USING TEMATH S VISUALIZATION TOOLS IN CALCULUS 1

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1 USING TEMATH S VISUALIZATION TOOLS IN CALCULUS 1 Robert E. Kowalczyk and Adam O. Hausknecht University of Massachusetts Dartmouth North Dartmouth, MA TEMATH (Tools for Exploring Mathematics) is a mathematics software package containing a set of easy-to-use computational and visual tools. Its design takes full advantage of the Apple Macintosh interface and it encourages the user to freely explore and visualize the mathematical concepts being studied. In this paper, we will discuss many of TEMATH s visualization tools and we will give examples of how they can be effectively used in teaching calculus. One of TEMATH s visualization tools is the Parametric Tracker. This tool can be used by the calculus student to gain insight into the construction of parametric curves. As an example, let s consider the graph of the parametric equations x = t and y = t 2 which is the parabola shown in Figure 1. By examining this two dimensional cartesian graph, all the information about the parameter t is lost. However, when you drag the scroll box of the Parametric Tracker, a dot moves along the curve representing the (x(t), y(t)) coordinates for a particular value of t. Also, the current values of t, x, and y are shown in TEMATH s Domain & Range window. By using the Parametric Tracker, the student can visualize how the curve is drawn as the value of t changes. For example, in Figure 1 it is shown that when t = 4.04, x = 4.04 and y = 16.3 and the corresponding point on the curve is in the top-left portion of the graph. Figure 1 Parametric Tracker (x = t and y = t 2 ) With the Parametric Tracker the student can easily study different parametrizations of the same curve. For example, if the student graphs the parametric equations x = t and y = t 2, the same parabola is drawn. But this time, the Parametric Tracker shows that when t = 4.04, x = 4.04 and y = 16.3 and the corresponding point on the curve is in the top-right portion of the graph (see Figure 2). 1 This article appeared in "Proceedings of the Fifth Annual International Conference on Technology in Collegiate Mathematics", Addison-Wesley, 1994, p

2 Using TEMATH's Visualization Tools in Calculus 2 Figure 2 Parametric Tracker (x = t and y = t 2 ) Other interesting parametrizations of the parabola are: x = 5 sin(πt/5) and y = 25 sin 2 (πt/5) and x = 5 sin(πt/3) and y = 25 sin 2 (πt/3). This last parametrization traces the same parabola more than three times and it demonstrates that it is possible for a parametric curve to pass through the same point more than once. The Parametric Tracker can also be used to estimate points of intersection for two parametric curves by placing the dot at the point of intersection on each curve. In Figure 3, the point of intersection is estimated to be ( 3.6, 12.6), but it should be noted that the two curves pass through this point for different values of t. Figure 3 Finding Points of Intersection for Parametric Curves

3 Using TEMATH's Visualization Tools in Calculus 3 Additionally, students become very excited with the interesting curves that can be generated with parametric equations and they spend a great deal of time creating their own designs (see Figure 4). Figure 4 Designs with Parametric Equations The Polar Tracker is similar to the Parametric Tracker in that it is a visualization tool used to help the student gain an understanding of the constuction of curves in polar coordinates. The four-leaved rose r = sin(2t) is shown in Figure 5. When the student drags the Polar Tracker, a radial line is drawn representing the angle t and a dot is drawn on the curve representing the radius r. The current values of t and r are also displayed in the Domain & Range window. For example, when t = 4 the value of r is positive and the radial line and dot are both drawn in the third quadrant. Figure 5 Polar Tracker (r = sin(2t)) The graph of the polar equation r = sin( 2t) is also a four-leaved rose that appears to be identical to the previous one. However, this time the Polar Tracker traces the leaves in a

