chapter 4 answers edxcel fp2 textbook
TRANSCRIPT
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