Safe and Simple Calculus Activities

Transcription

1 Safe and Simple Calculus Activities This paper focuses on a few easy-to-implement technology-enabled activities for a first year Calculus course. Specific topics will include the derivative at a point, the derivative as a function, the idea of the integral, and the Fundamental Theorem of Calculus. Derivative at a Point lim h f a h f a h or lim xa f x f a x a When either of these limits exist, we say f is differentiable at x = a and identify the limit as the derivative of f at x = a. How can technology help? A simple way is to look at values of f x f a x a for x close to a. Zoom in to table. The graphical analog is zooming in to the graph of y sin xat x. 3

2 The slope of the "line" is the value of the derivative at the point. What re some good questions to ask students as they are engaged in these activities? Continuity and Differentiability First year Calculus students are taught that differentiability at a point implies continuity at the point. But do they know why that is the case? This activity focuses on the definition of derivative as it relates to that conditional statement. Again, what are some good questions? Consider the following.

3 Suppose x 1, x 1 2 f x 2 x, x 1 4 Evaluate each of these expressions, or explain why the expression fails to exist: 1 f x f lim x1 x 1 f x f lim x1 x 1 f x f 1 lim x1 x 1 lim f x x1 f 1 1 x 1, x 1 2 Now, consider g x 2 x 5, x 1 4 Again, evaluate each of these expressions, or explain why the expression fails to exist: g x g lim x1 x 1 g x g lim x1 x 1 g x g lim x1 x 1 lim g x x1 g

4 Derivative as a Function This activity works well with the class divided into groups of two or three. Choose a function whose derivative will result in a recognizable pattern between the inputs and outputs. Give each group a different value for x (some negative, some positive). Have each group zoom in (to the table, as in the first activity, or to the graph, as in the second) and calculate the slope at their x, following the procedure outlined above. Tabulate the results. x Slope at x Then make a scatter plot of the ordered pairs consisting of (x, slope at x). An Aplet, DerivFxn, built for the HP4gs automates the process, but removes the responsibility of the student from determining slope at individual points. It could be use to explore other derivative rules, once students have begun to catch on to the idea that the derivative is a "slope machine." To use the Aplet, you must first enter the function whose derivative you are investigating into F1(X) in the Function Aplet. Then start the DerivFxn Aplet. (Note: You need to first go to PLOT view and select FIT before selecting points).

5 Students should see the pattern. Then you can enter the derivative function into Fit2 in the Symbolic View, and see the function slam though all the points! Cool! What should the graph of x x sin.1 sin y look like? Why?.1

6 The Idea of the Integral and Approximating Net Change from a Rate of Change: Calculus Car Lab Purpose: You will get practical experience with using numeric information about a rate of change to compute net change. Specifically, with knowledge about the velocity of a car, you will approximate distance traveled. Collecting the Data: Go for a 15 minute drive, with someone else (preferably a parent or guardian) as the driver. Record your initial odometer reading at the start of the drive, and every 3 seconds, record the car's speedometer reading on the data sheet. Note the final odometer reading at the end of the drive. Also note any relevant traffic or other conditions during the drive. Computations: Use your car's initial and final odometer readings to calculate distance traveled. Then use the midpoint rule (with a t of 1 minute, so you use the 3 second readings) to approximate distance traveled. Finally, use the trapezoid rule (again with t of 1 minute) to approximate distance traveled. Make a scatter plot of velocity versus time, and show the trapezoids and midpoint rectangles on the scatter plot. Preparing the Report. Hand in a final lab report that contains all of the following items: 1. A statement of purpose and setup (do not simply copy mine!) 2. The data you collected, in a nice table (use the table attached) 3. A graph of velocity versus time, with the rectangles and trapezoids. 4. Neatly displayed computations 5. Summary and conclusions In the last section, discuss why your midpoint and trapezoid calculations did not agree exactly with your odometer calculated distance. What assumptions are made about velocity when the midpoint rule is used? What assumptions are made about velocity when the trapezoid rule is used? Be specific! Also, discuss how you could change this experiment to get better results (besides using cruise control!). Explain why the results would be better. Grading: You will be graded on completeness, presentation, accuracy of computations, and clarity of writing. In addition, any velocity reading over 55 mph will result in a grade of! A table for the data is attached. Due Dates:

7 Time : :3 1: 1:3 2: 2:3 3: 3:3 4: 4:3 5: 5:3 6: 6:3 7: 7:3 8: 8:3 9: 9:3 1: 1:3 11: 11:3 12: 12:3 13: 13:3 14: 14:3 15: Velocity (miles/hour) Conditions Initial Odometer Reading: Final Odometer Reading:

8 World's Shortest Riemann Sum Program Lots of people have written elaborate programs for calculators and computers that compute Riemann Sums. Regrettable, most of these obscure what's going on inside, and to students, the program is a black box. Activities like the Car Lab help. And activities like this one enable students to begin to see how a Riemann Sum can converge as the number of uniform sub-intervals gets large. First, a dictionary of variables: A is the lower limit of integration B is the upper limit of integration N is the number of uniform partitions F1(X) is the integrand R is a variable between and 1. R = corresponds to the left-endpoint rule; R = 1 corresponds to the right endpoint rule; R =.5 corresponds to the midpoint rule H is the width of each sub-interval, or (B A)/N. Now, all you have to do is store ever larger values into N to increase the number of subintervals. And you can choose a different method by storing an appropriate value into R.

