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Submission
• You must submit your completed Written Assignment on Crowdmark by 6:00pm (EST)
Friday February 26, 2021. You will be emailed a link from Crowdmark with information on how to
submit your solutions.
• Late assignments (even by a couple seconds) will not be accepted.
• Consider submitting your assignment well before the deadline.
• You do not need to print out this assignment; you may submit clear pictures/scans of your work on lined
paper, or screenshots of your work.
• You do not need to submit the cover page, the grading scheme, or the hint page.
• You must correctly orient/rotate and order your submission.
• If you require additional space, please insert extra pages.
Additional Instructions
You must justify and support your solution to each question. You should use full sentences.
Academic Integrity
You are encouraged to work with your fellow students while working on questions from the written assignments.
However, the writing of your assignment must be done without any assistance whatsoever. Do not post partial or
complete solutions to Piazza.
I affirm that this assignment represents entirely my own efforts. I confirm that:
• I have not copied any portion of this work.
• I have not allowed someone else in the course to copy this work.
• This is the final version of my assignment and not a draft.
• I understand the consequences of violating the University’s academic integrity policies as outlined in the
Code of Behaviour on Academic Matters.
By submitting solutions for grading I agree that the statements above are true. If I do not agree with the
statements above, I will not submit my assignment and will consult the course coordinator (Mike Pawliuk) immediately.
This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 2.5 Canada License.
Original Authors: Mike Pawliuk, Qun Wang
1
2 MAT136H5 S - WINTER 2021 - WRITTEN ASSIGNMENT 2
Need help?
This problem set is designed to make you think, and it contains problems you’ve never seen before. We expect
you’ll need to come back to this assignment multiple times and try different approaches; we don’t expect you to
solve everything in one sitting. It’s normal to get stuck! Every time you get stuck that means you’re about to
learn something when you get unstuck. Look for those moments!
There are hints to some questions on the final page of the assignment.
If you’re stuck for more than a day or two, you may want to ask for help. Here are some places to do that:
• Ask on Piazza. (If you want to post some of your work, please make it a private post.)
• Office hours. See Quercus for times and locations. There are about 20 hours a week, and you can attend
the office hours of any instructor, not just the one for your LEC section.
Good luck, have fun!
Grading Scheme
This is the grading scheme that TAs will use when grading this assignment. You do not need to submit this page.
Question 1. [5 points].
• 1 point for correct, relevant geometry.
• 1 point for setting up a relevant integral.
• 1 point for correctly evaluating this integral.
• 1 point for a correct answer, given in a full sentence.
• 1 point for a clear presentation (including full sentences).
Question 2 [5 points].
• 2 points for correct computations in part 1.
• 1 point for a correct answer to part 2.
• 1 point for appropriate justification to part 2.
• 1 point for a correct answer to part 3 given in a full sentence.
Question 3 [5 points].
• 3 points for giving the main idea to their solution. (This does not have to be long.)
• 2 points for a clear explanation, with justification, including full sentences.
Question 4 [5 points].
• 1 point for finding P by direct calculations.
• 1 point for finding Q by direct calculations.
• 1 point for computing bP + aQ and −aP + bQ.
• 1 point for solving for P and Q in part 2.2.
• 1 point for a clear explanation, using full sentences where appropriate.
MAT136H5 S - WINTER 2021 - WRITTEN ASSIGNMENT 2 3
Question 1. How many identical “Alices” could stand on the surface of the earth between the equator and the
60th parallel north, using up all available space?
For more background on circles of latitude, read the first three paragraphs of https://en.wikipedia.org/
wiki/Circle_of_latitude, which are reproduced on the last page of this assignment.
For the purposes of this question you may assume:
• Alice is h = 1.75 m tall.
• Alice’s volume is volA = 0.06m3
.
• The Earth is a sphere with radius R = 6 371 000 m.
• The Alices can stand on any surface: land, water, etc.
4 MAT136H5 S - WINTER 2021 - WRITTEN ASSIGNMENT 2
Question 2. In this problem we give another characterization for the average function value introduced in lecture.
Let a < b be constants and let f be an integrable function. Consider the function
F : R → R, defined by F(x) = Z b
a
(f(t) − x)
2 dt.
(1) Find the critical point(s) of F (i.e., the point(s) where dF
dx
= 0).
(2) On such critical point(s) does the function F achieve a minimum, maximum, or neither?
(3) Based on parts 1 and 2, characterize the average function value 1
b − a
Z b
a
f(t) dt in terms of the function
F.
MAT136H5 S - WINTER 2021 - WRITTEN ASSIGNMENT 2 5
Question 3.
Remark. Everyone in the class can solve this question; you have all the tools you need. It may take you multiple
cups of coffee/tea to get it, but don’t give up.
Hints for this question are contained on the last page of the assignment.
