Calculus Lab: Relationship between a Function and its Derivative

Goals

Given the graph of a function, this lab helps you visualize the graph of its derivative.

Acknowledgement

This lab is based on a lab of the same title from Learning by Discovery: A Lab Manual for Calculus, edited by Anita Solow, MAA Notes, 1993.

Background

In this laboratory, you will be asked to compare the graph of a function like the one in Figure 1 to that of its derivative. This will help develop your understanding of the geometric information that f' carries.

You will need to bring an example of such a function into the lab with you, one whose graph meets the x-axis at four or five places over the interval [-3, 2]. During the lab, your partner will be asked to look at the graph of your function and describe the shape of its derivative (and you will be asked to do the same for your partner's function). One way to make such a function is to write a polynomial in its factored form. For example, f(x) = x² (x + l)(x - l)(x + 2) is the factored form of the function in Figure 1. Its zeros are at 0, ±1, and -2.

Figure 1: f(x) = x² (x + l)(x - l)(x + 2)

Before the Lab

1. Give another example of a polynomial g of degree at least 5 with four or five real zeros between -3 and 2. You will use this polynomial in Step 4.

   In your notes before the lab, complete the following:

   1. Your polynomial: g (x) =
   2. Its zeros:
   3. Its derivative: g' (x) =

In the Lab

2. Let f (x) = x²(x² - 1)(x + 2).

   1. Find the derivative of f.
   2. Plot the graphs of both f and f' in the same viewing rectangle over the interval [-3, 2].

   Answer the following questions by inspection of this graph:

   3. Over what intervals does the graph of f appear to be rising as you move from left to right?
   4. Over what intervals does the graph of f' appear to be above the x-axis?
   5. Over what intervals does the graph of f appear to be falling as you move from left to right?
   6. Over what intervals does the graph of f' appear to be below the x-axis?
   7. At what points does the graph of f appear to have a local maximum or a local minimum?
   8. At what points does the graph of f' appear to meet the x-axis?

3. Let f(x) = x/(1 + x²).

   1. Find the derivative of f. Plot the graphs of both f and f' in the same viewing rectangle over the interval [-3, 3]. Answer the same set of questions as in parts c-h above.
   2. On the basis of your experience so far, state a conjecture that relates where a function is rising, is falling, and has a local maximum or minimum to properties you have observed about the graph of its derivative.

4. Now let g be the function that your lab partner brought into the lab. (If you have no partner, just use your own function.) In this problem you will use your conjecture to predict the shape of the graph of g', given only the shape of the graph of g.

   1. Have your lab partner produce a plot of the graph of g over the interval [-3, 2]. Your partner may need to adjust the height of the window to capture all of the action. On the basis of this plot, use your conjecture to imagine the shape of the graph of g'. In particular, find where g' is above, where g' is below, and where g' meets the x-axis. Then carefully sketch a graph of g' on your data sheet.
   2. Now have your lab partner plot the graph of g'. Compare your graph with the computer drawn graph. How did you do?
   3. Reverse roles with your lab partner and do parts a and b again.

5. Consider the function f(x) = | x² - 4 |. A graph of the function may help you answer these questions.

   1. There are two values of x for which the derivative does not exist. What are these values, and why does the derivative not exist there?
   2. Find the derivative of f at those values of x where it exists. To do this, recall that f can be defined by

      f(x) =

      You can compute the derivative for each part of the definition separately.
   3. Give a careful sketch of f and f' (disregarding the places where f' is not defined) over the interval [-4, 4]. Does your conjecture still hold?

Further Exploration

6. Consider the function f(x) = 2^x .

   1. Draw an approximate graph of 2^x and write a few sentences about the expected shape of its derivative.
   2. Some people think that f' (x) = x2^x-1. Compare the graph of x2^x-1 with your description from part a, and explain why x2^x-1cannot be the derivative of 2^x.

7. This laboratory has given you experience in using what you know about the shape of the graph of a function f to visualize the shape of its derivative function f'. What about going backwards? Suppose that your partner had given you the graph of f', would you be able to reconstruct the shape of the graph of f ? If f' is positive, for example, does your conjecture enable you to rule out certain possibilities for the shape of f ? The graph in Figure 2 is a sketch of the derivative of f. Use your conjecture to construct a possible graph for the function f itself. The important part of this problem is not the actual shape that you come up with, but your reasons for choosing it.

   
   Figure 2: The graph of y = f'(x)

8. Figure 3 shows the graphs of three functions. One is the position of a car at time t minutes, one is the velocity of that car, and one is its acceleration. Identify which graph represents which function and explain your reasoning.

   
   Figure 3: Position, velocity and acceleration graphs

   Work to Turn In

   • Commentary for steps 1 (from you and your partner), 2, 3, 4 (from both you and your partner), 5, 6, 7, 8.

This document is available on the World Wide Web as

http://www.cs.grinnell.edu/~walker/courses/131.fa09/labs/lab-func-deriv.shtml

created 8 June 2009 last revised 1 October 2009
For more information, please contact Henry M. Walker at walker@cs.grinnell.edu.