# c 'L.du + .,9

25 FEB
2014 Math 101
Narne:
Page 2 of 10 pages
1' short Answer Questions. Each question is worth 3 points. put a box a,round your
final answer, but No OREDIT will be given for the answer without the correct accompanying
[24]
work.
(a)Evaruate
L,
rIr&quot;,&quot;&quot;dr:[l_CJU
i Jzn_ -_ -- z1r..t
?t= !fx2
Ju= zrrxdt
+
c
'L.du
+
zll
f-or* kLD4src
yyu'{- }&quot;&quot;h.r,6.ra
fil,t=*Jt
,(\4) ) (t:
(b) Evaluate
I
tsir(at) at,where o is a constant.
_t ol
t t. .,n t&quot;t) Jt 7
h
1l=
t
?o'{tt
Jv=
.,9 , ,U,
si^(ot)dt
da=l&amp; v=-l'col[o9
cos[ol)
+
I
Ot
I cos(ot) /t
25 FEB
2014 Math 101
(c) Write out the
Na,rre:
Page 3 of 10 pages
form of the partiat fractions decomposition of the function
xa
+l
G'])I;fiN;T
.l-
Do NOT determine the lalues of any of the coefficients.
t'+r
x4+ L
(x&quot;- r) 6y'+t Xx Wv\$WV,LX a
)
5*rnwwYg|
(x-r)a(r+-u) [x2+.t)
.r)
A t- b Z-* 'x+r *
(&quot;-.r)t
*;
- -&quot;&quot;L \$t&quot;t-r^*.*-l
naztec.f,W
4
',
( vtv/&quot;
*#
Dx+
3
E
^nrLt
16*0
for sorne- co.slanh A,6,C,D &quot;nl E.
(v'. n&quot;f
4&quot;!-i
&amp;&quot;-,.\.rr'tL
:4 wruL..V
in..t ff.^U.*XZ i^+.^k
&quot;&quot;,
A
#cular
\
(d)
water tank is shapeU like vertical right circular cylinder with height
heisht 6 metres
metr and
radius 1.5 metres. If the tank is full, write down, but do not evaluate, the integral that
represents the work needed to pump all of the water to the level of the top of the tenk.
Water has density of 1000 kgm-3. and gravitational acceleration is
g.g
g:
A-tWno** CoerJ
l{sLb
5(dof.^l^ _
l,l
11
6-X
= Fo&quot;cashl. Drsh&quot;ca ug \ ,
= r^r&quot;,trhl,ir, . DuL&quot;,,,6
=Aass&quot;65
=
H-o
6 I
6
5
\
itr
iJ&quot;c-.
A' 1
roo0.(n'&quot;.dy)'A Y
rooo.r t5;+ LS'-l|-
*+
u'ae4'rrrns-
A*ftW
r1 uot {ud
___*l \__;;t2
W= f l.E'rooo'\$zn1J1
o
\
'---,A W
=
J:
g
1000.VolunarLb
(4
+
-r-2.
qa +6bo 4.szrr,[6-E)Ja-
I
2014
25 FEB
(e) Express
f
*
4/\^c'rLs',
&quot;l5g
-If not- 4&quot;tt
,lacwQ-
Math 101
f;
Page 4
- r (;)'1
as a dennite inregrar.
x,--*
-t#-,
pa^fhri,.t^ t-.&quot;
'r!
t,*-&quot;,')Jx
irc&quot;,rax = -! Ltt*l
4 vrno,rl' -;.^i='
wq co,', la*e:
p&quot;tryt
OW1T
,,
of 10 pages
f tx)= ,(_3&quot;..
e=0, L=1.
_{;&quot;1&quot;
:
o
,r&quot;&quot;,ub,&quot; force of F(c)
ln(u + 2) newtons moves an object along a straight line when
the object is r'etres from the origin. calculate the work done in moving the object
from I metre to 10 metres from the origin.