4 Using TEMATH's Visualization Tools in Calculus 4 different order. For example, when t = 4 the value of r is negative and the radial line is moving in the third quadrant but the dot is tracing the leaf in the first quadrant (see Figure 6). The Polar Tracker can also be used to estimate the points of intersection for two or more polar curves. Figure 6 Polar Tracker (r = sin( 2t)) Another way to use TEMATH in the classroom is as a number-crunching demonstration tool to compare the rates of convergence for various numerical integration algorithms. As an example, let s consider the definite integral π/2 sin( x)dx =1 0 If we use rectangles to approximate the area under the curve, we can define the function R(n) = (π / 2) n n i =1 sin i(π / 2) n where R(n) is the sum of the areas of n rectangles. The Expression Calculator in TEMATH can be used to easily evaluate R(n) for n = 10, 100, These values are shown in Figure 7. Note that as n increases by a factor of 10, the approximation to the integral gains one significant digit of accuracy thus showing an O(1/n) rate of convergence. Similarly, if we use trapezoids to approximate the area under the curve, we can define the function T(n) = (π / 2) 2n n 1 i(π / 2) sin(0) + 2 sin n + sin(π / 2) i =1 where T(n) is the sum of the areas of n trapezoids. The Expression Calculator was used to evaluate T(n) for n = 10, 100, These values are shown in Figure 7. Note that as n increases by a factor of 10, the approximation to the integral gains two significant digits of

Using TEMATH's Visualization Tools in Calculus 5 accuracy thus showing an O(1/n 2 ) rate of convergence. Finally, we can use Simpson s rule to approximate the area under the curve.

5 Using TEMATH's Visualization Tools in Calculus 5 accuracy thus showing an O(1/n 2 ) rate of convergence. Finally, we can use Simpson s rule to approximate the area under the curve. Let s define the function S(n) = (π / 2) 3n n 2 1 n 2 2i(π / 2) sin(0) + 2 sin (2i 1)(π / 2) n + 4 sin i =1 n + sin(π / 2) i =1 where S(n) is the sum of the areas under the parabolas. The Expression Calculator was used to evaluate S(n) for n = 10, 100, These values are shown in Figure 7. Note that as n increases by a factor of 10, the approximation to the integral gains four significant digits of accuracy thus showing an O(1/n 4 ) rate of convergence. Figure 7 Using the Expression Calculator to Compare Rates of Convergence for Approximate Integration The above example demonstrates how rates of convergence for various numerical integration techniques can easily be compared both numerically and visually by seeing how the number of zeros or nines increase. With tools like these, the student is able to develop a good intuitive sense of what a rate of convergence is. In first semester calculus, much effort is spent trying to instill in our students the meaning of the derivative of a function. The following examples show how TEMATH can be used to visually demonstrate to students how the derivative can be defined from tangents and from limits. In our first example, we explore the derivative of f(x) = sin(x) through the idea of the tangent. Using TEMATH, we can graph f(x) = sin(x) on the interval 0 x 2 and using the Tangent tool, we can draw a sequence of tangents to the curve over this interval. See Figure 8.

6 Using TEMATH's Visualization Tools in Calculus 6 Figure 8 Drawing Tangents to the Sine Curve As we draw these tangents, their slopes are computed and written into the Report window. These values can be placed into a table and the graph of this data can be overlaid on top of the existing graph as is shown in Figure 9. Figure 9 Plotting the Values of the Slopes of the Tangents At this point, we can ask the students to conjecture what curve the plotted data looks like. As they make conjectures, we can plot their guesses along with the slope data. Of course, when they guess the cosine function they will see that the plot of cos(x) passes through all the data (see Figure 10). In this way, the student can truly develop a good understanding of the relationship between the derivative and the slope of the tangent.

Using TEMATH's Visualization Tools in Calculus 7 Figure 10 Fitting the Plotted Slopes Another way of looking at the derivative is through its definition f (x + h) f (x) f (x) = lim h 0 h In most

7 Using TEMATH's Visualization Tools in Calculus 7 Figure 10 Fitting the Plotted Slopes Another way of looking at the derivative is through its definition f (x + h) f (x) f (x) = lim h 0 h In most calculus courses, we evaluate this limit at some point x = a and talk about the derivative at the point x = a. However, we seldom evaluate this limit globally for all x in the domain of interest in order to reinforce the idea that the derivative is a function. In TEMATH, we can do this by setting up an experiment to visualize the convergence of the function f(x + h) f(x) h to the function f (x). We can define the constant h, the function f(x), and the slope of the secant line function msec(x) = 11.. f(x + h) f(x) h as is shown in Figure Figure 11 Setting Up the Derivative Experiment Now we can let h take on smaller and smaller values and plot the corresponding graphs of msec(x) to see if these functions converge to some function. Letting f(x) = sin(x), the graphs of msec(x) for h = 1, 0.5, 0.1, and 0.01 are shown in Figure 12. Again, if the students

