Cambridge Additional Mathematics

494 Answers

3 y=21andy=¡ 6 4 (^12 , 2

p

2)

5 k=¡ 5 6 y=¡ 3 x+1

7 a=¡ 4 , b=7 8 a=2,b=4

10 ax¡ 3 y=¡ 5 bx¡ 9 y=¡ 4

c x¡ 16 y=3 dy=¡ 4

11 ay=2x¡^74 by=¡ 27 x¡^2423

c 4 x+57y= 1042 dx¡ 2 y=¡ 1

12 a=4,b=3

13 ax+ey=2 bx+3y= 3 ln 3¡ 1

c 2 x+e^2 y=^2

e^2

¡e^2

15 ay=x by=x c 2 x¡y=¼ 3 ¡

p 3

2 dx=

¼

4

16 a(¡ 4 ,¡64) b (4,¡31)

17 a f^0 (x)=2x¡

8

x^3

b x=§

p

2 c tangent is y=4

18 Ais(^23 ,0),Bis(0,¡ 2 e)

19 ay=(2a¡1)x¡a^2 +9

by=5x, contact at(3,15),y=¡ 7 x, contact at(¡ 3 ,21)

20 y=0,y=27x+54 21 y=¡

p

14 x+4

p

14

22 y=eax+ea(1¡a) so y=ex is the tangent to y=ex

from the origin.

23 aHint: They must have the samey-coordinate atx=band

the same gradient.

c a=

1

2 e

d y=e

¡^12

x¡^12

24 ¼ 63 : 43 ±

25 aHint: y=f(a)+f^0 (a)(x¡a)

bHint: Expandf(x)=4¡8(x+1)¡(x+1)^2 +2(x+1)^3

c Notice the first 2 terms inbare the same as the tangent line

found in parta.

EXERCISE 14B

1aA - local max, B - stationary inflection, C - local min.

bi

ii

2a b

cd

ef

gh

ij

3 x=¡b

2 a

, local min if a> 0 , local max ifa< 0

4 a=9

5aa=¡ 12 ,b=¡ 13

b (¡ 2 ,3)local max., (2,¡29)local min.

6alocal maximum at(1,e¡^1 )

b local maximum at(¡ 2 , 4 e¡^2 ), local minimum at(0,0)

c local minimum at(1,e)

d local maximum at(¡ 1 ,e)

7ax> 0

10 a b

cd

-2 03 x

++-- f'(x)

-4 05

• 
  
  
  
  • 
    
    
    
    • 
      
      
      
      • x
      
      
      
      
    
    
    
    
  
  
  
  

f(x)

2

x

f(x)¡

()no stationary points

O

(-2 -27),

(1 0),¡

x

f(x)

-3 -3

local min.

stationary

inflection

x O

f(x)¡

local min.

1

(¡¡¡Qr,-Qr)

O

(0 1),¡

x

f(x)¡

local max.

O

(-1 -9), (1 -9),

(0 -8),

2 x

f(x)¡

local min. local min.

local

max.

-2

O

(0 -2),

x

f(x)

-~`2 ~`2

local min.

O

(0 1),¡

x

f(x)¡

-1

stationary

inflection

O

(-1 4),¡

(1 0),¡

2

x

f(x)

local min.

local max.

-2 O

-~`2 ~`2

(-1 -1), (1 -1),

(0 0),¡

x

f(x)

local min. local min.

local

max.

O^1

x

y

y=sin 2 x

(0 0),

min.

¡ (0)¼,

min.

¡ (2¼,0)

min.

¡

(1)Es_p,

max.

(_ 1)_wp,

max.

O

¼ 2 ¼ x

1

O

y=esinx

y

max.(__wp,e)

()Es_p,

min.

Q.

1

-1

x

y

y=cos2x

¼¼ 22 ¼¼

(0 1),

max.

¡ (1)¼,

max.

¡ (2¼,1)

max.

¡

O

(_ -1)_wp,

min.

(Es_p,-1)

min.

¼ 2 ¼

1

-1

x

y

y=sinx

O

(Es_p,-1)

min.

(_ 1)_wp, max.

(2 1),¡

-7

x

f(x) stationary

inflection

O 1

1

8aGreatest value is 63 whenx=5,

least value is¡ 18 when x=2.

b Greatest value is 4 when x=3andx=0,

least value is¡ 16 when x=¡ 2.

9 P(x)=¡ 9 x^3 ¡ 9 x^2 +9x+2

cyan magenta yellow black

(^05255075950525507595)
100 100
(^05255075950525507595)
100 100 IB HL OPT
Sets Relations Groups
Y:\HAESE\CAM4037\CamAdd_AN\494CamAdd_AN.cdr Tuesday, 8 April 2014 8:57:16 AM BRIAN