CK-12-Calculus

http://www.ck12.org Chapter 2. Derivatives

Review Questions

1. Find the linearization of

f(x) =x

(^2) + 1
x
ata=1.

2. Find the linearization off(x) =tanxata=π.


3. Use the linearization method to show that whenx1 (much less than 1), then( 1 +x)n≈ 1 +nx. Hint: Let
   x= 0.


4. Use the result of problem #3,( 1 +x)n≈ 1 +nx, to find the approximation for the following:
   a. f(x) = ( 1 −x)^4
   b. f(x) =√ 1 −x
   c. f(x) =√ 15 +x
   d. Without using a calculator, approximate( 1. 003 )^99.


5. Use Newton’s Method to find the roots ofx^3 + 3 =0.


6. Use Newton’s Method to find the roots of−x+ 3

√

− 1 +x=0.

f′(x) =d fdx=d fdu·dudx=n(u)n−^1 ·( 1 ) =n( 1 +x)n−^1

Ifx<<1, we can usex 0 =0 and linearize around the pointf(x 0 ) =f( 0 ):

y=f(x 0 )+f′(x 0 )(x−x 0 )

y= ( 1 + 0 )n+n( 1 + 0 )n−^1 (x− 0 )

y= ( 1 )n+n( 1 )n−^1 (x)

y= 1 +nx

4..
a. 1− 4 x
b. 1−^12 x
c. 5−^52 x
d. 1. 297
5.x≈− 1. 442
6.x≈ 1 .146 andx≈ 7. 854

Texas Instruments Resources

In the CK-12 Texas Instruments Calculus FlexBook® resource, there are graphing calculator activities designed
to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9727.