.
CONFIDENTIAL
$T H'b
LH3
UNIVERSITI TUN HUSSEIN ONN MALAYSIA
FINAL EXAMINATION
SEMESTER I
sESr 2012t20r3
COURSE NAME
ENGINEERING MATHEMATICS I
COURSE CODE
BDA
PROGRAMME
BACHELOR OF MECHANICAL
ENGINEERING WITH HONOURS
EXAMINATION DATE
JANUARY 2013
DURATION
3 HOURS
INSTRUCTION
ANSWER ALL QUESTIONS
14003
THIS QUESTION PAPER CONTAINS SIX(6) PAGES
CONFIDENTIAL
BDA
51
14003
(a). The high tide Portland, Maine, on April 25,2011 was at midnight. The height of
the water in the harbor is a periodic function, since it oscillates between high and
t in hours since midnight, the height (in feet) is approximated by the
low tide
.lf
formula
!
(i)
(ii)
(iii)
(iv)
(v)
= 4.e + 4.4."r
(i r)
to t :24.
What was the water level at high tide?
When was low tide, and what was the water level at that time?
Sketch of this function from t =
O
What is the period of this function, ffid what does it represent in
terms of tides?
What is the amplitude of this function, and what does it represent in
terms of tides?
(
(b)
l0 marks)
Determine the domain and range for
(i) y=#
(-x,
(ii) f(x)- l-2,
I x2,
x<
2,
-2<x1t,
x)L.
(2 marks)
(c)
Given f (x) = x
(i)
(f"
*2, g(x) = 12 and h(x) -
1, calculate
l)
(ii) f "goh
(iii) hog"f
(3 marks)
(d)
Find
(i)
(ii)
(iii)
(iv)
limr-3 5 = a and lim x-s x:
b. Thus find
(a + b)
(b - a)
(ab)
(bta)
(4 marks)
BDA
(e)
14003
Find the limit for
(i) limr-,*
(ii) lima--
(2x^
-
xz
*
Bx)
zxa-x2+8x
-sx4+?
,lT4
(iii) liml** 5-2x
(3 marks)
(f)
Given f (x) =
fnu**,
Sketch the graph of
continuous at x: l.
x 1L,
,''
x)L.
/.r)
and find the value of constant fr so that
/x) is
(3 marks)
s2
(a)
Find the derivatives of the function
f '(-4).
(b)
lf
f (x) -
xz
+ "tffi7,
hence find
(4 marks)
x:t+rtandy-
(*,-
ur(H)'
tm+ t-^,wheret +O,showthat
: *,(y,- 4)
(6 marks)
(c)
The radius,
r cm of a spherical balloon at time t seconds is given by
r=3+-1-.
t+t
(i) What is the initial radius of the sphere?
(ii) Determine the rate of change in the radius when t : 2.Is the
radius decreasing or increasing?
(5 marks)
(d)
= xz + ft,* * 0,x G R, nndffanaffi.Verifythatthere is
one critical point at x = - 1 and determine whether the point is a maximum
Giventhat
or minimum. Show that there are no other critical points on the curve. Find
the coordinates of the point of inflection on the curve, ild sketch the graph
(10 marks)
BDA
(a)
(i)
Showthat Jstn3 x
(ii)
Express
dx:
f, costx-
cos
x*
C.
the form of partial fractions and prove that
ffiin
ys z(x + t)
J2
14003
1)
@-L)(zx-
dx
12561
=t"\u
)
(9 marks)
(b)
By using appropriate substitutions, prove that each of the following integrals
,,t/.lZ - :'
fL tt
dx'
Jo 1r+ *r1,
dx'
I-"-
ft7
equals to
It
I; snze ae
Hence, find the value of this integral.
(c)
By expressing
*in
(8 marks)
the form of partial fractions, show that
rh- 1rnffi+
where C is a constant of integration'
By using integration by partso show also that
txz
c,
r dx
Ifua*'
Ih--fi1+
(a)
)^.
(8 marks)
to p
Differentiate the following expressions with respect
(i)
tarrL(#)
(ii)
(sec-l(2p + L)
(iii)
psin-Lp+JG
4
(6 marks)
BDA
(b)
14003
Evaluate the following integrals
(i) I:,r-r*
(ii) I:#_*
(iii)
Il,rstn-t
ax
(6 marks)
(c)
Find the area of the region bounded by the curve
- -l and x: l. Sketch the region.
y = x3,x-axis with
lines x
(4 marks)
(d)
(e)
Find the volume of the solid of revolution when the region bounded by the
curves y2 = Br and ! = x2 revolves 360' about y-axis. Sketch the volume.
(4 marks)
Find the radii of curvature of y =
#
at the points (0, 0) and (3, 3).
(Sketch the radii of curvature)
(5 marks)
- Er\D OF QUESTION -
BDA
14003
FINAL EXAM
SEMESTER/SESSION
PROGRAM :IBDD
:II2OI2I2OI3
COURSE NAME : ENGINEERING MATHEMATICS
d r.
_tuf:
4*Lsm
frrrrr-'ul
=
,lm
.+)ut
ax
lul'lF
.+)ut
ax
-1
: t*uz
*ko'.'u|
ax
I#=sin-1
f
)a
rureo=T
-Iu(#)',|'''
Radius of curva'iri^
d
>
_tuf
*ltan
I4OO3
=
du
L*u2'dx
d
1
;ax [cos
)
akottc-Luf
ax
'd*
c'
KOD KURSUS : BDA
1
du
I1 sec-'(;)
-,1r\ *
d*
I p5tfr=
(1)*
<
I
:
-L
lul.lFt
.+)ut
dx
>
1
lxl <a
c'lxl
curvatureK: l#\-r,u
l,*(#)'l'''
Iuau:'trv- tvdu
r
fI --du
Jnt>dx:1pAt4xla,= [ra>
sinzx* coszx:L
tanzx= seczx-t
cotzx= coseczx-L
a"