DIGMATH: Dynamic Investigatory Graphical Displays of Math:
Graphical Explorations for Numerical Analysis Using 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:

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.

 Some of Excel's special features -- sliders and check boxes (for data input) and Notes (suggestions for how to use the files and such) may not be functioning correctly with Windows 11. We are currently adapting all the files, though it may take several months to correct all of them. To avoid the difficulties with Windows 11 and the later versions of Excel, we are converting most to utilize an older version of Excel. A few still require the use of macros and are marked below with M. The following DIGMath explorations in numerical analysis are currently (March 2026) completed and ready for use. Please feel free to use any or all of these files (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, just click on the link: numericalanalysis.7z or you can send an e-mail to gordonsp@farmingdale.edu and we will try sending you a zip file with all of them. If you have any problems downloading or running any of these files, please contact us at gordonsp@retiree.farmingdale.edu for assistance. If you have any suggestions for improvements or for new topics, please pass them on also.

Root Finding Methods

1. 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.
2. 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.
3. The Regula Falsi (or False Position) Method  This DIGMath module illustrates the convergence of the Regula Falsi (or False Position) 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.
4. 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.
5. 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.
6. Bairstow's Method for Complex Roots This DIGMath spreadsheet lets you investigate Bairstow's Method for approximating the complex roots of a polynomial by creating a sequence of quadratic functions that converges to an irreducible quadratic factor of the polynomial. You can use any polynomial up to 8th degree and any choice of initial estimate for the quadratic factor. The spreadsheet draws the first five quadratic approximations to illustrate the convergence both graphically and numerically.

