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 161

(–4, 6)

(3, 2)

f(x)

y

x

• 42 – 3 – 2 – 1 10 3

Figure 8.4-5

Example 5

Iff(x)=

∣∣

ln(x+1)

∣∣

, find lim

x→ 0 −

f′(x). (See Figure 8.4-6.)

[–2, 5] by [–2, 4]

Figure 8.4-6

The domain offis (−1,∞).

f(0)=

∣∣

ln(0+1)

∣∣

=

∣∣

ln(1)

∣∣

= 0

f(x)=

∣∣

ln(x+1)

∣∣

=

{

ln(x+1) ifx≥ 0

−ln(x+1) ifx< 0

Thus,f′(x)=

⎧

⎪⎪⎨

⎪⎪⎩

1

x+ 1

ifx≥ 0

−

1

x+ 1

ifx< 0

Therefore, lim
x→ 0 −

f′(x)=lim

x→ 0 −

(

−

1

x+ 1

)

=−1.