Cambridge Additional Mathematics

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Answers 493

3a 2 xcos(x^2 ) b¡^1
2
p
x
sin(
p
x) c ¡ sinx
2
p
cosx
d2 sinxcosx e¡3 sinxcos^2 x
f¡sinxsin(2x) + 2 cosxcos(2x)
gsinxsin(cosx) h¡12 sin(4x) cos^2 (4x)
i ¡cosx
sin^2 x
j 2 sin(2x)
cos^2 (2x)
k¡8 cos(2x)
sin^3 (2x)

l ¡^12
cos^2 (x 2 ) tan^4 (x 2 )
4a¡^98 b 0
EXERCISE 13L
1af^00 (x)=6 bf^00 (x)=^3
2 x

(^52)
cf^00 (x)=12x¡ 6 df^00 (x)=
12 ¡ 6 x
x^4
ef^00 (x)=24¡ 48 x ff^00 (x)=^20
(2x¡1)^3
2a
d^2 y
dx^2
=¡ 6 x b
d^2 y
dx^2
=2¡
30
x^4
c
d^2 y
dx^2
=¡^94 x
¡^52
d
d^2 y
dx^2


8
x^3
e d
(^2) y
dx^2
=6(x^2 ¡ 3 x)(5x^2 ¡ 15 x+9)
f d
(^2) y
dx^2
=2+^2
(1¡x)^3
3af(2) = 9 bf^0 (2) = 10 c f^00 (2) = 12
5ax=1
bx=0,§
p
6
6 x ¡^101
f(x) ¡ 0 +
f^0 (x) + ¡ +
f^00 (x) ¡ 0 +
7bf^00 (x) = 3 sinxcos 2x+ 6 cosxsin 2x
8a
d^2 y
dx^2


1
x^2
b
d^2 y
dx^2


1
x
c
d^2 y
dx^2


2
x^2
(1¡lnx)
9af(1) = 0 bf^0 (1) = 3 c f^00 (1) = 0
10 Hint: Find
dy
dx
and
d^2 y
dx^2
and substitute into the equation.
REVIEW SET 13A
1a¡ 1 b¡ 1 c 8
2af^0 (x)=2x+2 b
dy
dx
=¡ 6 x
3af^0 (t)=¡ 9 : 6 tms¡^1
bf^0 (2) =¡ 19 : 2 ms¡^1
(the negative sign indicates travelling downwards)
4af(3) =¡ 17 bf^0 (3) =¡ 17 c f^00 (3) =¡ 6
5ady
dx
=6x¡ 4 x^3 b dy
dx
=1+^1
x^2
6 (0,0) 7a
dy
dx
=3x^2 ex^3 +2 b
dy
dx


1
x+3
¡
2
x
9a5+3x¡^2 b4(3x^2 +x)^3 (6x+1)
c 2 x(1¡x^2 )^3 ¡ 6 x(x^2 + 1)(1¡x^2 )^2
10 (¡ 2 ,19)and(1,¡2)
11 a
dy
dx
=¡2(5¡ 4 x)
¡^12
b
d^2 y
dx^2
=¡4(5¡ 4 x)
¡^32
12 a5 cos(5x)lnx+
sin(5x)
x
bcosxcos(2x)¡2 sinxsin(2x)
c¡ 2 e¡^2 xtanx+
e¡^2 x
cos^2 x
13
p 3
2
14 af^0 (x)=8x(x^2 +3)^3
bg^0 (x)=
1
2 x(x+5)
¡^12
¡2(x+5)
(^12)
x^3
15 af^00 (2) =^234 b f^00 (2) =¡ 8 p^12
16 a 10 ¡10 cos(10x) b tanx
c5 cos(5x)ln(2x)+sin(5x)
x
REVIEW SET 13B
1a¡ 3 b 3 c ¡ 1 2 f^0 (1) = 3
3ady
dx
=4x bwhen x=4, gradient=16
cwhen gradient=¡ 12 , x=¡ 3
4a
dy
dx
=3x^2 (1¡x^2 )
(^12)
¡x^4 (1¡x^2 )
¡^12
b
dy
dx=
(2x¡3)(x+1)
(^12)
¡^12 (x^2 ¡ 3 x)(x+1)
¡^12
x+1
5a
d^2 y
dx^2
=36x^2 ¡
4
x^3
b
d^2 y
dx^2
=6x+^34 x
¡^52
6 (1,e) 7af^0 (x)= e
x
ex+3
b f^0 (x)=^3
x+2
¡^1
x
8 When x=1,
dy
dx
=0.
9ady
dx
=^3 x
(^2) ¡ 3
x^3 ¡ 3 x
b dy
dx
=e
x(x¡2)
x^3
10 x=¡^12 ,^32
11 af(¼)=¼+1 bf^0 (¼ 2 )=2 cf^00 (^34 ¼)=¡
p 2
2
12 af^0 (x)=^12 x
¡^12
cos(4x)¡ 4 x
(^12)
sin(4x),
f^00 (x)=¡^14 x
¡^32
cos(4x)¡ 4 x
¡^12
sin(4x)
¡ 16 x
(^12)
cos(4x)
bf^0 ( 16 ¼)¼¡ 0 : 455 , f^00 (¼ 8 )¼¡ 6 : 38
14 ax=¡ 6 §
p
33 b x=§
p
3 c x=0,§ 3
15 af(x)=¡5 sin 4x
bf^0 (x)=0when x= 8 ¼,^38 ¼,^58 ¼,^78 ¼, 06 x 6 ¼
16 dy
dx
=3bcos(bx)+2asin(2x),a=2, b=§ 1
EXERCISE 14A
1ay=¡ 7 x+11 bx¡ 4 y=¡ 8 c y=¡ 2 x¡ 2
dy=¡ 2 x+6 ey=¡ 5 x¡ 9 f y=¡ 5 x¡ 1
2ax+6y=57 bx+7y=26 c x¡ 3 y=¡ 11
dx+6y=43
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Sets Relations Groups
Y:\HAESE\CAM4037\CamAdd_AN\493CamAdd_AN.cdr Tuesday, 8 April 2014 8:39:31 AM BRIAN

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