6 MAT136H5 S - WINTER 2021 - WRITTEN ASSIGNMENT 2
Question 4. In this exercise, we will find the indefinite integrals
by two different methods. We assume that a 6= 0, b 6= 0 and both are constants.
(1) Strategy I: Direct Method
(a) Find P by direct calculation.
(b) Find Q by direct calculation.
Hint: You might find the substitution u = tan x helpful.
(2) Strategy II: “Two-for-one”
(a) Compute bP + aQ and −aP + bQ.
(b) From there, solve P and Q at the same time.
Remark. An interesting phenomenon in integration is that sometimes things become easier when shown in pair.
MAT136H5 S - WINTER 2021 - WRITTEN ASSIGNMENT 2 7
Circle of Latitude.
“A circle of latitude on Earth is an abstract east–west circle connecting all locations around Earth
(ignoring elevation) at a given latitude.
Circles of latitude are often called parallels because they are parallel to each other; that is, planes
that contain any of these circles never intersect each other. A location’s position along a circle of
latitude is given by its longitude. Circles of latitude are unlike circles of longitude, which are all
great circles with the centre of Earth in the middle, as the circles of latitude get smaller as the
distance from the Equator increases. Their length can be calculated by a common sine or cosine
function. The 60th parallel north or south is half as long as the Equator (disregarding Earth’s
minor flattening by 0.3%). A circle of latitude is perpendicular to all meridians.
The latitude of the circle is approximately the angle between the Equator and the circle, with
the angle’s vertex at Earth’s centre. The Equator is at 0°, and the North Pole and South Pole are
at 90° north and 90° south, respectively. The Equator is the longest circle of latitude and is the
only circle of latitude which also is a great circle.”
Circle of latitude. (2021, January 15). Retrieved February 03, 2021, from https://en.wikipedia.org/wiki/
Circle_of_latitude
Hints for Q2.
(1) Most of the work is in part 1. Complete part 1 before you move on to parts 2 and 3.
(2) This question may look scary, but if you do the integral one step at a time, everything works out nicely.
(3) Are you integrating with respect to x or t?
Hints for Q3.
(1) You are not asked to compute the integrals individually (which is impossible to do).
(2) This question takes more thinking than writing; a complete solution can be written in 1-2 sentences.
(3) What does the integrand (i.e. the stuff you’re integrating) look like? What does it remind you of?
Hints for Q4.
(1) Part 2.2. Remember that a, b are constants. How many unknowns do you have? How many equations do
you have?
Submission
• You must submit your completed Written Assignment on Crowdmark by 6:00pm (EST)
Friday February 26, 2021. You will be emailed a link from Crowdmark with information on how to
submit your solutions.
• Late assignments (even by a couple seconds) will not be accepted.
• Consider submitting your assignment well before the deadline.
• You do not need to print out this assignment; you may submit clear pictures/scans of your work on lined
paper, or screenshots of your work.
• You do not need to submit the cover page, the grading scheme, or the hint page.
• You must correctly orient/rotate and order your submission.
• If you require additional space, please insert extra pages.
Additional Instructions
You must justify and support your solution to each question. You should use full sentences.
Academic Integrity
You are encouraged to work with your fellow students while working on questions from the written assignments.
However, the writing of your assignment must be done without any assistance whatsoever. Do not post partial or
complete solutions to Piazza.
I affirm that this assignment represents entirely my own efforts. I confirm that:
• I have not copied any portion of this work.
• I have not allowed someone else in the course to copy this work.
• This is the final version of my assignment and not a draft.
• I understand the consequences of violating the University’s academic integrity policies as outlined in the
Code of Behaviour on Academic Matters.
By submitting solutions for grading I agree that the statements above are true. If I do not agree with the
statements above, I will not submit my assignment and will consult the course coordinator (Mike Pawliuk) immediately.
This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 2.5 Canada License.
Original Authors: Mike Pawliuk, Qun Wang
1
2 MAT136H5 S - WINTER 2021 - WRITTEN ASSIGNMENT 2
Need help?
This problem set is designed to make you think, and it contains problems you’ve never seen before. We expect
you’ll need to come back to this assignment multiple times and try different approaches; we don’t expect you to
solve everything in one sitting. It’s normal to get stuck! Every time you get stuck that means you’re about to
learn something when you get unstuck. Look for those moments!
There are hints to some questions on the final page of the assignment.
If you’re stuck for more than a day or two, you may want to ask for help. Here are some places to do that:
• Ask on Piazza. (If you want to post some of your work, please make it a private post.)
• Office hours. See Quercus for times and locations. There are about 20 hours a week, and you can attend
the office hours of any instructor, not just the one for your LEC section.
Good luck, have fun!
Grading Scheme
This is the grading scheme that TAs will use when grading this assignment. You do not need to submit this page.
Question 1. [5 points].
• 1 point for correct, relevant geometry.
• 1 point for setting up a relevant integral.