19
x
x+dr
WoerRa'dr
Al0 wrtilt-
!0
W= I f(v)Jx = J lnlr*r) Jx -- (r'
I
ln[r+z) +
.L
il^rr-ro -ln3-tln3+r
J--
z' l&quot;[r+z)
-x) [
l^JC'- lt,l&quot;
\lo
-
'l
\$&amp;h*
(LJr = r.l',(;+z)-!(t -k\Ar
dx; r'l^tx+z\ - J x+z
+z l&quot;(,*+z) -x*C
J \^tx+z)
Anl;dariua!'o:
x.ln(Y+z)
['
gor+J:
l'.(r+a\ Jv'
u
=*
)r=
a--
fi/*
25 FEB
2014 Math l0l
(g) Evaruate
ft' '/TP
r'
Jrla
5\$*=+. f 'F
r= f,s;ntt
ra=f s;',tll
Na,me:
Page 5 of 10 pages
tu.
7l# Ja =2lfue,
E-.|** at =
= ?.\rr:&quot;Y:;{u
=2\ffisa-z[ta
-LSr'n?[
Tan?t-Zu+c =-zca!'
=
lx= lcost lu
au
-27'l+c
I
- 2rlr-;g'a
- zLl+c
Sin?l
- /arcsia(z*)
Tb,
[-=
d,
=
(-ry
I Avcroa\J'c=t'ft-o
d
\16
\
0
[email protected]):
cos2
(t)
on the i
trlc
X'
I t+osLzt)
cr,&quot;\$/, =9 Jz
II o
e (t .
=9r (t, *f, s,^(zr1)
-
-!,
2
c
-z^.u.e)(',r = o- 2a,csin\$) *\$- -Zoot( i)
(h) Find the average value ofthe function
J.w,olJ.,-
+
? \Iz'
T
3ft
z
4Tt
-f
4 wcu,hS*
L
+
s'&quot;\$)
A'lv'dk
25 FEB
[6]
2014 Math 101
Page 6 of 10 pages
Name:
2. Find the area of the region R
4 N'6fu
enclosed by the curves y2
t:,2
\$c^&quot;
: r*
1 and A2
:2(r - l).
Ir14za5.ctbr' P*;*s
x-+ 4= Z(x- t\
)
5E
q
=) f,t='3+4: 2.1h,
,v
2'=
,
T-=
-rW
)c+4
z''
4
In1'6'-Ae 4{&quot;&quot;
h.^
\,nPtrP
&quot;L
=
fl-
azrcits:
jKf
+4) -
(u'-&quot;]*
2 ''Yno&amp;t
-2-
(-_4
-TU z rz
= J^z
I
,lI
b
4b
3
Ju
d
--
23 r 2'2
[u=
+ ta
+ LcfL
L
-z
zGz\
,#*kfu
r/'th\&quot;\$-
25 FEB
[6]
2014 Math 101
3. Consider the functions
allz)0.
(&quot;)
Name:
Page 7
grven by
f (*)
: 21/i and g(r) : Jie'/2
of 10 pages
, which are defined for
(Z marks) Find the r-coordinates of the two points of intersection of the graphs of
A : [email protected]) and g : g(r), and determine whether f(r) &lt; g(r) or f(r) &gt; g(r) in the
interval between these two points.
Su[,re. 2,8
=
F:o1 \= &amp;
Llet-A6
J; ed- Wz7-7r O.
.J-+
s-(.r-\$'lazr.
x+ o ) +4,&quot;,, ,
(b) (4 marks) Evaluate the volume of the solid obtained by rotating the area bf th&quot; firrit&quot;
region from part (a) enclosed by the graphs of y : /(r) and y : g(r) aborl_{ tbe r-axis
e.y\
Lo,2 l-s&quot;\7
W-
?%
\:cr,J l- {,est ,t F;t lyr t(is lu,(et,ra-t sMaa..f6 Yrr.
t 4C,)= 2 't' aC4,)- d
ore- co,nevLo\*g 6\^rt' *'dr*'=4
O
r- 10, t1 A&quot;'VIF
alutA
&gt;
v=
j'
lJ'
(,n e)
(ztz)'-r
Lt
2-.'.^J&quot;
o
tt570-n1-
[Jo
(tnx.
