# DIGMATH: Dynamic Investigatory Graphical Displays of Math: Graphical Explorations for Calculus in Excel

## Sheldon P. Gordon

Most of the following graphical explorations require the use of macros to operate.  In order to use these spreadsheets, Excel must be set to accept macros.  To change the security setting on macros:

With Excel 2007 and later:

1. When you open the spreadsheet, a new bar appears near the top of the window that says "Security Warning: Active content has been disabled".
2. Click on Options.
3. Click on "Enable this Content" and then click OK.

1. 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).
2. 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.
3. 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.
4. 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.
5. 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.
6. 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.
7. 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.
8. 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.
9. 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.
10. 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.
11. 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.
12. 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.
13. 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.
14. Derivative of the Exponential Function  This DIGMath spreadsheet allows you to investigate the derivative of exponential functions of the form y = bxand 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
15. 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 = xp 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.
16. 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.
17. 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.
18. 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.
19. 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.
20. 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.
21. 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.
22. 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).
23. 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).
24. 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.
25. Visualizing l'Hopital's Rule: 0/0 at x = a 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. 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'.
26. Visualizing l'Hopital's Rule: ∞/∞ at x = a 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 ∞/∞ 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'.
27. 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'.
28. Visualizing l'Hopital's Rule: ∞/∞ 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 ∞/∞ as x approaches infinity. You can provide any two functions f and g you want that both become infinite as x approaches infinity. 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'.
29. The 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.
30. 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.
31. 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.
32. 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.
33. 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.
34. 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.
35. 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.
36. 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.
37. 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.
38. 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.
39. 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.
40. 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.
41. 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.
42. 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.
43. The Conical Pile of Sand 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.
44. The Length of Shadow 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.
45. 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.
46. 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.
47. 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.
48. 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.
49. 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.
50. 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.
51. 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.
52. 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.
53. 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.
54. 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.
55. 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.
56. 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 Pn+1 = aPn - bPn2 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 Pand 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.
57. 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.
58. 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.
59. 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.
60. 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.
61. 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: xp e cx , xp sin (cx), and xp 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.
62. 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.
63. 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.
64. 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.
65. 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, ak, 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.
66. 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.
67. 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 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.
68. 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.
69. 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.
70. 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).
71. 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.
72. Power Series Approximations This DIGMath spreadsheet allows you to investigate the successive polynomial approximations to a power series Σ ak (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).
73. 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 = eand y = e-x.
74. 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.
75. 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.
76. 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.
77. 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.
78. 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
79. 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.
80. 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).
81. 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).
82. 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).
83. 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.
84. 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.
85. 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.
86. 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.
87. Osculating Circle 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.
88. 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.
89. 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.
90. 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.
91. 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.
92. 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.
93. 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.
94. 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.
95. 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.
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