The goal of this activity is to learn how to use the
graph of a function to produce a graph of the derivative without
having a formula for the function or for the derivative.
 |
Start
by visiting the "maths online" web page linked
here. Once at this page, click on the big red
button labeled "Applet: definition of the
derivative". Play with the applet a little
bit and then within the applet click on the button labeled
exercises. Work through these exercises and use
the help button if you need to. Your goal here
is to be able to understand the derivative of this function
well enough to graph it. Try to "See" the answer
to each of the following questions:
- Where is the derivative (the slope of the tangent line) equal to 0?
- Where
is the derivative positive?
- Where
is the derivative negative?
- Where
is the derivative the largest (most positive)?
- Where is the derivative the smallest (most negative)?
- Where does the slope of the tangent line stop increasing and begin to decrease?
- Where does the slope of the tangent line stop decreasing and begin to increase?
While playing with this applet, take out a piece of paper
and a pencil and see if you can sketch a graph of the
derivative function... can you determine what it must look like?
|
|
Now
to reinforce this, use this link to visit the surfing
derivative web site. Drag the red button on
the surfer to see how the slope of the tangent line varies
as x changes. With your paper and pencil try to
sketch the derivative of this function. Now click
on the "trace" button on the left side of the
applet and drag the red button back and forth along the
graph. The applet records the values of the derivative
in the graph at the bottom ... does this graph match
the one you produced? It is difficult to create the exact
graph of the derivative, but you should get a similar
shape with similar features.
|
|
Now, revisit the "maths online" web page. Scroll
down a bit on the page and visit the three derivative puzzle applets. In
each puzzle, your goal is to assemble the 9 graphs
into a tic-tac-toe square so that each graph is the
derivative of the function graphed above it. Use
the help menu available on the page to assist you.
|
Once you have learned how to create the graph of the derivative from the graph of a function, send your instructor an email with the answer to the following questions:
Consider the function with the following graph:
- At which values of x does the function have horizontal tangent lines? (Assume each tick mark is one unit on the graph).
- Where is the derivative positive? (list the intervals).
- Which of the three graphs below contains the derivative of the function?
- Did you find this web activity useful?
If your email program is configured properly, you can click on the name of your instructor (below) to send the email. Use the subject line `Your Name, Math 207, Graph the Derivative Activity'.
|
