5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 17:21

Graphs of Functions and Derivatives 157

[–4, 4] by [–1, 2]

Figure 8.3-3

TIP • When evaluating a definite integral, you do not have to write a constant C,

e.g.,

∫ 3

1 2 xdx=x

2 ∣∣^3

1 =^8 .Notice, noC.

8.4 Graphs of Derivatives

The functions f, f′, and f′′are interrelated, and so are their graphs. Therefore, you

can usually infer from the graph of one of the three functions (f, f′,orf′′) and obtain

information about the other two. Here are some examples.

Example 1

The graph of a functionf is shown in Figure 8.4-1. Which of the following is true forf

on (a,b)?

0

y

f

a b x

Figure 8.4-1

I. f′ ≥0on(a,b)

II. f′′>0on(a,b)

Solution:

I. Since f is strictly increasing,f′≥0on(a,b) is true.

II. The graph is concave downward on (a, 0) and upward on (0,b). Thus, f′′>0on

(0,b) only. Therefore, only statement I is true.