Math1o3}u\q Ttttu\
Name:
1. Multiple-choice problems
(F\rll marks for correct answer. No partial marks.)
(u) (g points) Given the following general terms ar, determine whether the corresponding
sequences{o,')n>t are converging,diverging, and/or bounded. Check a,ll boxes that apply.
(do not calculate the limit of converging sequences.)
conueTgxnq d'iuerg'ing
i.
..
ll.
(-I)"n:
/
(-1)"(rr,+ 1).
------
.
----a-:
/
/
n+ ,-J
n2 +2n
lll.
bounded
t/
i
/
(b) (3 points) Determine whether the following seriesconvergeor diverge. Check appropriate
box.
(do not calculate the value of converging series.)
conuergxng d'iuerging
@
i
'
\- u
:i
rr'
\-
Rn
,t't
,
/
?5"+3n
oo
/
.,
t
\n + r)n' .
(-
.L
4
iii.
t
-o
co
SI
)
4^
\tu
-*
/
rI 9
\4
uI
n' lnn
:
/
(") (S points) Determine whether the following integrals converge or diverge. Check appropriate
box.
(do not calculate the integrals.)
conuerqLng diuerg'ing
f*
r-ll
''
J , * +* * U * d r :
ii'
f*1
ar'
J' lrtlrto'
^1
ii i . | ,-t a *,
Jo
Final Exam
Page 1 of 13
Name:
Math 103
(d) (4 points) Consider the following differential equations. Check the appropriate box.
true
i.
r(t) : -& i s
as ol u t io n . r c: t r 2
ff
b
ii. y(t):ln(1 * $1 i" usolution
#:
false
/
te-a
/
tT")
:,
t
, L
e
b )ii 7y t' -\,
-t"
ttb'/>
b " - l=
L a-/n/tt
=.L
7 t' /t
t , , . ' ^ Le-t
(") (6 points) Match Taylor series and functions. Check the appropriate box.
@
i. t #-r" - 1-"*r.
/
1\n
-2
n:0
(-r)-'
t-7 : t- -3 + -5
.,
r r' $
-r,
i
h
*2 (zn - 1 )!*
i i i . i n *'-t:
1 * 2 r * 3r 2...
n_o ll
-il\ [l=
4
t^o
Final Exam
,
4 J-^v
n')"
h. l-t
A^,y)"
Page2 of 13
Math L03
Name:
2. Short-answer-problems
(F\rll marks for correct answer. Work must be shown for partial marks.)
t/
(u) (Z points) Find the telescopingsum ,Sry: )- (An
2\
- /k)
k:L
/
P-
t,
-- z l-yi
{li,
'It=t
Ynt
' P-
lIl
=2
_ 4 ,,tt
tAzT
-'
M
AN S W E R : ,* :
t
(b) (3 points) Consider the function and its series f (r) :
:":
Find aee and 7(loo)10) (i.e. the 100thderivativeot f(r) evaluatedat r:0).
A
.
o0 * atr * azn2 *....
P
iz"
|: n*lzLfl'' nsd
ra
oQ't= I
-rh
In^n)
6: alre = |"")H/ort
ANSWER:
Final Exam
ass:
:-.) t' "b):o
_
./1e(100)161
\"/ -
Page3 of 13
Math L03
Name:
fT
(") (3 points) Evaluate the following limit: I:lim'o
- t) at
l^' (o'
'
s-+0
trr2
e*'- )
= lrm
Tato
J ,I-
= b nr
n1a
-
e
e4-)_
6
(r, n')
o
3
ANSWER:
(d)
,/g
L -
(2 points)Express[' yplo**
fb ^ , . .
J" f@)a"
ANSWER:
t"f
.
_
),
t 14
|
ft f (r) d,r + [* f {*) d,r as asingle integral of the form
Itr2t't,
+f
= i +-t ^f'-)y '
2?a^
b-
(") (Z points) The concentration of a certain hormone in the blood changesperiod,i,callyover
24h. The graph of the production rate p(t) and removal rate r(t) of the hormone are
shown below.
