The University of Sydney
School of Mathematics and Statistics
Assignment 1
MATH1062: Mathematics 1B Semester 2, 2024
Lecturers: Tiangang Cui, Jack Freestone, Brad Roberts, and Zhou Zhang
This individual assignment is due by 11:59pm Sunday 25 August 2024, via
Canvas. Late assignments will receive a penalty of 5% per calendar day until the
closing date. After 10 days, the mark will be down to 0. The submission should
include your SID. Please make sure you review your submission carefully. What
you see is exactly how the marker will see your assignment. Submissions can be
overwritten until the due date. To ensure compliance with our anonymous marking
obligations, please do not under any circumstances include your name in any area of
your assignment; only your SID should be present. 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. If you have
technical difficulties with your submission, see the University of Sydney Canvas Guide,
available from the Help section of Canvas.
This assignment is worth 5% of your final 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 effort to master. The marker will give you feedback
and allocate an overall mark to your assignment using the following criteria:
Copyright © 2024 The University of Sydney 1
Submission Instructions
Solutions to Part A must be prepared in handwritten form, and uploaded as a single pdf file to
https://canvas.sydney.edu.au/courses/59770/assignments/553027. Although we allow
iPad/tablet writing for assignments, it’s highly recommended for students to rely on traditional
pen and paper writing to mimic the exam situation.
Solutions to Part B must be prepared as a single html file and submitted to https://canvas.
sydney.edu.au/courses/59770/assignments/553028. You should use the following file to
complete Part B.
• An R Markdown worksheet Assignment1Worksheet.Rmd at https://canvas.sydney.
edu.au/courses/59770/files/38709347.
You need to write your solutions as either embedded R code or text answers in the provided
worksheet, and then generate the html file using Knit in R Studio.
Part A: Calculus Questions
Please justify your answers by showing all relevant works. Correct answers without adequate
justification will not receive full marks.
1. Write the following differential equations for the unknown function, y(x) with the variable
x, in the standard form and illustrate the direction field by drawing the tangent intervals
at the four points: (1, 0), (0, 1), (−1, 0) and (0,−1) in the (x, y)-plane.
For each differential equation, find the general solution, then find the particular solution
satisfying y(0) = 1, finally sketch the solution curve in the above illustration of direction
field. All drawing and sketch must be done manually by hand.
(a)
x
dy
dx
− 2x2ex2 = 0.
(b)
y
dy
dx
+ x = 0.
2. Consider a cubic water tank with the side length 4, placed horizontally. There is a hole
at the bottom. Due to a pump, the water will leave the tank according to the following
differential equation:
dV
dt
= −(1 + sin t)
√
V
where V (t) is the volume of water remaining in the tank at the time t.
(a) Find the general solution of this differential equation.
(b) Find the particular solution if the tank is full to start with, that is, V (0) = 43 = 64.
(c) For the solution from (b), let T be the time when the tank becomes empty, that
is V (T ) = 0. Is T > 14 or T < 14? Explain your choice.
2
Part B: Statistics Questions
Part B questions are provided in the R Markdown worksheet at https://canvas.sydney.edu.
au/courses/59770/files/38709347. You should answer these questions using the worksheet
and use the Knit button in R Studio to generate the html file. We can only mark the html
file.