Lesson 9: The Derivative As A Function (worksheet Solutions)

Solutions to Worksheet for Section 2.8 The Derivative as a Function Math 1a October 15, 2007 1. Match the graph of each function in A–D with the graph of this derivative in I–IV. Give reasons for your choices. (A) y

(B) y

x

(C) y

x

y (I)

x

y (II)

x

(D) y

x

y (III)

x

y (IV)

x

x

Solution. The function in (B) is not differentiable at two points, and the function in (IV) is the only one that is not continuous at two points. So they match. For the rest, count the number of horizontal tangents of the function. They are zeroes of the derivative! Function (C) has only one, and only (I) achieves the value 0 exactly once. Function (A) has two, so it matches with (II), and finally (D) matches with (III). 2.

Graphs of f , f 0 , and f 00 are shown below. Which is which? How can you tell? y

x

Solution. Again, look at the horizontal tangents. The short-dashed curve has horizontal tangents where no other curve is zero. So its derivative is not represented, making it f 00 . Now we see that where the bold curve has its horizontal tangents, the short-dashed curve is zero, so that’s f 0 . The remaining function is f .