2022/6/5 下午10:34 Quiz: MATH1021 Past Exam 2021 Semester 1
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MATH1021 Past Exam 2021 Semester 1
Started: Jun 6 at 0:34
Quiz Instructions
Note about past paper: this paper was set by James Parkinson as the final exam in 2021,
Semester 1. The text below this is the text that was included in that exam. It is not necessarily the
same as you will have on your final exam this semester.
_______________________________________________________________
All exam questions will appear only in this quiz. Do not click submit until you
have finished answering all questions.
Section 1
20 multiple choice questions
Section 2
3 extended answer questions
Section 2 questions should be handwritten on white paper in blue / black pen. You must scan /
photograph your work only after you have submitted the quiz as use of your mobile phone during the
quiz is prohibited. You must upload a separate document for each question's specified assignment
dropbox within 25 minutes after the quiz submission.
You must not handle your mobile device at any point prior to submitting this quiz. Use of a
mobile device during the quiz will be flagged as an integrity breach by ProctorU.
Permitted resources
Handheld calculator, Microsoft Word, Multiple sheets of blank scratch paper and a pen/pencil, online
and paper-based resources.
During this exam, you will NOT be allowed to access wolfram alpha, or online mathematical
computing software.
Section 1
Multiple-Choice Section (50%)
There are 20 multiple-choice questions.
Each question has exactly one correct answer.Processing math: 0%
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Each question is of equal value.
There are no penalties for wrong answers.
No working will be marked.
1 ptsQuestion 1
x^{\cosh x}\left(\frac{\cosh x}{x}+\ln x\sinh x\right)

x^{\cosh x}\left(\frac{\sinh x}{x}+\ln x\cosh x\right)
x^{\cosh x}\left(\frac{\cosh x}{x}-\ln x\sinh x\right)
x^{\cosh x}\left(-\frac{\sinh x}{x}+\ln x\cosh x\right)
If f ( x ) = x cosh x then d f d x equals:
1 ptsQuestion 2
2\pi
-2\pi
- \pi
\pi
Find ∫ 0 π x sin x d x .
1 ptsQuestion 3
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\displaystyle-\frac{5}{3}\cos \left(\frac{3x}{5}\right) + C
\displaystyle-\frac{3}{5}\cos \left(\frac{3x}{5}\right) + C
\displaystyle \frac{5}{3}\cos \left(\frac{3x}{5}\right) +C
\displaystyle\frac{3}{5}\cos \left(\frac{3x}{5}\right) + C
Find ∫ sin 3 x 5 d x .
1 ptsQuestion 4
\text{$f:\{-2,1,2\}\to \{1,2,4\}$ with $f(x)=x^2$}
\text{$f:\{-2,0,4\}\to\{0,4,16\}$ with $f(x)=x^2$}
\text{$f:\{-1,1,3\}\to\{0,1,9\}$ with $f(x)=x^2$}
\text{$f:\{-1,0,1\}\to \{-1,0,1\}$ with $f(x)=x^2$}
Which one of the following functions is surjective?
1 ptsQuestion 5
\text{$x=-1$ and $x=2$}
\text{$x=1$ and $x=2$}
The critical points of the function f ( x ) = 2 x 3 + 3 x 2 - 12 x + 1 are
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\text{$x=-1$ and $x=-2$}
\text{$x=1$ and $x=-2$}
1 ptsQuestion 6
\text{$\sinh^2(x)$}
\text{$f(x) = x^2-1$}
\text{$(x-1)^3$}
\text{$\cosh(x)$}
Which one of the following functions is injective on the domain ℝ ?
1 ptsQuestion 7
A=-2 and B=1
A=2 and B=1
A=-2 and B=-1
A=2 and B=-1
Find A and B if
x + 8 ( x - 2 ) ( x + 3 ) = A x - 2 + B x + 3 .
1 ptsQuestion 8
The area of the region of the x y -plane bounded by the curve y = 1 x , the x -axis,
and the lines x = 1 2 and x = 2 is equal to:Processing math: 0%
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\frac{1}{2} \ln(2)

\ln(2)
\frac{1}{4} \ln(2)

2\ln(2)
1 ptsQuestion 9
\text{The inverse of $f(x)$ is $f^{-1}(x)=-2x+3$.}
\text{The inverse of $f(x)$ is $f^{-1}(x)=2x+3$.}
\text{The inverse of $f(x)$ is $f^{-1}(x)=(x-3)/2$.}
\text{The inverse of $f(x)$ is $f^{-1}(x)=(x+3)/2$.}
Let f : ℝ → ℝ be the function with rule f x = 2 x - 3 . Which one of the following
statements is true?
1 ptsQuestion 10
\frac{4}{N}
\frac{8}{N}
Let U n and L n be, respectively, the upper and lower Riemann sums, with N being
equal to the number of sub-intervals of the function f x = x 3 in the interval 0 , 2 .
Compute U N - L N .
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\frac{16}{N}
\frac{2}{N}
1 ptsQuestion 11
2
\ln(2)
\ln(3)
e
Let A denote the area under the curve y = e x over the interval 0 , b . Which one of
the following is the value of b for which we have A = 2 ?
1 ptsQuestion 12
\text{$M=1$ and $m=\frac{1}{8}$}
\text{$M=\frac{1}{9}$ and $m=-1$}
\text{$M=\frac{1}{8}$ and $m=-1$}
\text{$M=1$ and $m=\frac{1}{9}$}
The global maximum M and global minimum m of the function
f ( x ) = x - 1 ( x + 1 ) 2
on the interval 0 , 5 are:
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\text{$(-\infty,-1) \cup (1, \infty)$}
\text{$(-\infty, -1 ]\cup [1 , \infty)$}
\text{$(-\infty, -\sqrt{2})\cup (\sqrt{2}, \infty)$}
(-\infty ,- \sqrt{2}] \cup [\sqrt{2}, \infty)
What is the natural domain of the real valued function f ( x ) = ln ( x 2 - 1 ) , where
x is a real variable?
1 ptsQuestion 14
{2,4,5,8}
{1,2,4}
{2,4}
Let A = 1 , 2 , 3 , 4 , B = 2 , 3 , 4 , 5 , 6 , 8 , and C = 1 , 3 , 6 , 9 . Find ( A ∪ B ) \
C .


