33130 Mathematical Modelling 1: Threshold Test 3
45 minutes – Competency requirement – 70%
NO Calculators. Leave all final answers as whole numbers.
WORKED SOLUTIONS MUST BE GIVEN or no marks will be awarded.
If you attempt this test, the rules on this page apply.
First Name: ________________________________Last Name:______________________________
Student Number:_____________________ Mark:______/40
Examination Conditions during COVID-19:
It is your responsibility to fill out and complete your details in the space provided on all the
examination material provided to you.
You are not permitted to have on your desk or on your person any unauthorised material. This
includes but not limited to:
 Mobile phones
 Electronic devices
(including calculators)
 Smart watches and bands
 Draft paper (unless provided)
 Textbooks (unless specified)
 Notes (unless specified)
You are not permitted to obtain assistance by improper means (including copying from another
student) or ask for help from or give help to any other person.
If you breach any of the above rules:


You will be given zero for the test.
Misconduct action may be taken against you.
Submission Instructions
1. Login to UTSonline and download the test between 5pm and 7.59pm (every student must login
independently so there is a record of your access).
2. Start immediately and take 45 minutes to do the test.
3. Create a SINGLE PDF file of the test with filename:
MTT32_FirstName_LastName_StudentNumber.pdf
4. Set up an email with the subject line:
Threshold Test 3.2 FirstName LastName StudentNumber
and attach the completed test file (SINGLE PDF) and send to: Science.MathsMM1@uts.edu.au
immediately after creation. (Should take about an hour– this is your completion time)
5. Upload the test (SINGLE PDF) to the submission portal for MTT3.2 in the 'Submit Here' folder.
Just click submit once, if it does not go through immediately, just leave it as ITD will be
investigating. I will reopen the portal later in the day.
UPLOADED test is marked. No uploaded copy, no mark. Email copy is only kept to check if
the uploaded copy has a defect and to have a record of completion time-emailed copy is not
marked SO CHECK IN THE MORNING.
MTT3.2C
1
1. - 14 points (1+2+2+2+2+3+1+1) continues over-page
Given a  3 2  3 2i , and b  5e
i
2
3
find the following :
(a) a
(b) a
(c) Plot a, and b on the Complex plane below
(number the axes!)
(d) Convert a to exponential polar form where
−𝜋 < 𝜃 ≤ 𝜋.
MTT3.2C
2
1. Continued…
Given a  3 2  3 2i , and b  5e
(e)
c
i
2
3
a
Leave in exponential polar form
b
 1
sin 1    
 2
Give your answer as an exact value in
radians.
(i)
MTT3.2C
find the following :
(f)
Find b 7 .
(Give your answer in Cartesian form).
1
cos 1   
2
Give your answer as an exact value in
radians.
(j)
3
2. - 6 points (1+2+3)
Use the graph of f(x) to evaluate each integral.
45
∫ 𝑓(𝑥)𝑑𝑥
0
135
∫
165
𝑓(𝑥)𝑑𝑥
45
∫
𝑓(𝑥)𝑑𝑥
0
3. - 6 points
A spring has natural length 7 cm. Compare the work W1 done in stretching the spring from
9 cm to 11 cm with the work W2 done in stretching it from 13 cm and 14 cm. How are W1
and W2 related?
MTT3.2C
4
4. – 6 points (3+ 3)
(a) Use Implicit differentiation to find
x2 y3  y3  6x2
dy
if
dx
(b) Use logarithmic differentiation to find
y   3x 
dy
if
dx
sin x
5. - 4 points
s(t )  t 4  4t 3  20t 2  30t , t  0
The position function of a particle is :
When does the particle reach a velocity of 30 m/s given velocity is found from the first
derivative of the position function?
MTT3.2C
5
6. - 4 points
Each side of a cube is increasing at a rate of 4 cm/s. At what rate is the surface area of the
cube increasing when the surface area of the cube is 486 cm2?
Extra Space
MTT3.2C
6
MTT3.2C
7
Table of Integrals
x
 sin 1    C
a
dx

a x
2
2
x2  a2

 x
 sinh 1    C1
a
dx
a x
2
dx
 1 x
2
dx
a x
2
1
 x
tan 1    C
a
a

2
x
 cosh 1    C1  ln x  x 2  a 2  C2
a
dx


2


 ln x  x 2  a 2  C2
1
 x
tanh 1    C if x  a.

dx
a
a
 a 2  x2   1 1 x 
 coth    C if x  a.
 a
a
1 x 1
 tanh 1 x  C1  ln
 C2
2 x 1
 cosh x dx  sinh x  C
 sinh x dx  cosh x  C
 tanh x dx  ln cosh x  C
Formula sheet
𝑏
𝑥
𝑛
∫ 𝑓(𝑥) 𝑑𝑥 = lim ∑ 𝑓 (𝑥𝑖 )∆𝑥
𝑛→∞
𝑎
𝐴(𝑥 ) = ∫ 𝑓 (𝑡)𝑑𝑡
𝑖=1
𝑎
𝑏
𝑏
1
∫ 𝑓 (𝑥 )𝑑𝑥
𝑦̅ =
(𝑏 − 𝑎 )
∫ 𝑓 (𝑡)𝑑𝑡 = [𝐹(𝑥)]𝑏 = 𝐹(𝑏) − 𝐹 (𝑎)
𝑎
𝑎
𝑎
𝑏
b
RMS 
2
1
f  x   dx


ba a
∫ 𝑓 (𝑥) 𝑑𝑥 ≈
𝑎
ℎ
[𝑓(𝑎) + 𝑓(𝑏)]
2
𝑏
∫ 𝑢 𝑑𝑣 = 𝑢𝑣 − ∫ 𝑣 𝑑𝑢
𝑊 = ∫ 𝑘𝑥 𝑑𝑥
𝑎
∫
𝑓′(𝑥)
𝑑𝑥 = ln|𝑓(𝑥)| + 𝐶
𝑓(𝑥)
𝑑𝑇
= −𝑘(𝑇 − 𝑇0 ), 𝑘 > 0
𝑑𝑡
𝑑𝑦
𝑑𝑥
𝑑𝑦
= 𝑓(𝑦)𝑔(𝑥 ) and ∫ 𝑓(𝑦) = ∫ 𝑔(𝑥 )𝑑𝑥
𝑑𝑦 2
𝐴𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ = ∫ √1 + ( ) 𝑑𝑥
𝑑𝑥
𝑎
𝑏
Summary integrating factor Method:
Write the DE in the form
𝑦 ′ + 𝑝𝑦 = 𝑞
Find the integrating factor, which is
𝐼 = 𝑒 ∫ 𝑝𝑑𝑥 ,
Multiply through by 𝐼 and use the fact that the LHS is now the derivative of a product
MTT3.2C
8