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

MA 3972-MA-Book May 7, 2018 9:52

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

Example 2

Determine the intervals on which the given function is continuous:

f(x)=

⎧

⎨

⎩

x^2 + 3 x− 10

x− 2

, x= 2

10, x= 2

Check the three conditions of continuity atx=2:

Condition 1:f(2)=10.

Condition 2: lim

x→ 2

x^2 + 3 x− 10

x− 2

=lim

x→ 2

(x+5)(x−2)

x− 2

=lim

x→ 2

(x+5)=7.

Condition 3:f(2)=limx→ 2 f(x). Thus, f(x) is discontinuous atx=2.

The function is continuous on (−∞, 2) and (2,∞). Verify your result with a calculator.

(See Figure 6.3-2.)

[–8, 12] by [–3, 17]

Figure 6.3-2

TIP • Remember that d

dx

(

1

x

)

=−

1

x^2

and

∫

1

x

dx=ln|x|+C.

Example 3

For what value ofkis the function f(x)=

{

x^2 − 2 x, x≤ 6

2 x+k, x> 6

continuous atx=6?

Forf(x) to be continuous atx=6, it must satisfy the three conditions of continuity:

Condition 1:f(6)= 62 −2(6)=24.

Condition 2: limx→ 6 −(x^2 − 2 x)=24; thus limx→ 6 −(2x+k) must also be 24 in order for the limx→ 6 f(x)

to equal 24. Thus, limx→ 6 −(2x+k)=24 which implies 2(6)+k=24 andk=12.

Therefore, ifk=12,

Condition (3):f(6)=limx→ 6 f(x) is also satisfied.

Example 4

Givenf(x) as shown in Figure 6.3-3, (a) findf(3) and limx→ 3 f(x), and (b) determine iff(x)

is continuous atx=3? Explain your answer.

(a) The graph off(x) shows thatf(3)=5 and the limx→ 3 f(x)=1. (b) Sincef(3)=limx→ 3 f(x),

f(x) is discontinuous atx=3.