigate the orthogonal trajectories associated with the solutions of the differential equation y ' = f ( x ). The orthogonal trajectories are the curves of solutions of the differential equation y ' = -1/ f ( x ), so that at each point the slope is the negative reciprocal of the slope of the solution to the original differential equation. As a result, at any point where an orthogonal trajectory crosses a solution curve, the tangent lines to the two curves will be perpendicular. You can enter your choice of the function, the initial condition This DIGMath spreadsheet lets you investigate the orthogonal trajectories associated with the solutions of the differential equation y ' = f ( x ). The orthogonal trajectories are the curves of solutions of the differential equation y ' = -1/ f ( x ), so that at each point the slope is the negative reciprocal of the slope of the solution to the original differential equation. As a result, at any point where an orthogonal trajectory crosses a solution curve, the tangent lines to the two curves will be perpendicular. You can enter your choice of the function, the initial condition x₀ and y₀ , the extent over which solutions are displayed, and the choice of height for an intermediate solution curve.

Orthogonal Trajectories for the Differential Equation y' = f(x, y) This DIGMath spreadsheet lets you investigate the orthogonal trajectories associated with the solutions of the differential equation y ' = f ( x, y ). The orthogonal trajectories are the curves of solutions of the differential equation y ' = -1/ f ( x, y ), so that at each point the slope is the negative reciprocal of the slope of the solution to the original differential equation. As a result, at any point where an orthogonal trajectory crosses a solution curve, the tangent lines to the two curves will be perpendicular. You can enter your choice of the function, the initial condition This DIGMath spreadsheet lets you investigate the orthogonal trajectories associated with the solutions of the differential equation y ' = f ( x, y). The orthogonal trajectories are the curves of solutions of the differential equation y ' = -1/ f ( x, y ), so that at each point the slope is the negative reciprocal of the slope of the solution to the original differential equation. As a result, at any point where an orthogonal trajectory crosses a solution curve, the tangent lines to the two curves will be perpendicular. You can enter your choice of the function, the initial condition x₀ and y₀ , the extent over which solutions are displayed, and the choice of height for an intermediate solution curve.

Dr. Sheldon Gordon

DIGMath Software Packages

Books

Articles (since 2000)

Dynamic Investigatory Graphical Displays of Math: DIGMath Graphical Simulations for Statistics and Probability using Excel

Dynamic Investigatory Graphical Displays of Math: DIGMath Graphical EXPLORATIONS for College Algebra and Precalculus using Excel

Dynamic Investigatory Graphical Displays of Math: DIGMath Graphical EXPLORATIONS for Calculus using Excel

Dynamic Investigatory Graphical Displays of Math: DIGMath Graphical EXPLORATIONS for Numerical Analysis using Excel

Dynamic Investigatory Graphical Displays of Math: DIGMath Graphical Investigations for DIFFERENTIAL EQUATIONS using Excel

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