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

MA 3972-MA-Book April 11, 2018 15:57

262 STEP 4. Review the Knowledge You Need to Score High

Example 4

If

∫k

−k

f(x)dx= 2

∫k

0

f(x)dxfor all values ofk, then which of the following could be the

graph off? (See Figure 12.3-1.)

yy

yyy

x x

xxx

0

(A) (B)

(D) (E)

(C)

0

000

Figure 12.3-1

∫k

−k

f(x)dx=

∫ 0

−k

f(x)dx+

∫k

0

f(x)dx

Since

∫k

−k

f(x)dx= 2

∫k

0

f(x)dx, then

∫k

0

f(x)dx=

∫ 0

−k

f(x)dx.

Thusfis an even function. Choice (C).

12.4 Rapid Review

1. Evaluate

∫x

π/ 2

costdt.

Answer: sint]xx/ 2 =sinx−sin

(

π

2

)

=sinx−1.

2. Evaluate

∫ 1

0

1

x+ 1

dx.

Answer: ln(x+1)]^10 =ln 2−ln 1=ln 2.

3. IfG(x)=

∫x

0

(2t+1)^3 /^2 dt, findG′(4).

Answer: G′(x)=(2x+1)^3 /^2 andG′(4)= 93 /^2 =27.

4. If

∫k

1

2 xdx=8, findk.

Answer: x^2

]k

1 =^8 ⇒k

(^2) − 1 = 8 ⇒k=±3.