In the figure, clearly mark the times
i. 11 at which the total concentration
of the hormone is highest, and
ii. tz at which the hormone concentration is increasing at the fastest rate.
Final Exam
Page 4 of 13
Math 103
Name:
3. The equation of a circle of radius r centred at the origin is 12 + U2 : 12.
(Workmustbe shownfor full marks.)
(u) (Z points) The areaA of the circleis givenby A:2
Y
ffit
ar.
[' url;;
r_,
Draw a sketch, which clearly identifies the integrand,
and shade the area given by
['
1r,
- *, a,
J-r
(b) (1 point) Give a reasonfor the simplification, z [' @
-
J -r
"z
ar : 4 [' Url;;
Jo
a*.
Ir aven
Itil, W
r7'
(") (6 points) Evaluate A:
4 |
t/r-
JO
"'ar.
(Hint: use trigonometricsubstitution.)
y = rilh e
n)z
=*
rLc- ps*adb
|
o
nlt
, ht^L
I
tt%t-
/y;
ntco@ 1 J ,'-.ttcz
n/etg
lo
,o.
4
n r'>
ANSWER:
Final Exam
A_
n-
rr
4
Page 5 of 13
Math L03
Name:
4. (6 points) Consider a swarm of ants distributed over a circular region. At distance r from the
centre of the region, the density of the ant population is observedto be b(r) :
h,measured
in units of one thousand per square meter. What is the total number of ants witfiin a radius
of 2m? (Workmustbe shownfor full marks.)
6gtr
yY
| / ")tr
l{
\\
rv/
vJ-Jtn A/n) rE QI,r' )r (n^tt /\
'' or
anL, trr
bl anD'
t>0"
^tvt
.
/At'il,
u^.TuLt'l
Pf'lelaa
/l
i ,!\ d'n
l"'o"1
ie
':''"4
/
o
=
t
lrl r
t
"t'l
Ldal
T-=
u=rkl
J,./r r r .d m
I
-
A NSW E R :
Final Exam
lflni
f Tlnf
x l pp o
Page 6 of 13
Math 103
Name:
5. Consider the differential equation * : y2 - 1. (Workmustbe shownfor full marks.)
-dt
(") (Z points) Find a// steady states (equilibria).
eP4)
V*-l
,
EfiI
VF'l
--
(b) (2 points) Determine the stability of all steady states (equilibria).
(Note: formal as well as graphical reasoning is acceptabie.)
4|
-l ! o s t aLle)
I
L i t 4 r' a ta i l ?
a Q):2 .
(") (0 points) Solve the differential equationrus ingit h
[t\
=
l4n
^, IYtt)^ a la
)tV
I
ffftyrr)
-t+L
Y't Ytt
);
!al
YF
Llllo
4)
f:
-L+-I^
-
=)B , -j
I
,
rr,)=
t(ttJr-lh 4) 2- l L n
i-
.^ ln l#t
YJ
yrl
r-h
ANSWER:
Final Exam
e ld L
zt
t.x
I
2
ce
I
*C
P 7'
ul
o ,( + \ -
=<)
L
2
v J-l
=') ltl 7
?4rst
*l
l. - ce*
2
ffitt
b
a Y=
*)
i - estb
L
-3
bapi 2 i. 3"l E) Lr
I-d
Page 7 of 13
Math 103
Narne:
6. Considerthe function F(r):
\tre-', where I > 0 is a parameter. Now considerthe iterated
map lr111: F(rt) for I : 0,I,2,3, ' ' .. (Workmustbe shownfor full marks.)
(u) (Z points) Find all fixed points (steady states, equilibria).
1zrflr )f- 4)
,.2. e"'rs L
=' $
$e-t'- I )
f,a & o { \ e L ' }
4 (*
P
i=-
F) , [ . x O6/ L= lu.(b) (5 points) Determine the range of ) for which each fixed point (steady state, equilibrium)
is stable.
)L zP:
s tt' n )e -r'
f ' [ v \ ' f l" - t - ) L e ' t )=
tuile trn ?: ,'t
s
ia
9e
0
).
