series exercise

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  xk

 1  kx 

kk  1 2 kk  1k  2 3 kk  1k  2k  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  22  x  23  

a)

ln3  x   x  2 

b)

x  2  x  2   2  x   2 ln3  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%