1

Answers to Chapter 4

Exercise 4A 1 y x2 c where c is constant

x

yy � x2 � 3

y � x2 � 2

y � x2 � 1

y � x2

y � x2 � 1

y � x2 � 2

�2

�1

1

2

3

O

2 y Aex where A is constant

x

y y � 3ex

y � �3ex

y � 2ex

y � �2ex

y � ex

y � �ex

1

�1

�2

�3

2

3

O

3 y x3 __

3 c where c is constant

y � � 2

O x

x3

3

y

�4

�3

�2

�1

1

2

3

4

�2 2

y � x3

3

y � � 1x3

3

y � � 1x3

3

y � � 2x3

3

4 y ln Ax, where A is constant

5 y Ax2, where A is constant

O x

yy � 2x2

12

y � x2

y � �2x2

y � �x2

y � x2

12y � � x2

6 y2 x2 2c, where c is constant

O x

y

y2 � x2 � 1

y2 � x2 � �1

y2 � x2 � 16

y2 � x2 � �16

y2 � x2 � 0

y2 � x2 � 4

�4

�3

�2

�1

1

2

3

4

�4 �2�3 �1 2 41 3 y2 � x2 � �4

Answers to Chapter 4

O x

y

�4�3�2�1

1234

�4�6

y � ln 3xy � ln 2xy � ln x

y � ln x

�2 2 4 6

13

y � ln x

y � ln (�3x)y � ln (�2x)

y � ln (�x)

y � ln (� x)y � ln (� x) 1

213

12

2

ANSWERS

7 y ln 1 ________ (x c)

, where c is constant

O x

yy � ln

�4

�3

�2

�1

1

2

3

4

�4�6 �2�3�5 �1 21

12 � x

y � ln 1�x

y � ln 11 � xy � ln 1

(�1 � x)

y � ln( )1�2 � x

8 y Ax _____ x 1 , x 0, where A is constant

O x

y

y �

2

1

3

44x

x � 1

y � 3xx � 1

y � 2xx � 1

y � xx � 1

9 y sin x c, where c is constant

O x

y

y � sin x � 1

y � sin x

y � sin x � 1

y � sin x � 2

10 y A sin x, where A is constant

O x

y

y � 3 sin x

y � �3 sin x

y � �2 sin xy � �sin x

y � 2 sin xy � sin x

�4

�3

�2

�1

1

2

3

4

π2 π

11 x tan t c for __ 2 t __

2 , where c is constant

O t

x � tan t � 2

x � tan t � 2

x � tan t � 1

x � tan t � 1

x � tan t

x

�4

�6

�2

2

4

6

� π2

π2

12 x Aet _______

1 Aet , where A is constant

O t

x

x �

0.5

1

3et

1 � 3et

x � et

1 � et

x � et

1 � et

12

12

13 a

O x

y

y2 � x

y2 � 4x

y2 � 9xy2 � 16x

b y2 9x14 a y2 5x2 45

ANSWERS

3

b

O x

y

�4

�5

�6

�7

�3

�2

�1

1

2

3

4

5

6

7

�3 3

Exercise 4BIn the following answers, c and A are constants.

1 y 1 __ x sin x c __ x

2 y xe2x e2x cex

3 y 3x cosec x c cosec x 4 y xex cx

5 y ln [ 1 __ 2 c __

x2 ] 6 y √

___________

( 1 __ 6 x2 c ___

2x ) 7 a y 1 1 __

x c __

x2

b 1 1 __ x 1 ___

4x2

1 1 __ x 1 __

x2

y 1 1 __ x 5 __

x2

O x

y

y � 1

y � 1 � �

(�2, )

� 12

1x

14x2

y � 1 � �1x

1x2

34

y � 1 � �1x

5x2

8 a y ln

A(x 1) ________

(x 2) __________

ln x

b ln 16 ___ 3 (x 1)

_______ (x 2)

_________

ln x

Exercise 4CIn the following answers, c is a constant. 1 y 1 _ 3 e

x ce2x

2 y cot x c cosec x 3 y xecos x cecos x

4 y e2x cex

5 y ( x2 __

2 c ) cos x

6 y 1 __ x ln x c __ x

7 y x ln (x 2) cx 8 y 1 _ 4 x cx 1 _ 3

9 y (x 2) ln (x 2) c (x 2)

10 y 1 __ x3 e x 1 __

x4 e x c __ x4

11 y 1 __ x e x 1 __ x2 e x 1 __

x2

12 y 1 ___ 3x2 4x ___

3

13 a y 1 _ 3 (x2 1)2 c _______

(x2 1)

b y 1 _ 3 (x2 1)2 2 _________

3(x2 1)

14 a y 1 c ____________ sec x tan x

b y 1 1 ____________ sec x tan x or y 1 cos x _________ 1 sin x

Exercise 4DIn the following answers, c is a constant and A is a positive constant. 1 y2 2x2 (ln x c) 2 y3 3x3 (ln x c)

3 y x _______ ln x c

4 y3 x3 (A x 1)

5 y √______

cos x ______ 4x c

6 x (1 c e 1 _ 6 t3

)2

7 y 4x ____________ 4c (x 1)4

8 a y √______

x c ______ 1 x2

b y √_______

x 4 ______ 1 x2

4

ANSWERS

Mixed exercise 4E 1 (x2 16 )

1 _ 2 c, where c is a constant

x

y y � x2 � 16 � 2

y � x2 � 16 � 4

y � x2 � 16

4

6

O

2 a y e 1 _ 2 x

2

b y 2e 1 _ 2 x

2

c y 3e 1 _ 2 x

2

x

y

y � e x2

2

12

1

3

O

y � 2e x212y � 3e x21

2

3 v 1 __ k [(g ku)ekt g]

t

�gk

v

O

4 y 2 sin x c cos x, where c is a constant 5 y 5 c (1 x2 )

1 _ 2 , where c is a constant

6 y x __ 2 c __ x , where c is a constant

7 y 2 __ 5 x

3 _ 2 c __ x , where c is a constant

8 y 1 _ 2 c e x2

, where c is a constant

9 y 2x c x √______

1 x2 , where c is a constant

10 a q k e t

__ Rc , where k is a constant

b q Ec k e t

__ Rc , where k is a constant

c q c ___________ (1 R2p2c2)

(cos pt Rpc sin pt) k1 e t

__ Rc ,

where k1 is constant

11 a y k _____ a ex ceax, where c is a constant

b y (kx c)eax, where c is a constant

c y kx n 1 ______

n 1 eax ceax, where c is a constant

12 y 1 ____________ 1 cx ln x , where c is a constant

13 y √_____

c x _____ cos x , where c is a constant

14 2y2 (ln y c) x2 0, where c is a constant

15 y ___

2x 1 __ 2 ln y 1 __

2 ln x c, where c is a constant

16 ln y 3x2 ___

2y2 c, where c is a constant

17 y 1 _____ x c x, where c is a constant

18 y x 2 1 Ae2x ________

1 Ae2x , where A is a positive constant