The University of Sydney
School of Mathematics and Statistics
Assignment 2 Part A: Calculus
MATH1062: Mathematics 1B Semester 1, 2025
Lecturers: Tiangang Cui and Jun Yong Park
This individual assignment is due by 11:59pm Sunday 11 May 2025, via Canvas.
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Copyright © 2025 The University of Sydney 1
Part A: Calculus
Submission instructions
Solutions to the calculus part (Part A) must be prepared in written form, and uploaded as a
single pdf file to https://canvas.sydney.edu.au/courses/64063/assignments/598719.
Calculus questions
1. (a) Find the solution of the second order homogeneous linear differential equation
x′′ + 9x = 0
with x(0) = 0 and x′(0) = 2. Show all your working and explain your derivation.
(b) Find the general solution of the second order inhomogeneous linear differential
equation
x′′ − 2x′ + x = 8e3t
Your solution must include both the particular solution to inhomogeneous part
and also the general solution to homogeneous part with 2 constants of integration.
Show all your working and explain your derivation.
2. (a) Find the parametric equations with domain which gives the Cartesian equation
16x2 + 9y2 = 144. Show all your working and explain your answer.
(b) Given the curve x4− 5xy+ y3 = −3, find the value of dy
dx
at the point (1, 1). If you
use Implicit Function Theorem, clearly state the formula you use and check that
you can use IFT as the conditions of the Theorem are met.
3. Consider the surface defined by z = f(x, y), where:
f(x, y) = ex
3+y2
(a) Calculate the first order partial derivatives fx(x, y) and fy(x, y) at the point
(x, y) = (1, 1). Show all your working.
(b) Calculate the equation of the tangent plane to the surface f(x, y) at the point
(x, y) = (1, 1). Show all your working.
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