Inquiry – The Relationship Between a Function and Its Derivative

In this activity you will discover for yourself the remarkable relationship between the graph of a function and the graph of its derivative.

Key question: If I know what the graph of a function looks like, can I draw the graph of its derivative?

The derivative of a function at a given point is the slope of the tangent to the graph at that point.

$$f'(x_0) = \text{slope of the tangent to } f \text{ at } x_0$$

Part A: Free Exploration

Instructions: Move slider a slowly from right to left and observe what happens.

1. When the point is on an increasing part of the function (blue graph), what is the sign of the slope?

Positive (+)
Negative (-)
Zero

2. When the point is on a decreasing part of the function, what is the sign of the slope?

Positive (+)
Negative (-)
Zero

3. When the point is at an extremum (maximum or minimum), what is the slope?

Positive (+)
Negative (-)
Zero

Now reveal the derivative graph (the red graph)!

Click the checkbox "Show derivative" in the application above.

Move the point and complete:

A graph of function f is given. The function is increasing on $$(-\infty, 2)$$ and decreasing on $$(2, \infty)$$.

a) What is the sign of the derivative on $$(-\infty, 2)$$? ______

b) What is the sign of the derivative on $$(2, \infty)$$? ______

c) What is the value of the derivative at $$x=2$$? ______

d) What type of point is $$x=2$$ for function f? ______

Given that $$f'(x) > 0$$ for all $$x < 1$$ and $$f'(x) < 0$$ for all $$x > 1$$.

a) On which intervals is f increasing? ______

b) On which intervals is f decreasing? ______

c) What happens at $$x=1$$? ______

If $$f(x) = x^3 - 3x$$, sketch (roughly) the graph of the derivative $$f'(x)$$ without computing!

Hint: find where f is increasing, where it is decreasing, and where the extrema are.