5 Steps to a 5 AP Calculus BC 2019

Graphs of Functions and Derivatives 123

[–4,4] by [–1,2]

Figure 7.3-3

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.

7.4 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 7.4-1. Which of the following is true forf

on (a,b)?

0

y

f

a b x

Figure 7.4-1

I. f′≥0on(a,b)

II. f′′>0on(a,b)

Solution:

I. Sincef 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.