Cambridge Additional Mathematics

366 Introduction to differential calculus (Chapter 13)

Review set 13B

1 Evaluate the limits:

a lim

h! 0

h^3 ¡ 3 h

h

b lim

x! 1

3 x^2 ¡ 3 x

x¡ 1

c lim

x! 2

x^2 ¡ 3 x+2

2 ¡x

2 Given f(x)=5x¡x^2 , find f^0 (1) from first principles.

3aGiven y=2x^2 ¡ 1 , find

dy

dx

from first principles.

b Hence state the gradient of the tangent to y=2x^2 ¡ 1 at the point where x=4.

c For what value ofxis the gradient of the tangent to y=2x^2 ¡ 1 equal to¡ 12?

4 Differentiate with respect tox: a y=x^3

p

1 ¡x^2 b y=x

(^2) ¡ 3 x
p
x+1
5 Find
d^2 y
dx^2
for: a y=3x^4 ¡
2
x
b y=x^3 ¡x+
1
p
x
6 Find all points on the curve y=xex where the gradient of the tangent is 2 e.
7 Differentiate with respect tox: a f(x) = ln(ex+3) b f(x)=ln
·
(x+2)^3
x
̧
8 Suppose y=
³
x¡
1
x
́ 4

. Find
dy
dx

when x=1.

9 Find

dy

dx

if: a y=ln(x^3 ¡ 3 x) b y=

ex

x^2

10 Findxif f^00 (x)=0and f(x)=2x^4 ¡ 4 x^3 ¡ 9 x^2 +4x+7.

11 If f(x)=x¡cosx, find

a f(¼) b f^0 (¼ 2 ) c f^00 (^34 ¼)

12 a Find f^0 (x) and f^00 (x) for f(x)=

p

xcos(4x).

b Hence find f^0 ( 16 ¼) and f^00 (¼ 8 ).

13 Suppose y= 3 sin 2x+ 2 cos 2x. Show that 4 y+d

(^2) y
dx^2
=0.
14 Consider f(x)=
6 x
3+x^2

. Find the value(s) ofxwhen:

a f(x)=¡^12 b f^0 (x)=0 c f^00 (x)=0

15 The functionfis defined by f:x 7 !¡10 sin 2xcos 2x, 06 x 6 ¼.

a Write down an expression for f(x) in the form ksin 4x.

b Solve f^0 (x)=0, giving exact answers.

16 Given thataandbare constants, differentiate y= 3 sinbx¡acos 2x with respect tox.

Findaandbif y+

d^2 y

dx^2

= 6 cos 2x.

cyan magenta yellow black

(^05255075950525507595)
100 100
(^05255075950525507595)
100 100 4037 Cambridge
Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_13\366CamAdd_13.cdr Tuesday, 7 January 2014 12:07:46 PM BRIAN