Solutions to Worksheet for Section 2.8

The Derivative as a Function

Math 1a

February 13, 2008

1. Match the graph of each function in A–D with the graph of this derivative in I–IV. Give reasonsfor your choices.

(A) (B) (C) (D)

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(I) (II) (III) (IV)

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Solution. The function in (B) is not differentiable at two points, and the function in (IV) is theonly one that is not continuous at two points. So they match. For the rest, count the number ofhorizontal 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), andfinally (D) matches with (III).

2. Graphs of f , f ′, and f ′′ are shown below. Which is which? How can you tell?

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