17 Maclaurin Series Qn

7/29/2019 17 Maclaurin Series Qn

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Topic: Macluarins Series

1) SRJC/BT2/10/2/4b

tan1

3x2

Given y e . Prove that (19x )d y 3(16x)

dy.

dx2 dx

Obtain the Maclaurins series of y etan1 3x up to and including the first three

terms. [4]

Ans: 1 3x 9

x2 2

2) IJC/BT2/10/1/1

Find the constants a and b such that, when x is small,

Ans: a 3

, b 11

cos 2 x

13x1 ax bx2 .

[4]

2 83) PJC/BT2/10/1/1

Given that is sufficiently small such that2cos

b 3

192 ,

a tan 25 250find the values of a and b. [5]

Ans: a 5;b 35

4) ACJC/07/2/4i)

Given that y tan x , show thatd

2y

dx2 2 ydy

. Hence find Maclaurins seriesdx

for y, up to and including the term in x3. [5]

ii) Using the standard series expansion for ln1x and Maclaurins series fory, find the series expansion of ln1 tan x, in ascending powers of x up to

iii)

and including the term in x3. [2]

sec2 2xHence show that the first three non-zero terms in the expansion of

1 tan 2xare 12x 8x2 . [3]

Ans: i) x 1

x3 ... , ii)3x

1x2

2x3 ...2 3


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5) NJC/07/2/2

Given that y

1

1x2, show that

i)1 x2

d yxy ;

d x [1]ii) d3 y 0

d x3

when x 0[3]

Obtain the Maclaurins expansion of y up to and including the term in x2. [1]

Hence find the Maclaurins expansion of y sin1 x up to and including the term

in x3 . (Integration is required) [2]

3

Ans: 11

x2 ,

26) NYJC/07/1/4

x x6

d3 y d2 y dy

If yln(cos x) , prove that dx32dx

2

0 .dx [2]

Hence or otherwise, obtain the Maclaurins expansion of y in terms of x up to and

including the term in x4. [3]

2 2

Usingx

, show that ln 2

1

.

4 16

96

x2 x4

[3]

Ans: ...2 12

7) MJC/BT2/10/2/11

Given that y 3 2e2x 4.i) Show that

4y3 dy 2 y4 6 0 .

dx [2]

ii) By further differentiation of this result, or otherwise, find Maclaurins series

iii)for y up to and including the term in x

2 . [3]1

Deduce the equation of the tangent to the curve y 32e2x 4 at the pointwhere x 0 . [1]

Ans: 1x 5

x2 ... ,

2y 1 x


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d

dx

2

x

8) AJC/10/1/3

The variables x and y are related by

ydy

etan1 2x

dxand the gradient of the curve of y = f(x) at the y-intercept is 2. Prove that

2

1 4x 2 y y dy dy2 y .

dx2 dx

By further differentiation, find the series expansion of y in terms of x up to and

including the term in x3. Hence write down the equation of tangent to the curve at

the y-intercept. [6]

Ans:12x 2x2

28x3 , y 1 2x

2 3 29) SAJC/BT2/10/1/8

It is given that y lncosx, where

x

.

4 4i) d3 y d2 y dy

Prove that 2dx3 dx2

0 .dx [2]

ii) Find the Maclaurins series forlncosx, up to and including the term in x4 . [3]iii) Deduce that the Maclaurins series for ln1cos2x up to and including the

term in x4 is ln 2 x2 1x4.

6 [2]

Ans: y 1

x2

1 x4

2 12

10) NYJC/BT2/10/2/3

It is given that y tan(1 ex ) .i) d2 y dy

Show thatdx2

(1 2e y)dx [3]

ii) Find the Maclaurins series for y in ascending powers of x, up to and including

the term in x2 . [2]

iii)Expand

tan(1ex )

1 2xas a series in ascending powers of x, up to and including

the term in x2 and state the range of x for which this expansion is valid. [3]

Hence, by using the expansion in (iii), find the range of x such thattan(1ex ) 1

x .1 2x 3 [2]

Ans: ii) x 1

x2 ... , iii) x3 x2 , x 1 , 0.5 x 0.471or 0.471 x 0.52 2 2


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11) IJC/BT2/10/1/7

The points Q and R are fixed in a plane, and S is a variable point which moves in the

plane so that QS kRS , where k is a constant such that 0 k1. The angles SQR andSRQ, measured in radians, are x and y respectively (see diagram).

S

x yQ R

i) Show that sin y ksin x . Deduce that y sin 1 k for all positions of S. [2]ii) By differentiation of the equation in part (i), or otherwise, show that

d

2y

dy

cos y sin y dx2 dx k sin x . [2]

iii) By using Maclaurins series, or otherwise, show that if x is sufficiently small

for powers of x above x3 to be neglected, then

kk2 1y kx

6x3 .

[4]

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17 Maclaurin Series Qn