Approximating a Function

1. 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.   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).
2. Approximating Sinusoidal Functions This DIGMath spreadsheet lets you investigate dynamically the idea of approximating a sinusoidal function with a polynomial. You can choose to work with either the sine or the cosine function and can use an approximating polynomial up to sixth degree in the form y = a + (1/b)x + (1/c)x²+ (1/d)x³ + (1/e)x⁴ + (1/f)x⁵ + (1/g)x⁶, where the seven parameters are all integers. You use sliders to change each of the parameter values and the associated graph shows how well the corresponding polynomial fits the sinusoidal curve. The spreadsheet also shows the value for the sum of the squares to provide a numerical measure for the goodness of the fit. This investigation is intended as a precursor to the notion of Taylor polynomial approximations.
3. Approximating the Exponential Function This DIGMath spreadsheet lets you investigate dynamically the idea of approximating the exponential function f(x) = b^x with base e = 2.71828... with a polynomial. You can use an approximating polynomial up to fifth degree in the form y = a + (1/b)x +(1/c) x²+ (1/d) x³+ (1/e) x⁴+ (1/f) x⁵, where the six parameters a, b, c, d, e, and f are all integers. You use sliders to change each of the parameter values and the associated graph shows how well the corresponding polynomial fits the exponential curve. The spreadsheet also shows the value for the sum of the squares to provide a numerical measure for the goodness of the fit. This investigation is intended as a precursor to the notion of Taylor polynomial approximations.
4. Approximating the Natural Logarithmic Function This DIGMath spreadsheet lets you investigate dynamically the idea of approximating the natural logarithmic function f(x) = ln x with base e = 2.71828 with a polynomial. You can use an approximating polynomial up to fifth degree in the form y = a + (1/b)(x - 1) + (1/c) (x-1)²+ (1/d) (x-1)³+ (1/e) (x-1)⁴+ (1/f) (x-1)⁵, where the six parameters a, b, c, d, e, and f are all integers. You use sliders to change each of the parameter values and the associated graph shows how well the corresponding polynomial fits the exponential curve. The spreadsheet also shows the value for the sum of the squares to provide a numerical measure for the goodness of the fit. This investigation is intended as a precursor to the notion of Taylor polynomial approximations.
5. 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.
6. 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.
7. 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.
8. 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.
9. 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.
10. 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.
11. 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.  
12. 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.
13. 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.
14. 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.
15. Creating Polynomial Approximations to the Cosine FunctionThis 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.
16. 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.
17. 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).
18. 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.
19. Approximating Polar Functions with Taylor Polynomials This DIGMath module lets you investigate how any polar coordinate curve r = f ( q ) on any interval [a, ß] can be approximated using Taylor polynomial approximations. 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 to enter the expression for the Taylor polynomial approximation. . The spreadsheet draws the polar curve (in blue) on the desired interval and superimposes the associated Taylor polynomial (in red). You can see how well the corresponding polynomial attempts to match the curve. You can also trace along both curves and see the corresponding coordinates of both displayed. M
20. Approximating Parametric Functions with Taylor Polynomials This DIGMath module lets you investigate how any parametric curve x = f (t ). y = f(t) on any interval [a, b] can be approximated using Taylor polynomial approximations. You enter your choice for both functions in terms of the variable t (instead of x) and the desired interval using a slider. You also have to enter the expressions for the two associated Taylor polynomial approximations, also in terms of t. The spreadsheet draws the polar curve (in blue) on the desired interval and superimposes the associated Taylor polynomial approximations (in red). You can see how well the corresponding polynomialx attempt to match the curve. You can also trace along both curves and see the corresponding coordinates of both displayed.
21. 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. 
22. Lagrange Interpolation Formula This DIGMath module lets you investigate the Lagrange Interpolating Polynomial based on any set of up to 8 data points. The spreadsheet draws the graph of the Lagrange polynomial as it passes through the interpolating points. It also displays the equation of the polynomial using the Lagrange formula for the linear, quadratic, and cubic cases.
23. Building Lagrange Interpolation Polynomials This DIGMath spreadsheet lets you investigate the way that the Lagrange Interpolating Polynomial is created as a combination of the individual component polynomials. You can look at the way that the linear, the quadratic, and the cubic Lagrange polynomials are built based on the sum of two linear, three quadratic, or four cubic polynomials, respectively. In each case, the individual components are drawn along with the Lagrange polynomial based on the data points entered and you can then examine the relationships between the roots of the polynomial components, the interpolating points, and the associated function values.
24. Newton Interpolation Formula This DIGMath module lets you investigate the Newton Forward Interpolating Polynomial based on any set of up to 8 uniformly spaced data points. The spreadsheet draws the graph of each of the successive linear, quadratic, ... Newton polynomials as they pass through the various interpolating points.
25. Difference Tables and the Newton Interpolating Polynomial This DIGMath spreadsheet lets you investigate the Newton Interpolating Polynomial and how it is based on a table of successive differences formed from a set of data. You can enter your data values with equally spaced x-values and the spreadsheet displays the corresponding table of differences, as well as the graph of the resulting Newton Interpolating Polynomial and the formula for that polynomial. M M
26. Approximating the Sine using Lagrange Interpolation This DIGMath spreadsheet lets you investigate how the sine function can be approximated on the interval [0, p/4] using Lagrange Interpolating Polynomials based on either three or four interpolating points given either in radians or degrees. The spreadsheet draws the graph of sine and the interpolating polynomial as well as the graph of the error function P(x) - sin x so you can see how well or poorly the polynomial fits the function as you change the interpolating points.
