13 – LINEAR APPROXIMATION
AND DIFFERENTIALS
MATHEMATICS 201-NYA-05
PHILIP FOTH
1. Find the linearization L( x) of a given function at a given point:
(b) g ( x) 
(a) f ( x)  x4  3x2 , a  1
(d) f ( x)  cos x , a 

2
(e) h( x) 
6
, a4
x
ex
, a0
ex  1
(c)
k ( x)  x 4/3 , a  8
(f) g ( x)  x  sec x , a  
2. Use a suitable linear approximation to estimate the following value.
(a)
26
(b) 1.9985
(c) e0.03
(d)
3
63.4
(e) sin 2o
(f) cot 93o
(g) 803/4
3. Find the differential dy and use it to approximate y :
(a) y  x 2  5 from x  2 to x  2.03
(c) y  tan x from x  45o to x  46o
(b) y 
2
from x  0 to x  0.04
1  x2
(d) y  e x /5 from x  0 to x  0.05
4. The edge of a cube was found to be 40 cm with a possible error in measurement of 0.5 cm.
Use differentials to estimate the maximum possible error, relative error, and percentage error in
computing: (a) the volume of the cube (b) the surface area of the cube.
5. The radius of a disk was measured at 24 m with a possible error in measurement of 0.2 m.
Use differentials to estimate the maximum error in the calculated area of the disk. Also find the
relative error.
6. The circumference of a sphere was measured at 80 cm with a possible error of 0.5 cm. Use
differentials to estimate the maximum possible error in computing:
(a) the volume of the sphere (b) the surface area of the sphere.
7. One leg of a right triangle is known to be 90 cm long and the opposite angle is measured at
60o, with a possible error in measurement of 1o. Use differentials to estimate the error in
computing the length of the hypotenuse. Also, find the relative error.
ANSWERS
3
(b) L( x)  3  ( x  4)
8
1 x
(e) L( x)  
2 4
1. (a) L( x)  2  2( x  1)


(d) L( x)    x  
2

2. (a) 5.1 (b) 31.84
3. (a) dy 
x dx
x2  5
(c) 1.03
, y  0.02
(c) dy  sec2 x dx , y 
4. (a) 2400 cm3,
5. 9.6  m2,
1
60
1600
cm3
6. (a)
7.

2


cm,
3
180 3

90
3
, 3.75 %
80
(b)
80

cm2
(d)
319
80
(e)
(b) dy 
(d) dy 

90
8
(c) L( x)  16  ( x  8)
3
(f) L( x)    ( x   )
(f) 

60
4 x dx
, y  0
(1  x 2 )2
e x /5 dx
, y  0.01
5
(b) 240 cm2,
1
, 2.5 %
40
(g)
107
4