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AP
Worksheet
Section 4.5
Name ________________________________________hr 4
Date ____/____/____ Score:________/
= ________%
For the following problems: (a) find the linearization L (x ) of f (x ) at x  a . (b) How accurate is the approximation
L(a  0.1)  f (a  0.1) ?
1
,a 1
x
1. f ( x)  x 3  2 x  3, a  2
2. f ( x)  x 
3. f ( x)  tan x, a  
4. f ( x)  cos 1 x, a  0
5. Show that the linearization of f ( x)  (1  x) k at x = 0 is L( x)  1  kx. .
6. Use the linearization (1  x) k  1  kx to approximate the following. State how accurate your approximation is.
(a) (1.002)100
(b)
3
1.009
Use the linear approximation (1  x) k  1  kx to find an approximation for the function f (x ) for the values of x near
zero.
(b) f ( x) 
7. (a) f ( x)  (1  x) 6
8. (a) f ( x)  (4  3x)
1/ 3
(b) f ( x) 
2
1 x
2 x
(c) f ( x) 
2
1
1 x
1 

(c) f ( x)  1 

 2 x
2
Approximate the root by using a linearization centered at an appropriate nearby number.
9.
101
10.
80
In the following problems (a) find dy , and evaluate dy for the given value of x and dx.
11. y  x 3  3x, x  2, dx  0.05
13. y  xy  x  0, x  0, dx  0.01
12. y  x 1  x 2 , x  0, dx  .2
14. Find the differential of d (e 5 x  x 5 ).
In exercises 15 – 16, the function f changes value when x changes from a to a  dx.
Find
(a) the true change f  f (a  dx)  f (a). .
(b) the estimated change df  f ' (a)dx.
(c) the approximation error f  df .
15. f ( x)  x 2  2 x, a  0, dx  0.1
(a) true change f  f (a  dx)  f (a).
16. f ( x)  x 4 , a  1, dx  0.01
(a) true change f  f (a  dx)  f (a).
(b) estimated change df  f ' (a)dx.
(b) estimated change df  f ' (a)dx.
(c) approximation error f  df .
(c) approximation error f  df .
In exercises 17 – 18, write a differential formula that estimates the given change in volume or surface area. Then use the
formula to estimate the change when the dependent variable changes from 10 cm to 10.05 cm.
17. The change in the volume V  x 3 of a cube when the edge lengths change from a to a  dx.
18.The change in the lateral surface area S  2rh of a right circular cylinder when the height changes from a to a  dh
and the radius does not change.
19.Linear Approximation: Let f be a function with f (0)  1 and f ' ( x)  cos( x 2 ). (a) Find the linearization of f at
x = 0. (b) Estimate the value of f at x = 0.1. (c) Do you think the actual value of f at x = 0.1 is greater than or less than the
estimate in part (b)? Explain.