Multivariate Calculus and ordinary differential equations
MATH 1052

Winter 2011
MATH1052
Assignment 3 Solutions
2008
Second Semester
1. (6 marks) I wish to make a cylindrical metal can with a base and lid. Although I can
make the can with a base of any radius and any vertical height the total surface area is fixed at 600 cm2 . Use La
Multivariate Calculus and ordinary differential equations
MATH 1052

Winter 2011
This paper is NOT to
be removed from the ex~
amination room.
INTERNAL '_ STUDENTS
ONLY 
THE UNIVERSITY OF QUEENSLAND
Second Semester Examination, November 2004
. a, ., ', MATH1052
 *.r(Multivariate Calculus and ODEs)
Time: 2 hours for working
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Multivariate Calculus and ordinary differential equations
MATH 1052

Winter 2011
VENUE:
SEAT NUMBER :.
STUDENT
NUMBER:
STUDENT NAME :
Family Name
First Name
FINAL EXAMINATION
St Lucia Campus
Semester Two 2011
MATH1052 Multivariate Calculus and ODEs
PERUSAL TIME
10 mins. During perusal, writing is not permitted at all
WRITING TIME
2:00
Multivariate Calculus and ordinary differential equations
MATH 1052

Winter 2011
MATH1052 Problem Sheet #1 Solutions
Semester 2, 2016
1. Without completing the square, make a guess at identifying each of the following conic
sections:
a) y = x2 2y 2 + 1, hyperbola.
b) x2 + y 2 2 = 0, circle.
c) y + 2x = x2 + 3, parabola.
d) x2 2y 2 = 4
Multivariate Calculus and ordinary differential equations
MATH 1052

Winter 2011
MATH1052
Extra Questions: Limits & Continuity
Semester 2 2016
1. Let D = R2 \cfw_(0, 0) and let f : D R be given by
f (x, y) =
1 exp(x2 y 2 )
.
x2 + y 2
Show that the lim(x,y)(0,0) f (x, y) does not exist.
SOLUTION:Let y = x (with x 6= 0). Then f (x, x) =
Multivariate Calculus and ordinary differential equations
MATH 1052

Winter 2011
MATH1052 Problem Sheet #6 Solutions
1. Find
Semester 2, 2016
dw
a) by using the appropriate chain rule and b) by first converting w to a function
dt
of t:
i) w = xy,
x = 2 sin t,
ii) w = xy tan z,
iii) w = xyz,
y = cos t,
x = t,
x = t2 ,
y = t2 ,
y = 2t,
Multivariate Calculus and ordinary differential equations
MATH 1052

Winter 2011
MATH1052 Problem Sheet #7 Solutions
Semester 2, 2016
1. Find and classify the critical points of the functions f given by:
a) f (x, y) = xy ex
2 y 2
,
1
b) f (x, y) = 8xy (x + y)4 ,
4
c) f (x, y) = sin x sin y,
d) f (x, y) = ex (1 cos y).
a)
f = (1 2x2 )y
Multivariate Calculus and ordinary differential equations
MATH 1052

Winter 2011
% (Underline family name)
it». OF QUEENSLAND STUDENTNAME:
w STUDENT NUMBER:
St Lucia Campus SEAT NUMBER: _________________ 
FINAL Examination
Second Semester, 2006
COURSE CODE MATH1052
COURSE TITLE MULTIVARIATE CALCULUS
Multivariate Calculus and ordinary differential equations
MATH 1052

Winter 2011
MATH1052
Semester 1, 2017
Assignment Questions
Questions from the following list, plus some additional questions, will form part
of your four assignments.
Closer to the due date for a given assignment you will be informed, in lectures,
via email and on
Multivariate Calculus and ordinary differential equations
MATH 1052

Winter 2010
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Multivariate Calculus and ordinary differential equations
MATH 1052

Winter 2010
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Multivariate Calculus and ordinary differential equations
MATH 1052

Winter 2010
Please note that LIFT does not warrant the correctness of the materials contained within the notes. Additionally, in some cases, these
notes were created for previous semesters and years. Courses are subject to change over time, both in content and scope
Multivariate Calculus and ordinary differential equations
MATH 1052

Winter 2010
Thus:
(
)
So this complicated parametrisation was actually just a straight line! Dont be fooled!
Position Vectors
The position vector ( )
( )
Similarly, the position vector ( )
( ( ) ( ) ( ).
( ) traces out the path given by the parameterisation ( ( ) ( )
Multivariate Calculus and ordinary differential equations
MATH 1052

Winter 2010
This has the discriminant:
(
)
Where
, the spring is underdamped, or weakly damped.
Where
, the spring is critically damped.
Where
, this the spring is overdamped, or strongly damped.
Might want to study the examples on this!
The pendulum
According to New
Multivariate Calculus and ordinary differential equations
MATH 1052

Winter 2010
(Turn into a general case)
Triangle method:
Any three points that lie in a plane (that do not lie on a single straight line) uniquely determine that plane.
Finding the equation of a plane given three points:
1.
2.
3.
4.
5.
Write out the plane equation usi
Multivariate Calculus and ordinary differential equations
MATH 1052

Winter 2010
Or:
(
)
CASE 2: ( ) is a polynomial times an exponential
Lets say youve got
No amount of differentiating will reduce that thing to a constant. Weve thus got to get rid of the exponential
first. The trick is to put:
( )
( )
So we want some UNKNOWN function
Multivariate Calculus and ordinary differential equations
MATH 1052

Winter 2011
Solutions to Module 1
Solutions to the exercises for Module 1 are provided below. The answers you submitted for the exercises are also shown, so that you can
compare your answers with the solutions.
Exercise 1
Question
In the Command Window use a colon (
Multivariate Calculus and ordinary differential equations
MATH 1052

Winter 2011
Semester One Final Examination, 2016
This exam paper must not be removed from the venue
* ' voitxom
Venue
THE UNIVERSITY Seamumber
OF QUEENSLAND Student Number
AUSTRALIA
Family Name
First Name
School of Mathematics & Physics
EXAMINATHON
Semester One Fin
Multivariate Calculus and ordinary differential equations
MATH 1052

Winter 2011
Venue
THE UNIVERSITY Seamumber
Q 9 OF QUEENSLAND Student Number ]_L_l_l_l_l_l__l
A U TRA L I A
S Family Name
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School of Mathematics & Physics
EXAMINATION
3 OLUCW 6 NS
PRACTICE Semester One Mid
Multivariate Calculus and ordinary differential equations
MATH 1052

Winter 2011
I
1
MATT11052 MULTIVARIATE CALCULUS 85 ODEs
1 Practice Semester Examination, Semester 1, 2019K
1. Let ~ 20 91 %
f(w y>=4eccp<x + =44 3
l (3.) Find g andgUmp) 9. y \l1 if M
' b L  3 M
at 1: [74;ka Q 2
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)Further let w 3273 2and y:
Multivariate Calculus and ordinary differential equations
MATH 1052

Winter 2011
Semester One Final Examination, 2016 MATH1052 MULTIVARIATE CALCULUS 85 ODEs
Venue . \
THE UNIVERSITY Seamumbe.
W01: QUEENSLAND Student Number I_I_l_._l_i_l_l_l
AUSTRALIA
Family Name
This exam paper must not be removed from the venue First Name
School o