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

MA 3972-MA-Book April 11, 2018 16:5

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

14.1 Average Value of a Function

Main Concepts:Mean Value Theorem for Integrals, Average Value of a Function on [a,b]

Mean Value Theorem for Integrals

Iffis continuous on [a,b], then there exists a numbercin [a,b] such that

∫b

a

f(x)dx=

f(c)(b−a). (See Figure 14.1-1.)

acb

x

0

y

(c, f(c))

f(x)

Figure 14.1-1

Example 1

Givenf(x)=

√

x−1, verify the hypotheses of the Mean Value Theorem for Integrals for

f on [1, 10] and find the value ofcas indicated in the theorem.

The functionfis continuous forx≥1, thus:

∫ 10

1

√

x− 1 dx= f(c)(10−1)

2(x−1)^1 /^2

3

] 10

1

= 9 f(c)

2

3

[

(10−1)^1 /^2 − 0

]

= 9 f(c)

18 = 9 f(c);

2 = f(c);

2 =

√

c−1;

4 =c− 1

5 =c.