Excel Graphical Simulations for Statistics and Probability

DIGMATH: Dynamic Investigatory Graphical Displays of Mathematics:

Graphical Explorations for Calculus Using Excel

Sheldon P. Gordon and Florence S. 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:

1. When you open any of the spreadsheets, a new bar appears near the top of the window that says something like: "Security Warning: Active content has been disabled", depending on the version of Excel you are using.

2. Click on Options.

3. Click on "Enable the Content" and then click OK.

The following are the 140 DIGMath explorations that are currently (January 2025 ) completed and ready for you to use in calculus. (Others are under development.) Please feel free to download (with Chrome or Edge, simply click on any of the links below; with Firefox, right click on any of the links and then select Save Link As ...). If you want all of them, click on the link: calculus.7z or you can send an e-mail to gordonsp@retiree.farmingdale.edu and we will try sending you a zip file with all of the files. If you have any problems downloading or running any of these Excel files, please contact us at gordonsp@retiree.farmingdale.edu or fgordon@nyit.edu for assistance. If you have any suggestions for improvements or for new topics, please pass them on also.

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) and 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 zooming in.
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 = bx and 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 = 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.
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 = ax² + 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 change 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 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: sin x This DIGMath module is the first in a series of 7 spreadsheets that lets you visualize and discover the chain rule for differentiating a composite function. This one starts with y = sin x to set the stage for the other spreadsheets by emphasizing the mathematical ideas and reasoning. It shows the graph of the function, the associated graph of its derivative based on the slopes of multiple tangent line segments (which clearly resembles the graph of y = cos x, a comparison of the graphs of the hypothesized derivative function and the actual derivative, and the graph of the error function to assess how close the approximate derivative function is to the actual derivative.
Discovering the Chain Rule: y = sin (2x) This DIGMath module is the second in a series of 7 spreadsheets that lets you visualize and discover the chain rule for differentiating a composite function. This one focuses on y = sin 2x. It displays the approximate derivative function, which has the shape of a cosine curve, but one with two full cycles between -p and p and oscillates vertically between roughly -2 and +2. It also shows a comparison of the graphs of the hypothesized derivative function and the actual derivative, and the graph of the error function to assess how close the approximate derivative function is to the actual derivative.
Discovering the Chain Rule: y = sin (3x) This DIGMath module is the third in a series of 7 spreadsheets that lets you visualize and discover the chain rule for differentiating a composite function. This one focuses on y = sin 3x. It displays the approximate derivative function, which has the shape of a cosine curve, but one with three full cycles between -p and p and oscillates vertically between roughly -3 and +3. It also shows a comparison of the graphs of the hypothesized derivative function and the actual derivative, and the graph of the error function to assess how close the approximate derivative function is to the actual derivative.
Discovering the Chain Rule: y = sin (4x) This DIGMath module is the fourth in a series of 7 spreadsheets that lets you visualize and discover the chain rule for differentiating a composite function. This one focuses on y = sin 4 x. It displays the approximate derivative function, which has the shape of a cosine curve, but one with four full cycles between -p and p and oscillates vertically between roughly -4 and +4. It also shows a comparison of the graphs of the hypothesized derivative function and the actual derivative, and the graph of the error function to assess how close the approximate derivative function is to the actual derivative.
Discovering the Chain Rule: y = sin (x²) This DIGMath module is the fifth in a series of 7 spreadsheets that lets you visualize and discover the chain rule for differentiating a composite function. This one focuses on y = sin x². It displays the approximate derivative function, which has sort of the shape of a cosine curve, but one with more and more cycles the further away from the origin in both directions and with oscillations that also keep growing in size. In fact, the graph of the approximate derivative shows that the successive peaks and valleys fall into a pair of linear patterns.
Discovering the Chain Rule: y = sin (x³) This DIGMath module is the sixth of the spreadsheets that let you visualize and discover the chain rule for differentiating a composite function. This one focuses on y = sin x³. It displays the approximate derivative function, which has sort of the shape of a cosine curve, but one with more and more cycles of greater and greater amplitude the further away from the origin in both directions. The graph of the approximate derivative shows that the successive peaks and valleys fall into a pair of non-linear patterns that are reminescent of a pair of parabolas.
Discovering the Chain Rule: y = sin (e^x) This DIGMath module is the last of the spreadsheets that let you visualize and discover the chain rule for differentiating a composite function. This one focuses on y = sin e^x. It displays the approximate derivative function, which has sort of the shape of a cosine curve, but one with more and more cycles of greater and greater amplitude the further to the right, but dies out quickly to the left. The graph of the approximate derivative shows that the successive peaks and valleys fall into a pair of non-linear patterns that are reminescent of a pair of exponential curves.
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 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.
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 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 sliders, 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 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 different 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 Rate 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 piecewise-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 watch 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 P₀ and 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 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.
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 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.
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 q or x = a/b tan q. 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^psin (cx), and x^pcos (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 associated 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 traveling 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 can 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 = 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 as x approaches a finite point x = a. 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: inf/inf 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'.
Visualizing l'Hopital's Rule: 0/0 as x approaches infinity 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: inf/inf as x approaches infinity 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'.
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 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.
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.
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 two numerical measures -- 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 S a_k (x - c)^k based on the coefficients of the series and the center point x = c . 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^x and y = e^-^x.
Taylor Approximations to Hyperbolic Functions This DIGMath module lets you investigate how the various Taylor approximations to the hyperbolic functions y = sinh x and y = cosh x compare to one another and how well they approximate both functions.
Fourier Series Approximations This DIGMath spreadsheet allows you to investigate the successive Fourierq) 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 (using Q instead of q) 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 Polar Curves 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.
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.
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.
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 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 Polar Curves with Newton Interpolation This DIGMath module lets you investigate how any polar coordinate curve r = f (q ) 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 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 (q ) 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 q ) 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 for the parameter t. 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 for 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 t 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 Interpolation This 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 Interpolation This 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 known as the poles. The linear function takes the form f =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.
Surface Plots 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 = p 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 = p 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 = 1/3p 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.

All of these files were developed under the support from a variety of grants from the National Science Foundation, to whom the author is very appreciative.