Cambridge Additional Mathematics

364 Introduction to differential calculus (Chapter 13)

EXERCISE 13L

1 Find f^00 (x) given that:

a f(x)=3x^2 ¡ 6 x+2 b f(x)=

2

p

x

¡ 1 c f(x)=2x^3 ¡ 3 x^2 ¡x+5

d f(x)=

2 ¡ 3 x

x^2

e f(x)=(1¡ 2 x)^3 f f(x)=

x+2

2 x¡ 1

2 Find

d^2 y

dx^2

given that:

a y=x¡x^3 b y=x^2 ¡

5

x^2

c y=2¡

3

p

x

d y=^4 ¡x

x

e y=(x^2 ¡ 3 x)^3 f y=x^2 ¡x+^1

1 ¡x

3 Given f(x)=x^3 ¡ 2 x+5, find:

a f(2) b f^0 (2) c f^00 (2)

4 Suppose y=Aekx whereAandkare constants. Show that:

a

dy

dx

=ky b

d^2 y

dx^2

=k^2 y

5 Find the value(s) ofxsuch that f^00 (x)=0, given:

a f(x)=2x^3 ¡ 6 x^2 +5x+1 b f(x)=

x

x^2 +2

x ¡ 1 0 1

f(x) ¡

f^0 (x)

f^00 (x)

6 Consider the function f(x)=2x^3 ¡x.

Complete the following table by indicating whether f(x), f^0 (x),

and f^00 (x) are positive (+), negative (¡), or zero ( 0 ) at the given

values ofx.

7 Suppose f(x) = 2 sin^3 x¡3 sinx.

a Show that f^0 (x)=¡3 cosxcos 2x. b Find f^00 (x).

8 Find

d^2 y

dx^2

given:

a y=¡lnx b y=xlnx c y=(lnx)^2

9 Given f(x)=x^2 ¡

1

x

, find:

a f(1) b f^0 (1) c f^00 (1)

10 If y=2e^3 x+5e^4 x, show that

d^2 y

dx^2

¡ 7

dy

dx

+12y=0.

11 If y= sin(2x+3), show that

d^2 y

dx^2

+4y=0.

12 If y= 2 sinx+ 3 cosx, show that y^00 +y=0 where y^00 represents d

(^2) y
dx^2
.
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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_13\364CamAdd_13.cdr Tuesday, 7 January 2014 11:34:40 AM BRIAN