8 Using TEMATH's Visualization Tools in Calculus 8 conjecture that the functions appear to be converging to cos(x), we can overlay the graph of cos(x) to verify this conjecture. Of course we can still mathematically prove that the derivative of sin(x) is cos(x). Figure 12 Looking at the Limit of the Slope of the Secant Line Function Calculus teachers are always looking for applications and real data to use in their calculus courses. One set of real data that can open up a lot of discussion is the population of the world data shown in Table 1. Year Population (millions) Table 1 World Population In this example, we want to find the best mathematical model to represent the world population data. Our first try is an exponential model (since the data represents a growth phenomenon). We can use TEMATH s Least Squares tool to find the least squares

9 Using TEMATH's Visualization Tools in Calculus 9 exponential fit to the data. This fit is given as f(x) = 0.014e x. The graphs of the world population data and the exponential fit are shown in Figure 13. Notice that the exponential function does not model the data very well. Figure 13 Least Squares Exponential Fit to the World Population Data Our next try is a power function model. TEMATH gives the power function fit as f(x) = x The graph of the power function is very similar to the graph of the exponential function and again it is not a very good fit (see Figure 14). Figure 14 Least Squares Power Fit to the World Population Data In both cases, we noticed that the rate of growth of the population appears to be faster than the least squares fit. Maybe this suggests that we try some type of rational function model. Using TEMATH s Least Squares Rational1 tool, we get the rational function fit f(x) = x. The graph of the data and the rational function are shown in Figure 15. Notice how well the rational function fits the population data. Now we can get into some interesting discussion concerning the use of this model. Do we use the model only for

10 Using TEMATH's Visualization Tools in Calculus 10 interpolating data? Do we use the model to extrapolate the data and predict the population of the world in some future year? If we do, what happens to the world in the year ? Not only are we introducing model building in this example, but we are also developing a sense of the appropriate use of a model. Figure 15 Least Squares Rational Fit to the World Population Data In this final example, we would like to show how TEMATH can be used to verify the need for the hypotheses given in the statement of a theorem. Most theorems in calculus give a list of hypotheses that are necessary for a result to be true and many times these hypotheses are glossed over or are mentioned only in the proof of the theorem. As an example, let s consider the mean value theorem: Let f be a function that satifies the following hypotheses: 1. f is continuous on the closed interval [a, b]. 2. f is differentiable on the open interval (a, b). Then there is a number c in (a, b) such that f (c) = f (b) f (a) b a Using the function f(x) = 2x x 2 which satisfies both of the hypotheses for the interval [0, 3], we can use TEMATH s Distance tool and Tangent tool to quite easily visualize that there is a number c that satisfies the mean value theorem (see Figure 16)..

11 Using TEMATH's Visualization Tools in Calculus 11 Figure 16 Verifying the Mean Value Theorem Now, what if we violate one of the hypotheses of the theorem, say the differentiability hypothesis. As an example, we can use the function f(x) = x 1. If we graph this function and draw the secant line between the endpoints, we see that it is impossible to find a tangent line to the curve that has the same slope as the secant line. See Figure 17. Figure 17 Violating the Differentiability Hypothesis x What if we violate both of the hypotheses? As an example, consider the function f(x) = x 1. If we graph this function and draw the secant line between the endpoints, we see that it is impossible to find a tangent line to the curve that has the same slope as the secant line since the function is decreasing and the slope of the secant line is positive. See Figure 18.

12 Using TEMATH's Visualization Tools in Calculus 12 Figure 18 Violating Both Hypotheses of the Mean Value Theorem All of the above examples are but a brief survey of the kinds of visual activities that can be done with a class of calculus students. We believe they serve to illustrate the power of visualization as an aid to understanding subtle mathematical concepts.

GENERATING AND MODELING DATA FROM REAL-WORLD EXPERIMENTS

GENERATING AND MODELING DATA FROM REAL-WORLD EXPERIMENTS Robert E. Kowalczyk and Adam O. Hausknecht University of Massachusetts Dartmouth Mathematics Department, 285 Old Westport Road, N. Dartmouth, MA