492 Answers

EXERCISE 13G

1af^0 (x)=2x¡ 1 bf^0 (x)=4x+2

c f^0 (x)=2x(x+1)

(^12)
+^12 x^2 (x+1)
¡^12
2ady
dx
=2x(2x¡1) + 2x^2
b dy
dx
= 4(2x+1)^3 +24x(2x+1)^2
c dy
dx
=2x(3¡x)
(^12)
¡^12 x^2 (3¡x)
¡^12
d dy
dx
=^12 x
¡^12
(x¡3)^2 +2
p
x(x¡3)
e dy
dx
=10x(3x^2 ¡1)^2 +60x^3 (3x^2 ¡1)
f dy
dx
=^12 x
¡^12
(x¡x^2 )^3 +3
p
x(x¡x^2 )^2 (1¡ 2 x)
3a¡ 48 b 40614 c^133 d^112
4bx=3or^35 c x 60 5 x=¡ 1 andx=¡^53
EXERCISE 13H
1a
dy
dx

7
(2¡x)^2
b
dy
dx

2 x(2x+1)¡ 2 x^2
(2x+1)^2
c
dy
dx

(x^2 ¡3)¡ 2 x^2
(x^2 ¡3)^2
d dy
dx

1
2 x
¡^12
(1¡ 2 x)+2
p
x
(1¡ 2 x)^2
e dy
dx
=^2 x(3x¡x
(^2) )¡(x (^2) ¡3)(3¡ 2 x)
(3x¡x^2 )^2
f
dy
dx=
(1¡ 3 x)
(^12)
+^32 x(1¡ 3 x)
¡^12
1 ¡ 3 x
2a 1 b 1 c ¡ 3247 d¡^2827
3b inever fdy
dx
is undefined at x=¡ 1 g
iix 60 andx=1
4b ix=¡ 2 §
p
11 iix=¡ 2
EXERCISE 13I
1af^0 (x)=4e^4 x bf^0 (x)=ex
c f^0 (x)=¡ 2 e¡^2 x df^0 (x)=^12 e
x 2
ef^0 (x)=¡e
¡x 2
f f^0 (x)=2e¡x
gf^0 (x)=2e
x 2
+3e¡x hf^0 (x)=e
x¡e¡x
2
i f^0 (x)=¡ 2 xe¡x^2 j f^0 (x)=e
x^1
£¡^1
x^2
kf^0 (x)=20e^2 x l f^0 (x)=40e¡^2 x
m f^0 (x)=2e^2 x+1 nf^0 (x)=^14 e
x 4
o f^0 (x)=¡ 4 xe^1 ¡^2 x
2
pf^0 (x)=¡ 0 : 02 e¡^0 :^02 x
2aex+xex b 3 x^2 e¡x¡x^3 e¡x
c xe
x¡ex
x^2
d^1 ¡x
ex
e 2 xe^3 x+3x^2 e^3 x f
xex¡^12 ex
x
p
x
g^12 x
¡^12
e¡x¡x
(^12)
e¡x h e
x+2+2e¡x
(e¡x+1)^2
3a 108 b¡ 1 c p^9194 k=¡ 9
5a
dy
dx=2
xln 2 6 P=(0,0)or(2,^4
e^2 )
EXERCISE 13J
1a
dy
dx

1
x
b
dy
dx

2
2 x+1
c
dy
dx

1 ¡ 2 x
x¡x^2
d dy
dx
=¡^2
x
e dy
dx
=2xlnx+x
f dy
dx
=^1 ¡lnx
2 x^2
g dy
dx
=exlnx+e
x
x
h
dy
dx

2lnx
x
i
dy
dx

1
2 x
p
lnx
j
dy
dx=
e¡x
x ¡e
¡xlnx k dy
dx=
ln(2x)
2 px +
1
px
l dy
dx
=plnx¡^2
x(lnx)^2
m dy
dx
=^4
1 ¡x
n
dy
dx
=ln(x^2 +1)+
2 x^2
x^2 +1
2a
dy
dx
=ln5 b
dy
dx

3
x
c
dy
dx

4 x^3 +1
x^4 +x
d dy
dx
=^1
x¡ 2
e dy
dx
=^6
2 x+1
[ln(2x+1)]^2
f dy
dx
=^1 ¡ln(4x)
x^2
g dy
dx
=¡^1
x
h
dy
dx

1
xlnx
i
dy
dx

¡ 1
x(lnx)^2
3a
dy
dx

¡ 1
1 ¡ 2 x
b
dy
dx

¡ 2
2 x+3
c
dy
dx
=1+
1
2 x
d dy
dx
=^1
x
¡^1
2(2¡x)
e dy
dx
=^1
x+3
¡^1
x¡ 1
f
dy
dx

2
x

• 
  

  1
  3 ¡x
  g f^0 (x)=
  9
  3 x¡ 4
  h f^0 (x)=^1
  x
  +^2 x
  x^2 +1
  i f^0 (x)=^2 x+2
  x^2 +2x
  ¡^1
  x¡ 5
  4a 2 b¡^535 a=3, b=¡e
  EXERCISE 13K
  1ady
  dx
  = 2 cos(2x) b dy
  dx
  = cosx¡sinx
  c
  dy
  dx
  =¡3 sin(3x)¡cosx d
  dy
  dx
  = cos(x+1)
  e
  dy
  dx
  = 2 sin(3¡ 2 x) f
  dy
  dx

  
  

  5
  cos^2 (5x)
  g dy
  dx
  =^12 cos
  ¡x
  2
  ¢

  
  


• 3 sinx h dy
  dx
  =^3 ¼
  cos^2 (¼x)
  i dy
  dx
  = 4 cosx+ 2 sin(2x)
  2a 2 x¡sinx b^1
  cos^2 x
  ¡3 cosx
  c excosx¡exsinx d¡e¡xsinx+e¡xcosx
  e cosx
  sinx
  f 2 e^2 xtanx+ e
  2 x
  cos^2 x
  g 3 cos(3x)
  h ¡^12 sin
  ¡x
  2
  ¢
  i
  6
  cos^2 (2x)
  j cosx¡xsinx
  k xcosx¡sinx
  x^2
  l tanx+ x
  cos^2 x
  cyan magenta yellow black
  (^05255075950525507595)
  100 100
  (^05255075950525507595)
  100 100 IB HL OPT
  Sets Relations Groups
  Y:\HAESE\CAM4037\CamAdd_AN\492CamAdd_AN.cdr Tuesday, 8 April 2014 8:39:25 AM BRIAN