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 145

Example 5

The derivative of a functionf is given asf′(x)=cos(x^2 ). Using a calculator, find the values

ofxon

[

−

π

2

,

π

2

]

such that f is increasing. (See Figure 8.2-7.)

[–π, π] by [–2, 2]

Figure 8.2-7

Using the [[ Zero] function of the calculator, you obtainx= 1 .25331 is a zero of f′on

0,

π

2

]

. Sincef′(x)=cos(x^2 ) is an even function,x=−1.25331 is also a zero on

[

−

π

2

,0

]

.

(See Figure 8.2-8.)

• 1.2533 1.2533 π


• 2 2
  π

f′ –

f

+ –

[]

decr. incr. decr.

x

Figure 8.2-8

Thus,fis increasing on [− 1 .2533, 1.2533].

TIP • Be sure to bubble in the right grid. You have to be careful in filling in the bubbles,
especially when you skip a question.

First Derivative Test and Second Derivative Test for Relative Extrema

First Derivative Test for Relative Extrema

Letf be a continuous function andcbe a critical number off. (See Figure 8.2-9.)

f′ > 0 f′ < 0 f′ > 0 f′ < 0 f′ < 0 f′ > 0 f′ > 0

f′ = 0

rel. min.

rel. max.

f′ = 0

f′ = undefined

rel. min.

f′ = undefined

rel. min.

f′ < 0

Figure 8.2-9