Cambridge Additional Mathematics

Introduction to differential calculus (Chapter 13) 365

Review set 13A

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1 Evaluate:

a lim

x! 1

(6x¡7) b lim

h! 0

2 h^2 ¡h

h

c lim

x! 4

x^2 ¡ 16

x¡ 4

2 Find, from first principles, the derivative of:

a f(x)=x^2 +2x b y=4¡ 3 x^2

3 In theOpening Problemon page 334 , the altitude of the jumper is given by

f(t) = 452¡ 4 : 8 t^2 metres, where 06 t 63 seconds.

a Find f^0 (t) = lim

h! 0

f(t+h)¡f(t)

h

.

b Hencefind the speed of the jumper when t=2seconds.

4 If f(x)=7+x¡ 3 x^2 , find: a f(3) b f^0 (3) c f^00 (3).

5 Find

dy

dx

for: a y=3x^2 ¡x^4 b y=

x^3 ¡x

x^2

6 At what point on the curve f(x)=

x

p

x^2 +1

does the tangent have gradient 1?

7 Find dy

dx

if: a y=ex

(^3) +2
b y=ln
³
x+3
x^2
́
8 Given y=3ex¡e¡x, show that
d^2 y
dx^2
=y.
9 Differentiate with respect tox:
a 5 x¡ 3 x¡^1 b (3x^2 +x)^4 c (x^2 + 1)(1¡x^2 )^3
10 Find all points on the curve y=2x^3 +3x^2 ¡ 10 x+3 where the gradient of the tangent is 2.
11 If y=
p
5 ¡ 4 x, find: a
dy
dx
b
d^2 y
dx^2
12 Differentiate with respect tox:
a sin(5x)ln(x) b sin(x) cos(2x) c e¡^2 xtanx
13 Find the gradient of the tangent to y= sin^2 x at the point where x=¼ 3.
14 Find the derivative with respect toxof:
a f(x)=(x^2 +3)^4 b g(x)=
p
x+5
x^2
15 Find f^00 (2) for:
a f(x)=3x^2 ¡
1
x
b f(x)=
p
x
16 Differentiate with respect tox:
a 10 x¡sin(10x) b ln
³
1
cosx
́
c sin(5x)ln(2x)
4037 Cambridge
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Y:\HAESE\CAM4037\CamAdd_13\365CamAdd_13.cdr Tuesday, 7 January 2014 12:06:11 PM BRIAN