5 Steps to a 5 AP Calculus BC 2019

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

[–3,8] by [–4,8]

Figure 5.3-3

Example 5

Ifg(x)=x^2 − 2 x−15, using the Intermediate Value Theorem show thatg(x) has a root in

the interval [1, 7].

Begin by findingg(1) andg(7), andg(1)=−16 andg(7)=20. Ifg(x) has a root, theng(x)

crosses thex-axis, i.e.,g(x)=0. Since− 16 ≤ 0 ≤20, by the Intermediate Value Theorem,

there exists at least one numbercin [1, 7] such thatg(c)=0. The numbercis a root ofg(x).

Example 6

A functionfis continuous on [0, 5], and some of the values off are shown below.

x 0 3 5

f − 4 b − 4

Iff(x)=−2 has no solution on [0, 5], thenbcould be

(A) 1 (B) 0 (C) − 2 (D) − 5

Ifb=−2, thenx=3 would be a solution for f(x)=−2.

Ifb=0, 1, or 3, f(x)=−2 would have two solutions forf(x)=−2.

Thus,b=−5, choice (D). (See Figure 5.3-4.)

y

x

f(x) = –2

3

2

1

(^0) 12 345
–1
–2
–3
–5
(3,1)
(3,0)
(3,–2)
(0,–4) (3,–5) (5,–4)
Figure 5.3-4