School of Mathematics and Statistics
Assignment 1
MATH1011: Applications of Calculus Calculus Semester 1, 2018
Web Page: http://sydney.edu.au/science/maths/u/UG/JM/MATH1011/
Lecturers: Emma Carberry, Clio Cresswell
This assignment is due by 11:59pm Thursday 22nd March 2018, via Turnitin.
A PDF copy of your answers must be uploaded in the Learning Management System
(Canvas) at https://canvas.sydney.edu.au/courses/6729. Please submit only
a PDF document (scan or convert other formats). It should include your SID, your
tutorial time, day, room and Tutor’s name. Do not include your name in the PDF
document, since anonymous marking will be implemented. It is your responsibility to
preview each page of your assignment after uploading to ensure each page is included
in correct order and is legible (not sideways or upside down) before con rming your
submission. After submitting you can go back and view your submission to check
it. The School of Mathematics and Statistics encourages some collaboration between
students when working on problems, but students must write up and submit their
own version of the solutions.
This assignment is worth 2.5% of your nal assessment for this course. Your answers should be
well written, neat, thoughtful, mathematically concise, and a pleasure to read. Please cite any
resources used and show all working. Present your arguments clearly using words of explanation
and diagrams where relevant. After all, mathematics is about communicating your ideas. This
is a worthwhile skill which takes time and e ort to master. The marker will give you feedback
and allocate an overall letter grade and mark to your assignment using the following criteria:
Mark Grade Criterion
5 A Outstanding and scholarly work, answering all parts correctly, with clear
accurate explanations and all relevant diagrams and working. There are
at most only minor or trivial errors or omissions.
4 B Very good work, making excellent progress, but with one or two substantial
errors, misunderstandings or omissions throughout the assignment.
3 C Good work, making good progress, but making more than two distinct sub-
stantial errors, misunderstandings or omissions throughout the assignment.
2 D A reasonable attempt, but making more than three distinct substantial
errors, misunderstandings or omissions throughout the assignment.
1 E Some attempt, with limited progress made.
0 F No credit awarded.
Copyright c 2018 The University of Sydney 1
1. As part of a mechanical device there is a circular cog, Cog1, that has a diameter of
50 cm. From the mechanic’s point of view, when the device is in action, Cog1 rotates
anticlockwise and makes two full turns per minute. There is a red marker on Cog1 to
help track this motion.
(a) Consider only the placement of the red marker on Cog1. It is de ned by two
variables, X and Y. X is the horizontal displacement from the vertical line through
the centre of Cog1 and Y is the vertical displacement from the horizontal line
through the centre of Cog1. Let be the angle from the positive horizontal axis
to the line between the centre of Cog1 and the red marker. Assume is measured
anticlockwise and in radians.
(i) Sketch a representation of this situation.
(ii) Let the red marker begin at = 0. Sketch the graphs X( ) and Y( ); the
horizontal and vertical position of the red marker as functions of , as the
marker rotates. Clearly label the axes, the maximums and minimums of the
functions, and any points where the graphs cross the axes.
(iii) Give explicit expressions for X( ) and Y( ).
(b) From a mechanical point of view it turns out to be advantageous to measure
the placement of the red marker from the centre of another cog, Cog2, placed
symmetrically directly below it. The two cogs are placed so they turn together.
Cog2 has diameter 75 cm. Let x be the horizontal displacement of the red marker
from the centre of Cog2 and y be its vertical displacement from the centre of Cog2.
Let be the angle from the line joining the centres of the two cogs to the line
between the centre of Cog1 and the red marker. Again, assume is measured
anticlockwise and in radians.
(i) Sketch a representation of this situation.
(ii) Let = 0 represent when the red marker is at the point of contact of the two
cogs. Sketch the graphs x( ) and y( ); the horizontal and vertical position
of the red marker as functions of , as the marker rotates. Clearly label the
axes, the maximums and minimums of the functions, and any points where
the graphs cross the axes.
(iii) Give explicit expressions for x( ) and y( ).
(c) Keeping with the two cog situation examined in part b) and recalling that Cog1
makes two full turns per minute, we now consider where the red marker is as a
function of time, t.
(i) Set t = 0 as representing when the red marker is at the point of contact of
the two cogs. Sketch the graphs x(t) and y(t); the horizontal and vertical
position of the red marker as functions of t, as the marker rotates. Clearly
label the axes, the maximums and minimums of the functions, and any
points where the graphs cross the axes.
(ii) x(t) and y(t) can be written in the form. x(t) = d + Acos(k(x )) and
y(t) = d + Asin(k(x )). Identify the values for the parameters for each
function x and y and hence write x(t) and y(t) explicitly.