Ftlo').=
)L^l4)
1-l"r
7 ' l to s J =) { /' l n }) it u'
rt r tnLb f I
9o lvttr
( t't r' h'ol3o,lo
1l 4 ) a < l n r/-2
t*
tn
l
l
I
l4 ) 1 e z
&)
( I= l
L tvl,a n la te J
Ar z
t et
(c) (4 points) Starting at 16, use cobwebbing to draw ur up to 15 in the two graphs below,
which show F(r) for two different values of ).
/i
/l
I
{
I
(/
(
0 roxl
Final Exam
v\
xt
o xoltt
h
Y)
v3
,r ,,
Page 8 of 13
Name:
Math 103
7. Consider the Normal (or Gaussian) probability density function (pdl) given by
t;
(Workmustbe shownfor full marks.)
f (r\ : ,l 1"-2"' fo, -oo ( r < oo.
Yn
&
^^_^
7. .
(u) (Z points) Find themean
z
|
"-rfr
'a
e-r- h
I ald /oa/r"",1
b,
b
ANSWER: r:
(b) (1 point) Find the median 11'
e nen
llvlit
tr 7 :
ANSWER:
'P
xl^o
o
2
(") (Z points) Using the fact that /(r)
is a pd,f,find the integral t :
[*
"-'""
dr.
,= e'rh 'b
i&
I -
ANSWER:
(d) (4 points) Find the variance V.
(Hint: what is the anti-derivative of ue-2*'?)
t/z i; E d'L'h - a
-e
=:thd'ttirrl7d'ttt
o *+
ANSWtrR:
Final Exam
rr-
v':
, c''e;
z)L
.l
y;
lte'Yh
={rl+t' ;'th
='h,e-zlLz
t/ry
Page 9 of 13
zl
Name:
Math 103
B. (7 points) Find all r suchthat the ,"ri". i
(n-I)s(r-2)
converges?
1+3n
(Work must be shown for full marks.)
tf 1u ' ll' nl It t a'J
tr = Irrn fu
J
nic,
{")
ln
ld
z lt *
ft40
<
ldr'hl t I
d,ivorTlf il /'Y'tl V 3,
v z' l'l t
ot
TPtzfi
F l at-l
ANSWER:
Final Exam
eJ
I at-zl' '
3
. 1 C r n rertlrfit'f lil'Al t 7l
=?
! r -r)
[ , -')t
1
,4
Z I o^! "' v
n^o
'
yivq
V n-f f'v')
nal
ffi4
b't 0 tterlcuta Ttrl
)iwrlrs
-.n ln'ti'
-"l'l' ;
t
di rt rltg lr
a
:r P
Ai'e rf,eY'tcT"P/'
1,{ry1-ta
/
I
'-lt vn
z e (-lt $\
Page 10 of 13
Math
Name:
103
(Work must be shown for full marks.)
10. Integration
f1J-P r
1 :-:-:I I-er
(u) (a points) Evaluate the integral /1' :
r,=l'ffi,)
ut/,1
,{tt
L,-h),4
rt =f
dr.
fi= e"
7+'*k
du*euh
tlt4 ? AI4 + 0il"u)
z -L +L ' = )
vI
t'4
j ,'l
t4 Ab e ) l= fr
2,
^Jt't
F4
ANSWER:
z)#
la z - zlrylt'u) +' tn lal
: -zln lt "cnl 't,l,z Ee
: - 214
il" exl i) L *
* 2 la lt^*l
It:
d
c
t y.1-(.
r
(b) (4 points) Evaluate the integral 12:
/ cos(ln r) d'r.
.= | lcet*1 ewl'n
: p.eiv)2* r $l!' n '')cn/u
.,*
z l L .0 /w ) E
(-
j
212=
ANSWER:
Final Exam
+ lot+u)ery
L*
r
r rt
s
g
t
'
e
n
)
1ta)w
*l
wt und * 9j'7ttu z ) )
12:
4
W: Jn d 7 - - Ew
h-e*/z
tl * Lb?t?I
y' r cb
tz
-gt4 *2
V -etu
Ma h lmt t "
v' refu
u ^ab
u
i"/J = t vg tt' ,
l.o*Qnu)+e/a(/"D) l' C
Page 12 of 13