MAST10005 Calculus 1

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Student
number
Semester 1 Assessment, 2021
School of Mathematics and Statistics
MAST10005 Calculus 1
Reading time: 30 minutes | Writing time: 3 hours | Upload time: 30 minutes
This exam consists of 22 pages (including this page)
Permitted Materials
• This exam and/or an offline electronic PDF reader, one or more copies of the masked
exam template made available earlier and blank loose-leaf paper.
• One double sided A4 page of notes (handwritten or printed).
• No calculators are permitted.
Instructions to Students
• If you have a printer, print the exam one-sided. If using an electronic PDF reader to read
the exam, it must be disconnected from the internet. Its screen must be visible in Zoom.
No mathematical or other software on the device may be used. No file other than the
exam paper may be viewed.
• Ask the supervisor if you want to use the device running Zoom.
Writing
• There are 12 questions with marks as shown. The total number of marks available is 110.
• You should attempt all questions and will receive partial marks for partial answers.
• Write your answers in the boxes provided on the exam that you have printed or the masked
exam template that has been previously made available. If you need more space, you can
use blank paper. Note this in the answer box, so the marker knows. The extra pages can
be added to the end of the exam to scan.
• If you have been unable to print the exam and do not have the masked template write
your answers on A4 paper. The first page should contain only your student number, the
subject code and the subject name. Write on one side of each sheet only. Start each
question on a new page and include the question number at the top of each page.
Scanning
• Put the pages in number order and the correct way up. Add any extra pages to the end.
Use a scanning app to scan all pages to PDF. Scan directly from above. Crop pages to
A4. Make sure that you upload the correct PDF file and that your PDF file is readable.
Submitting
• You must submit while in the Zoom room. No submissions will be accepted after
you have left the Zoom room.
• Go to the Gradescope window. Choose the Canvas assignment for this exam. Submit
your file. Wait for Gradescope email confirming your submission. Tell your supervisor
when you have received it.
©University of Melbourne 2021 Page 1 of 22 pages This is the masked template
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MAST10005 Calculus 1 Semester 1, 2021
Question 1 (8 marks)
(a)
(b)
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MAST10005 Calculus 1 Semester 1, 2021
Question 2 (11 marks)
(a)
(b)
(c)
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MAST10005 Calculus 1 Semester 1, 2021
(d)
(e)
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MAST10005 Calculus 1 Semester 1, 2021
Question 3 (11 marks)
(a)
(b)
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MAST10005 Calculus 1 Semester 1, 2021
(c)
(d)
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MAST10005 Calculus 1 Semester 1, 2021
Question 4 (8 marks)
(a)
(b)
(c)
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Question 5 (7 marks)
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Question 6 (7 marks)
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(b)
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(b)
(c)
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MAST10005 Calculus 1 Semester 1, 2021
(e)
(f)
(g)
x
y

AssignmentTutorOnline

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MAST10005 Calculus 1 Semester 1, 2021
Question 8 (7 marks)
(a)
(b)
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MAST10005 Calculus 1 Semester 1, 2021
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(b)
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MAST10005 Calculus 1 Semester 1, 2021
(c)
(d)
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MAST10005 Calculus 1 Semester 1, 2021
Question 10 (8 marks).
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(b)
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MAST10005 Calculus 1 Semester 1, 2021
(c)
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MAST10005 Calculus 1 Semester 1, 2021
Question 11 (10 marks)
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MAST10005 Calculus 1 Semester 1, 2021
(b)
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Question 12 (11 marks)
(a)
(b)
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MAST10005 Calculus 1 Semester 1, 2021
Extra space for solution of 12(b), if needed.
End of Exam | Total Available Marks = 110
Turn the page for appended material
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MAST10005 Calculus 1 Semester 1, 2021
Useful Formulae
Pythagorean identity
cos2(x) + sin2(x) = 1
Compound angle formulae
sin(x + y) = sin(x) cos(y) + cos(x) sin(y)
cos(x + y) = cos(x) cos(y) – sin(x) sin(y)
tan(x + y) = tan(x) + tan(y)
1 – tan(x) tan(y)
Derivatives of inverse trigonometric functions
arcsin0(x) = p1 1- x2
arccos0(x) = -p1 1- x2
arctan0(x) = 1
1 + x2
Antiderivatives from inverse trigonometric functions

xs Z ps2–1 x2 dx = arccos xs + C
Z s2 +1 x2 dx = 1s arctan xs + C
where s is a positive constant, and C is an arbitrary constant of integration.
Complex exponential formulae
eiθ = cos(θ) + i sin(θ)
cos(θ) = 1
2
eiθ + e-iθ
sin(θ) = 1
2i
eiθ – e-iθ
Vector projections
• The vector projection of v onto u is vk = (u^ · v)u^ = ku; where k 2 R is the unique solution
of u · (v – ku) = 0.
• The vector component of v perpendicular to u is v? = v – vk:
Complex roots
The n-th roots of w = seiφ are sn1 ei( n1 (φ+2kπ)) for k = 0; 1; : : : ; n – 1.
Changes in speed
Provided r0(t) 6= 0, the speed function kr0(t)k is decreasing when r0(t) · r00(t) increasing when r0(t) · r00(t) > 0.
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