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KOLEJ UNIVERSITI TEKNOLOGI TUN HUSSEIN ONN

PUSAT PENGAJIAN SAINS

SEMESTER I 2002/2003

BSM 1613 TEST 1 (15%) Time: 50 MIN

Instruction: Answer all of the questions below.

Q1 Find the given limits. Do notuse LHopitals rule.

(a)4

2

11

lim4

x

xx

[hint: ))(()( axaxax += ]

(6 marks)

(b)xx

xx

x sin2

cos2lim

+

[hint: 0sin

lim = x

x

x]

(6 marks)

Q2 Find a value ofAsuch that is continuous for allx.

>

=

1,

1,1)(

2

2

xx

xAxxf

(6 marks)

Q3 (a) By using natural logarithm ln, finddx

dyif

xx

ay

x

4tanh3sinh= , where ais a scalar.

(8 marks)

(b) If txty sin),cos(cos == is a parametric equation for a curve, then find

dx

dyand

2

2

dx

ydin terms of t.

(10 marks)

Q4 Given3

2 3

x

xy

= .

(a) Show thatyhas a vertical asymptote at 0=x , and a horizontal asymptote at.0=y

(4 marks)

(b) Doesyhas an oblique asymptote (asimptot condong)? Explain why.

(3 marks)(c) Show that the critical points foryare (3, 2/9) and (-3, -2/9). Determine

the maximum and minimum point.

(7 marks)

(d) If yhas inflection points at 2.4=x , draw the graph of yand show theinflection points in that graph.

(10 marks)

SELAMAT MAJU JAYA

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Marking Scheme

Q1(a)

16

1

)2(2

1lim

)2)(2(

2

2

lim4

2

11

lim444

=

+

=

+

=

xxxx

x

x

x

xxxx

Q1(b)

1sin

lim21

cos

lim21

sin21

cos

21lim

sin2

cos2lim =

+

=

+

=+

x

xx

x

x

xx

x

xx

xx

x

x

xx

Q2

211)1(

1lim1lim

2

2

1

2

1

==

==+

AA

xAxxx

Q3(a)

=

=

=

x

xxa

xx

a

dx

dy

x

x

x

xa

dx

dy

y

xxaxy

x

4tanh

4sech43coth3ln

4tanh3sinh2

1

4tanh

4sech4

3sinh

3cosh3ln

2

11

)4tanhln3sinhlnln(2

1ln

2

2

Q3(b)

t

ttttt

ttt

dx

d

dx

yd

ttt

tt

dx

dy

tdtdx

ttttdtdy

cos

tansin)cos(cossec)sin(cos

cos

1]tan)[sin(cos

tan)sin(coscos

sin)sin(cos

cos/

sin)sin(cos)sin)(sin(cos/

2

2

2 ==

==

=

==

Q4(a) when vertical asymptote at0;0 3 == xx 0=x

01

/3/1lim

3lim

3

3

2

=

=

xx

x

x

xx horizontal asymptote at 0=y

Q4(b)0

1

/3/1limlim

42

=

==

xx

x

ym

xx there is no oblique asymptote

Q4(c) 30)3)(3(09

'4

2

==+=

= xxxx

xy

5

2 362"

x

xy

=

when 0":9/2:3