Derivatives

Target Audience: Calculus I students, especially after they have learned how to compute derivatives of common functions.

Objective: At the completion of this exercise, the successful student will understand the correlation between a function and its first and second derivatives.

Created by: Robert Brown (Carroll Community College) for Project Synergy

This exercise takes advantage of a Java applet that allows you to interact with the graph of a function and to display the graphs of this function and its first and second derivatives at the same time! The first set of instructions below directs you to a specific website and starts our Derivatives Java applet. Once you have the program started, answer the questions below using this program.

How to Start “Derivatives” applet.

1.        Start your favorite web browser and visit the Derivatives Applet web page (http://math.hws.edu/javamath/config_applets/Derivatives.html).

2.       
Scroll halfway down the web page and click the button called “Launch Derivatives.” This will load the Derivatives applet in a separate window (see below). The applet loads automatically by graphing a function they called T(x). Take a minute to look over the interface to familiarize yourself before proceeding.

Additional Notes on using this applet:

1)       If you left click on any of the 3 graphs, you zoom in on that portion for all three graphs. Successive left mouse clicks continues the “zoom in.”

2)       For better control of “zoom in” on a particular rectangular portion of the graph, hold down the left mouse button on the graph window and drag out a rectangle on the portion you want and release the mouse button. The graphs will automatically resize.

3)       To change the view of the graph window without zooming, hold down the right mouse button on a graph window and move in the direction you are interested in viewing. Release mouse button when you have the view you want.

4)       Return to the default window settings on the graphs by clicking the  “Restore Limits” button.

5)       When you type in your own functions, enclose arguments of functions in parenthesis and use the standard symbols for the operations. You must hit the ENTER key on the keyboard to graph the function.

a)       *              Multiplication

b)       /               Division

c)       +              Addition               

d)       -               Subtraction

6)       To view a specific value for x, just type the value in the box to the right of  “x”. You can also move the x value by dragging the box in the horizontal scroll bar at the bottom of the window.

7)       Play! Try several functions and drag the horizontal bar. Watch the instantaneous changes in the graphs of f’(x) and f’’(x).

Calculus Exercise:

Complete the following problems and hand in for credit.

1)       Graph . (Don’t forget the “*” to represent multiplication.)

a)       At what x-value do you have a horizontal tangent line to the graph of ?

b)       For this x-value, what is ?

c)       To the left of this x-value on the graph of , is the graph increasing, decreasing, or constant?

d)       To the left of this x-value on the graph of , are the slopes of the tangent lines positive, negative, or constant?

e)       To the left of this x-value on the graph of , what do all of these y-values have in common?

f)        To the right of this x-value on the graph of , is the graph increasing, decreasing, or constant?

g)       To the right of this x-value on the graph of , are the slopes of the tangent lines positive, negative, or constant?

h)       To the right of this x-value on the graph of , what do all of these y-values have in common?

2)       Repeat all parts to problem 1 above with the following functions.

a)      

b)      

c)      

d)      

e)      

3)       What conclusions can you make from the graphs of  and  about the graph of ?