27. Approximating the Sine using Newton Interpolation This DIGMath spreadsheet lets you investigate how the sine function can be approximated on the interval [0, p/4] using Newton Interpolating Polynomials based on either three or four uniformly spaced interpolating points given either in radians or degrees. The spreadsheet draws the graph of sine and the interpolating polynomial as well as the graph of the error function P(x) - sin x so you can see how well or poorly the polynomial fits the function as you change the interpolating points.
28. Approximating the Cosine using Lagrange Interpolation This DIGMath spreadsheet lets you investigate how the cosine function can be approximated on the interval [0, p/4] using Lagrange Interpolating Polynomials based on either three or four interpolating points given either in radians or degrees. The spreadsheet draws the graph of cosine and the interpolating polynomial as well as the graph of the error function P(x) - cos x so you can see how well or poorly the polynomial fits the function as you change the interpolating points.
29. Approximating the Cosine using Newton Interpolation This DIGMath spreadsheet lets you investigate how the cosine function can be approximated on the interval [0, p/4] using Newton Interpolating Polynomials based on either three or four uniformly spaced interpolating points given either in radians or degrees. The spreadsheet draws the graph of sine and the interpolating polynomial as well as the graph of the error function P(x) - cos x, so you can see how well or how poorly the polynomial fits the function as you change the interpolating points.  
30. Approximating the Exponential Function using Lagrange Interpolation This DIGMath spreadsheet lets you investigate how the exponential function y = e^x can be approximated on the interval [0, 1] using Lagrange Interpolating Polynomials based on either three or four interpolating points. The spreadsheet draws the graph of exponential function and the interpolating polynomial as well as the graph of the error function P(x) - ex so you can see how well or poorly the polynomial fits the function as you change the interpolating points.
31. Approximating the Exponential Function using Newton Interpolation This DIGMath spreadsheet lets you investigate how the exponential function y = e^x can be approximated using Newton Interpolating Polynomial.   One page lets you work with a quadratic polynomial based on three uniformly spaced points on the exponential curve and another page lets you use a cubic polynomial based on four uniformly spaced points. The spreadsheet calculates and draws the graph of the interpolating function as it passes through the interpolating points and seeks to match the exponential curve. It also draws the graph of the error function -- the difference between the polynomial and the exponential function.
32. Approximating the Natural Logarithm using Newton Interpolation This DIGMath spreadsheet lets you investigate how the natural logarithm function y = ln x can be approximated using Newton Interpolating Polynomials.  The first page lets you compare the effectiveness of the quadratic, the cubic, the quartic, and the quintic Newton polynomials based on uniformly spaced points on the curve.  The spreadsheet calculates and draws the graphs of the four interpolating polynomials as they pass through the interpolating points on the natural logarithm curve. It also lets you select a particular degree for the polynomial and draws the graph of the associated error function -- the difference between the polynomial and the exponential function.  A second page shows all four interpolating polynomials separately and all four error functions separately.
33. Approximating Polar Curves with Newton Interpolation This DIGMath module lets you investigate how any polar coordinate curve r = f ( q ) on any interval [a, ß] can be approximated using Newton's forward 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 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.
34. Approximating Polar Curves with Lagrange Interpolation This DIGMath module lets you investigate how any polar coordinate curve r = f ( q ) on any interval [a, ß] 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.
35. 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.
36. 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.
37. Chebyshev Nodes This DIGMath module lets you investigate the so-called Chebyshev nodes that are often the best points to use to construct an interpolating polynomial based on a set of points. The Chebyshev nodes have the property that they provide the minimum value of the maximum error in the approximation at the interpolating points. The spreadsheet lets you enter your choice of either three or four interpolating points on the interval [-1, 1] and compares, both numerically and graphically, the maximum error to the corresponding maximum error using the Chebyshev nodes.
38. Cubic Splines This DIGMath spreadsheet lets you investigate the notion of cubic splines, a way to construct a smooth curve determined by a set of points in such a way that the curve is made up of a series of smoothly connected cubic curves. The spreadsheet has two components. In the first, the data points are grouped three at a time subject to the condition that the slope at the third point must be equal to the slope of the following cubic at the first point. In the second, the points are grouped two at a time subject to the two conditions that the slope and the value of the second derivative must agree at each of the overlapping points.

Approximating the Derivative

1. 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.
2. Newton Difference Quotients This DIGMath module lets you investigate the notion of the Newton difference quotient as an approximation to the slope of the tangent line to a curve at a point. You can use any function, any interval, any desired step-size, and any point of tangency. The spreadsheet displays graphically the graph, the associated secant line and the triangle based on the point of tangency. If also displays numerically slope of the secant line compared to the slope of the tangent line, so you can see what happens to the agreement as the step-size is decreased or as the point of tangency moves along the curve.

Numerical Integration

1. Numerical Integration Methods This DIGMath spreadsheet allows you to investigate four different methods to approximate the value of a definite integral -- using left and right-hand Riemann Sums, using the Trapezoid Rule, the MidPoint Rule, and Simpson's Rule for any function of the form y = f(x) on any desired interval [a, b]. You control the number of subdivisions for each method using a slider and the spreadsheet draws the graph of the function, draws the approximating subdivisions, and displays the associated approximation to the definite integral.
2. 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. 

Approximating Solutions of Differential Equations

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

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.