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8/13/2019 sem0203test1
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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