-ITL^1
I
TI
l&quot;
?4
A*X
dru=
)t
2 .Ln L
=
eL Jx-
=&amp;
-dxe
&quot;['&quot;lre'-l]'
{ l&quot;tt LtC - Brrl&quot;^ L
+
25 FEB
2014 Math 101
Page 8
Name:
of 10 pages
to launch an object of mass rn from the surface of Planet X into
* the gravitational force
outer space is given by W : [:^^F(r) d.r, where F(r): *
Jnooo
between Planet X and the object, and K : 4 x 1913 *3 t-2.
[6] A. The work required
(u)
(S marks)
Find the work required to launch an object in terms of its mass m'
rR
^
VV= ,[,* I
Tx'tol3&quot;n
R+oo J&quot;|l{oa
-,^
4h^ utL&amp;.
t'Urt
,;.Jr
\r /rm lR+*\
Ll
L\
J:c -
,'{^*
[a2rc
vo
R
4 u^nJt
Nlwr
+(4o43nA\
4000o I
4oo
,.4rjjwt
4 unrt J*
{r*&quot;[
{&lt;^tw
OJ,tt]&lt;'rr'd
oA
P-
r&quot;
(b) (1 mark) To leave the gravitational attraction of Planet X, m object must reach a'n
a kinetic energy
escape velocity, u&quot;. What escape velocity is required to give an object
ot!*o? equal to the work, W, required to launch the object?
2'
9&quot; lue
L{n
+
L= L*&quot;f
2*
4 ,t'r^l^
ono
,
=)
-ll
Kwt s
25 FEB
[O]
r.
point
VEe
2014 Math 101
Name:
Page 9 of 10 pages
(a) (a marks) Find the slope of the tangent line to the curve
(o,o).
+!*
FtC
:
Cnrdt clu:t, Aut)
er'
dt
at
the
4Ju
3*'D*,
z.
=
d-x7
vna-uVg
\ r^r,L
1:la
)
iJ&quot;..
, : Jr-'
[t*&quot;
&gt;r\
p,Tc
A,
dro,i,,N
/u^U
ft ptu56rrvt6 vl^ 7--'o
(b) (2marks) Let f(r) beacontinuousfunctionsuch thate?tttt)at):rn+rrl2r*Ifor
every real number r. Find /(0), the value of (r) at x:0.
f
-To t'&quot;tol.
€iJes
h
l,
we
u,,ar-
+fl.
Frc,
di#&quot;&quot;&quot;.cSzl-
b.t&amp;.
6aL
Cf*&quot;dt)
,
e
So aL x-=O&gt;
I
e(J;
_\
4C)
- **3 +2x- + z, 4vw,t^l-
tlr. &quot;+trJa
o
+l-&quot;:dA
.lc^ = L
&amp;cn&gt; = 2'
a
4 t,u.,,t/*
25 FEB
[6]
6.
2014 Math 101
Page 10 of 10 pages
Name:
Evaluate
2
lo-ta-*
or explain why you cannot evaluate this integral.
ttrv
al *=4, \$o \P'L (c\ws\de-r
Wu- no{&lt;.
inlegra'^rud
*t...-T
t&lt; &lt;l*Vv'!a'r 4 *&quot;^L
4
:t
[^5a&quot; &quot;\,.6fut
[' ,r , .d.,c Joct-n)
-'[
\rcffir.*'v
^L
,
a'
@
@
+o a+&quot;,46
2 -vv'cuJ*t^ 4Y
p&quot;tt/ {a'\*s
,
Lr,= l^ \--^\
:
t^^otk
-ftl.^\t-a\
B,^t
v
t-&gt;
0l^\'-t\
\iwr
^-
diuer5rs. 1l&quot;'tts -,
5k**
-Al&quot;l-t\
{ll&quot;[ z-,'\
ff..Jn
[n
I zr-- 4
-o
criu'rx&quot;s'
4wil&quot;-(Y
a
\
cu^!&quot;Sl