{1,3}
1 ptsQuestion 15
125\pi
The volume of the solid formed when the area enclosed by the x -axis, the line y =
x and the line x = 5 is rotated about the y -axis is:

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\frac{125\pi}{3}
250\pi
\frac{250\pi}{3}
1 ptsQuestion 16
\text{$f$ is continuous at $0$ if $a=c$ or $b=c$.}
\text{None of the other items are true. }

\text{$f$ is continuous at $0$ if $a=b$.}
\text{$f$ is continuous at $0$ if $a$, $b$, and $c$ are finite.}
Let f x be a function of one real variable, such that
lim x → 0 - f x = a , lim x → 0 + f x = b , f 0 = c ,
for some real numbers a , b , c . Which one of the following statements is true?
1 ptsQuestion 17
30
24
2
12
Let P ( x ) = 1 + 2 x + 4 x 2 + 2 x 3 + 4 x 4 + 10 x 5 be the fifth-degree Taylor
polynomial for the function f ( x ) about x = 0 . What is the value of
f'''\left(0\right) ?
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1 ptsQuestion 18
One
There is not enough information to be able to decide.
Two
Three
You are given that a sixth order polynomial f z with real coefficients has six
distinct roots. You are also given that z = 2 + 3 i , z = 1 - i , and z = 1 are solutions
of f z = 0 . How many real solutions to the equation f z = 0 are there?
1 ptsQuestion 19
-1-i
-1+i
1+i
1-i
Write 5 + i 2 + 3 i in Cartesian form.
1 ptsQuestion 20
F\left(-2\right)=0
F'\left(2\right)=0
Let g t be a differentiable function and let
F ( x ) = ∫ - 2 x g t d t.
You are given that g 2 = 0 and that g t > 0 , for all t ∊ - 2 , 1 . Furthermore, assume
that g t is strictly decreasing on the interval - 1 , 1 . Which one of the following
statements is false?
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F\left(0\right)>0
F''\left(0\right)>0
Section 2
Extended Answer Section (50%)
There are three extended answer questions, each with multiple parts.
The questions are of equal value.
Working must be shown.
These questions must be answered individually on your own
paper, scanned/photographed, and uploaded into the each canvas assignment:
MATH1021 Final Exam Submission Question 1
MATH1021 Final Exam Submission Question 2
MATH1021 Final Exam Submission Question 3
The questions ONLY appear in this quiz. Do not submit this quiz until you have
finished answering ALL extended response questions.
Your document should be labelled with your SID and unit code in the format
‘SID_MATH1021’. Do not include your name.
Extended Answer Question 1
(a) Let z = - 1 2 + 3 2 i .
(i) Calculate the modulus of z and the principal argument of z ,
showing all working.
(ii) Write z in polar exponential form.
(iii) Write z 2021 in Cartesian form, showing all working.Processing math: 0%
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(b) Use the substitution t = ln x to compute the following definite integral,
showing all working
∫ 1 e ln x x d x .
(c) Let f ( x ) = 2 x 3 - 3 x 2 + 1 . Find the global maximum and global minimum
values of f ( x ) on the interval 0 , 2 . Show all working.
Extended Answer Question 2
(a) Compute the following limits, showing all working.
(i) lim x → 0 2 cos x - 2 + x 2 x 4
(ii) lim x → ∞ sin ( ln x ) x 2
(b) Let f ( x ) = 1 + x .
(i) Compute the second order Taylor polynomial P 2 ( x ) of f ( x ) about
x = 0 , showing all working.
(ii) Use Lagrange's formula to find an expression for the remainder term
R 2 ( x ) = f ( x ) - P 2 ( x ) . Show all working.
(iii) Use the previous part to prove that if x ≥ 0 then
1 + x ≥ 1 + 1 2 x - 1 8 x 2 .
Extended Answer Question 3
(a) Let b > 0 . Let R be the region of the x y -plane bounded with 0 ≤ x ≤ b and 0
≤ y ≤ e - x . Let V ( b ) be the volume of the solid obtained by rotating the region
about the y -axis.
(i) Sketch the region R .
(ii) Compute V ( b ), showing all working.
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(iii) Calculate the limit lim b → ∞ V b . Show all working.
(b) Let
m ( z ) ≤ 3 π . Let f : A → ℂ \ 0 be the function with f ( z ) = e z .
(i) Sketch the region A in the complex plane.
(ii) Prove that the function f : A → ℂ \ 0 is bijective.
(iii) Find a formula for the inverse function f - 1 : ℂ \ 0 → A . Show all
working.
0 ptsQuestion 21
Yes
I confirm that:
I have answered all exam questions including the extended response
questions as instructed. (If not, go back and complete the extended response
questions now).
I also understand that:
I will first need to submit this quiz, and then I have 25 minutes to (scan), save
and upload my extended response question answers to the Canvas
assignment. Submissions of the extended response questions after this upload
time will receive no marks.
You must upload a separate document for each question's specified assignment
dropbox within 25 minutes after the quiz submission.
MATH1021 Final Exam Submission Question 1
MATH1021 Final Exam Submission Question 2
MATH1021 Final Exam Submission Question 3
You must not handle your mobile device at any point prior to submitting this
quiz as it will be flagged as an integrity breach by ProctorU.
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