9 Fundamental Theorem Investigation HP4gs In this activity, you will explore the Fundamental Theorem of Calculus from numerical and graphical x perspectives. The exploration will give you additional practice with functions of the form F( x) f ( t) dt. This activity requires you to have two applets: FTCFXNS and FTCSTATS. The former contains the function definitions shown in the screens below. 2 X T Note that F2 is a function defined as a definite integral of F1. That is, F2( X ) cos dt 2. Make sure that F1 is the only function selected for graphing and look at the graph. Your PLOT SETUP and graph should look like the ones shown: If your graph is different, check that your window is correct, and that you are in radian mode. Now start the FTCSTATS aplet. In the numeric view, C1 consists of X values from the graph screen. These serve as inputs into F1, F2, and F3. The outputs from F1, the integrand, are in column C5. The outputs from the integral in F2 are in C2. The outputs from the integral in F3 are in C3. Each element of C2, then, results from the evaluation of a definite integral. For example, when.1, the second value of C1, is plugged into F2, the value of.1 cos T 2 2 dt results, which is the second value in C2. Calculate that integral on your HOME screen and verify the result. 1. Explain why the first value in C3, which results from evaluating F2(), is. Look at the graph of the integrand, F1(X) (you can use S1 in the FTCSTATS aplet, or go back to the FTCFXNS aplet and look at a continuous plot). Use the graph to answer questions 2 and Why is the second value in C2 greater than?

10 .2 3. Why is the third value in C2, which represents cos T 2 2 dt, greater than the second value in C2, which.1 represents cos T 2 2 dt? Go back to the numeric view in the FTCSTATS aplet and look at the table of inputs and outputs. Remember that C1 contains the X values; C5 contains the values of the integrand; cos T 2 2 ; and C2 contains the values of the integral. 2 X T 4. From the list of data values stored in C2, for what value of X (stored in C1) does F2( X ) cos dt 2 appear to reach its first local maximum? 5. Explain why the 2 th value in C2 is smaller than the 19 th value in C2. That is, why is 1.9 cos T 2 2 dt 1.8 cos T 2 2 dt? 2 X T 6. From the list of data values stored in C2, for what positive value of X does F2( X ) cos dt 2 appear to reach its first local minimum? Go back to the FTCFXNS aplet and look at the graph of F1. Find the smallest two x-intercepts on the graph of the integrand, not4d as points A and B below, using the Root command in the FCN menu. 7. (A) and (B)

11 8. Compare the answers to 7 (A) and 7 (B) to the answers to numbers 4 and 6 respectively. Explain why they are similar. 9. Find the next positive X-intercept of the graph of F1(X) (close to X = 4). 1. What happens with F2(X)at the point found in number 9? (Verify your answer numerically by looking at C2 in the numeric view of the FTCSTATS aplet) 2 X T Now you will take a look at F3( X ) cos dt. Note that the only difference between F2 and F3 is the 1 2 lower limit of integration. The values of F3 are in C3, and appear in scatter plot S3. Press SYMB and check scatter plot S3. Then press PLOT to see the graph. 2 2 T 1 T Note that the first element of C3 represents F3() cos dt cos dt Why is the graph of C3 versus C1 ( scatter plot S3) below the graph of C2 versus C1 (scatter plot S2)? (Remember, the only difference between the two is the lower limit of integration.) 12. It appears that S2 and S3 differ by a constant, i.e. that F3(X) F2(X) is the same for all values of X. Find the value of this constant. (You could calculate this in several different ways. If you get stuck, try expressing F3(X) F2(X) in terms of an integral.) 13. Explain why plots S2 and S3 have the same shape. Look at the table of values for C2 and C3, and notice that the locations of the extrema are the same for both functions (as you would expect from an inspection of the scatter plots).

12 14. Look at the locations of the local minima and the local maximum on the graph of F1(X) (scatter plot S1). At these values of X, describe what appears to happen with the concavity of the graphs from S2 and S Write an equation with a derivative on one side that shows how F1(X) and F2(X) are related. 16. Recall that the derivative of a function f at x = a can be defined by f (x) f (a) f (a) lim. Approximate F21 by evaluating the expression F21.1 F21 /.1 xa x a. Your answer should be close to the value of F1(1). Why? 17. Approximate F3 1 by evaluating the expression F F your answer should be close to the value of F1(1). Why? /.1.. Again, Extension: Approximate the derivative of F2(X) by forming difference quotients using the List command: List(C2)/.1 STO> C4. Then define and look at a scatter plot of C4 versus C1. The result is stunning.