MATH1013 - Mathematics and Applications
Mathematics and Applications
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MATH1013 - Mathematics and Applications I
Book B - Calculus
Reading time: 15 minutes Writing time: 90 minutes
Instructions To Students:
• Set up ANU Zoom on your device with video on and audio muted.
• Save a copy of the exam so you can keep working if the internet goes down.
• You can print the exam during reading time and write your answers in the spaces
provided (if you need more room add a page.) Or you can use a tablet. Or you can
view the exam on your screen and write on blank paper.
• Two A4 sheets handwritten on both sides are permitted, to cover both exams. A
trig cheat sheet is also provided. No calculators or books are permitted, with the
exception of paper dictionaries. The exam must be your own work. During the
exam you must not use any resources on the internet or have any examination help
from any source.
• Total is 50 points for this calculus exam. There are six questions in all, with point
value indicated.
• You must justify your answers. Do not expect credit for a correct answer with no
justification. Write clearly and legibly.
• A good strategy is not to spend too much time on any question. Read them through
first and attack them in the order that allows you to make the most progress.
• You have 15 minutes reading time for the exam. You can make notes on scrap paper
during this time (but must not start writing your final answers.)
• You then have 90 minutes writing time.
• At the end of the 90 minutes you will be given additional time to make a PDF and
upload to Wattle.
MATH1013 Final Exam Book B (Calculus), June 2021 Page 1 of 8
Solutions
Question 1 (Derivatives and applications) 8 points
a). Find the following derivatives: (2+2=4 points)
i).
d
dx
(x3 + 3x)
ii).
d
dx
(tan1
p
x 1)
b). Consider the function P defined by P (x) = 5 x x3 x5. (4 points)
i). Show that P is one-to-one.
ii). Find (P1)0(2).
MATH1013 Final Exam Book B (Calculus), June 2021 Page 2 of 8
3
2 this 3
l Ii
it
2 Ziff 2K
i p x 5 x x3 XS
cts with DomCp R
P x I 35 5
4 C O V KE IR
s P is a decreasing function
and therefore one to one
E From i we know that P is defined
I inverse fu thoup i 2 pi P 27
observing that Pcc 2
we have p z
P 2 pig gt
Question 2 (Applications, Di↵erential Equations) 8 points
a). A radioactive substance decays according to M = M0ekt. Assume that it takes
600 years for 1000g of the substance to decay to a remaining mass of 125g. Find
the half-life of the substance. Justify your work. (4 points)
b). Find the solution to the following di↵erential equation,
satisfying the given initial condition. (4 points)
dy
dx
=
ln x
xy
, y(1) = 3
MATH1013 Final Exam Book B (Calculus), June 2021 Page 3 of 8
Full Given Me Moe
Kt
MARKS
forthis
µ we
have 125 ooo e
600k
e
600k If
we could
600k ln8
be led
also just 600
note that To determine half life we use
600years Mo Moe ht
is three
e
ht z let lerhalf lives
So C big he 600 31.600 200lu8half life
200yrs Half life is 200 years
Sydy Y doc
Eye CluxJtc
q2 flux
Z t Cz
y 3 a 9
Solution is
y n t9 satisfying y 3
Question 3 (Limits) 8 points
Find the following limits if they exist. Justify your work.
a). lim
x!0
✓
sin 3x
tan 7x
◆
b). lim
x!⇡
✓
sin x cos x
sin x+ cosx
◆
c). lim
x!0+
✓
1
ex 1
1
x
◆
MATH1013 Final Exam Book B (Calculus), June 2021 Page 4 of 8
Form we can try L'H
3cos3x
his.kz 9EsoE I
Ot C c
l directsubstitution
x e t I
lui Formeo
ot x ex D c'Hospitals
him Again 8
of xe te I c y
eion
X 70T x exte te
I
I
Question 4 (Hyperbolic functions, integration by parts) 7 points
a). Given: sinh x =
ex ex
2
, cosh x =
ex + ex
2
, show that:
i).
d
dx
(sinh x) = cosh x. (1 point)
ii).
d
dx
(cosh x) = sinh x. (1 point)
b). Use integration by parts to evaluate
Z 1
0
t cosh t dt.
Simplify your answer. (5 points)
MATH1013 Final Exam Book B (Calculus), June 2021 Page 5 of 8
da sialic
e _t e cosh x
2
ex e
dfa cosh
e tt
2
sinha
2
To evaluate of'Ecoshtdt
we use integrationbypants fudu au fudu
Take u t dir coshE dt
du It Sinh E
c
And f'tcosh Edt f sinht Isiah
Edt
fetsinht cosh E o
sink 1 cosh o
cosh0
e e i
2 2
I te
Question 5 (Volumes) 8 points
a). Find the volume generated by rotating about the y-axis the region bounded by the
curves: y = x(2 x), y = 0. (4 points)
(Hint : Use of cylindrical shells is recommended: V =
R b
a 2⇡xf(x) dx. )
b). Set up and evaluate an integral to find the volume of a tetrahedron with three
mutually perpendicular faces and three mutually perpendicular edges with lengths
4cm, 5cm, and 6cm. (4 points)
MATH1013 Final Exam Book B (Calculus), June 2021 Page 6 of 8
A sketch is helpful
µ
we use 2Txfcxjdx
2
2K fx x 2 x doc
2 2
2 x3 dsc
2K Fsd 14
4
Zit C'S 4 o
8
Again a sketch is helpful
is
at one corner on the side
D length 4cm could
use
any of the
sides
we can take a slice at a point K
as shown Area of slice 15
And V Isda de
f x2dx f f xD 20
Question 6 (Integrals) 11 points
a). i). Use an appropriate substitution to find:
Z
sin2 ✓ cos ✓ d✓. (3 points)
ii). Use an appropriate substitution to find:
Z
dx
x2
p
x2 4 (3 points)
MATH1013 Final Exam Book B (Calculus), June 2021 Page 7 of 8
Set u since with du cos Odo
Then fsinocosOdo fatale
f us
since c
use x ZsecO O E o t U et EI
with doc 2 Sec Otano do
Then f de f
2secOtau
4sed Offset
If secotamodosedO Ztano
do
tea
foos do
I since
4 Hi
Seco_Is
HI xc siuO x
4K
b). Use the method of partial fractions to find:
Z
1
x3 + x2
dx.
(Show the steps required to find the partial fraction decomposition.) (4 points)