• 1 point for correctly evaluating this integral.
• 1 point for a correct answer, given in a full sentence.
• 1 point for a clear presentation (including full sentences).
Question 2 [5 points].
• 2 points for correct computations in part 1.
• 1 point for a correct answer to part 2.
• 1 point for appropriate justification to part 2.
• 1 point for a correct answer to part 3 given in a full sentence.
Question 3 [5 points].
• 3 points for giving the main idea to their solution. (This does not have to be long.)
• 2 points for a clear explanation, with justification, including full sentences.
Question 4 [5 points].
• 1 point for finding P by direct calculations.
• 1 point for finding Q by direct calculations.
• 1 point for computing bP + aQ and −aP + bQ.
• 1 point for solving for P and Q in part 2.2.
• 1 point for a clear explanation, using full sentences where appropriate.
MAT136H5 S - WINTER 2021 - WRITTEN ASSIGNMENT 2 3
Question 1. How many identical “Alices” could stand on the surface of the earth between the equator and the
60th parallel north, using up all available space?
For more background on circles of latitude, read the first three paragraphs of https://en.wikipedia.org/
wiki/Circle_of_latitude, which are reproduced on the last page of this assignment.
For the purposes of this question you may assume:
• Alice is h = 1.75 m tall.
• Alice’s volume is volA = 0.06m3
.
• The Earth is a sphere with radius R = 6 371 000 m.
• The Alices can stand on any surface: land, water, etc.
4 MAT136H5 S - WINTER 2021 - WRITTEN ASSIGNMENT 2
Question 2. In this problem we give another characterization for the average function value introduced in lecture.
Let a < b be constants and let f be an integrable function. Consider the function
F : R → R, defined by F(x) = Z b
a
(f(t) − x)
2 dt.
(1) Find the critical point(s) of F (i.e., the point(s) where dF
dx
= 0).
(2) On such critical point(s) does the function F achieve a minimum, maximum, or neither?
(3) Based on parts 1 and 2, characterize the average function value 1
b − a
Z b
a
f(t) dt in terms of the function
F.
MAT136H5 S - WINTER 2021 - WRITTEN ASSIGNMENT 2 5
Question 3.
Remark. Everyone in the class can solve this question; you have all the tools you need. It may take you multiple
cups of coffee/tea to get it, but don’t give up.
Hints for this question are contained on the last page of the assignment.
6 MAT136H5 S - WINTER 2021 - WRITTEN ASSIGNMENT 2
Question 4. In this exercise, we will find the indefinite integrals
by two different methods. We assume that a 6= 0, b 6= 0 and both are constants.
(1) Strategy I: Direct Method
(a) Find P by direct calculation.
(b) Find Q by direct calculation.
Hint: You might find the substitution u = tan x helpful.
(2) Strategy II: “Two-for-one”
(a) Compute bP + aQ and −aP + bQ.
(b) From there, solve P and Q at the same time.
Remark. An interesting phenomenon in integration is that sometimes things become easier when shown in pair.
MAT136H5 S - WINTER 2021 - WRITTEN ASSIGNMENT 2 7
Circle of Latitude.
“A circle of latitude on Earth is an abstract east–west circle connecting all locations around Earth
(ignoring elevation) at a given latitude.
Circles of latitude are often called parallels because they are parallel to each other; that is, planes
that contain any of these circles never intersect each other. A location’s position along a circle of
latitude is given by its longitude. Circles of latitude are unlike circles of longitude, which are all
great circles with the centre of Earth in the middle, as the circles of latitude get smaller as the
distance from the Equator increases. Their length can be calculated by a common sine or cosine
function. The 60th parallel north or south is half as long as the Equator (disregarding Earth’s
minor flattening by 0.3%). A circle of latitude is perpendicular to all meridians.
The latitude of the circle is approximately the angle between the Equator and the circle, with
the angle’s vertex at Earth’s centre. The Equator is at 0°, and the North Pole and South Pole are
at 90° north and 90° south, respectively. The Equator is the longest circle of latitude and is the
only circle of latitude which also is a great circle.”
Circle of latitude. (2021, January 15). Retrieved February 03, 2021, from https://en.wikipedia.org/wiki/
Circle_of_latitude
Hints for Q2.
(1) Most of the work is in part 1. Complete part 1 before you move on to parts 2 and 3.
(2) This question may look scary, but if you do the integral one step at a time, everything works out nicely.
(3) Are you integrating with respect to x or t?
Hints for Q3.
(1) You are not asked to compute the integrals individually (which is impossible to do).
(2) This question takes more thinking than writing; a complete solution can be written in 1-2 sentences.
(3) What does the integrand (i.e. the stuff you’re integrating) look like? What does it remind you of?
Hints for Q4.
(1) Part 2.2. Remember that a, b are constants. How many unknowns do you have? How many equations do
you have?