EXERCISE: SERIES
1.
2.
a)
Derive the first three nonzero terms of Taylor’s Series of ln 3 x centered at 2.
b)
2 x Expand ln 3 x until the third term.
Given that P( x )
5 3 .Expand P( x ) as a series in ascending power of x , 2( x 1) 2( x 3)
up to the term in x 3 . 1 3 x)
3.
Derive the Maclaurin series for (1
4.
a)
Give the Maclaurin series of the function e3x and cos(2x) .
b)
Find the first non-zero terms of Maclaurin series of e3 x cos( 2x ) .
5.
until the fourth term.
Show that by using Maclaurin series
1 xk
1 kx
kk 1 2 kk 1k 2 3 kk 1k 2k 3 4 x x x ... 2! 3! 4! 7
6.
7.
1 Ratio of the seventh coefficient to the fifth coefficient in x 2 expansion is 1:20 px when the series is written in ascending power of x. Determine the possible values of p.
Without using calculator, find the exact value of
4
5 1
320
5 1
4
.
1
8.
a)
1 x 3 3 Expand until the term of x . 1 2x
b)
Approximate the value of
3
11 by substituting x
1 in question (a). 10 1
9.
Using binomial expansion, estimate the percentage of change in the gradient of y x 3 if x is decreasing 3.5%.
10.
Each side length of a cube is decreased by 1.2%. Approximate the percentage of change in the volume and surface area.
EXERCISE: SERIES
ANSWERS: 1.
x 22 x 23
a)
ln3 x x 2
b)
x 2 x 2 2 x 2 ln3 x x 2 2 3
2
3
3
4
7 23 2 67 3 x x x 3 9 27
2.
P( x) 3
3.
(1 x ) 3 1
4.
a)
1
1 1 5 3 x x2 x 3 9 81
e3 x 1 3x
9 2 27 3 x x 2 6
cos(2x) 1 2x2
2 4 4 6 x x 3 45
b)
e3 x cos(2x) 1 3x
6.
p
1 2
7.
6
5 2 x 2
1
8.
a)
1 x 3 2 16 3 x 1 x x 9 1 2x
b)
5003 2250
or
2.224
9.
gradient of y increased by 2.4%
10.
volume decreased by 3.6% ; surface area decreased by 2.4%