Chapter 4: Double Integrals
It is easy to just view integration as a symbolic procedure which is the reverse of differentiation. Remember however:
The integral of f over [a, b] is dened,
via a limit of rectangles, as the area
under the graph of f .
Find
Week 10 Version 2
Time allowed: 25 minutes.
1. (0 marks if correct, -1 mark if incorrect)
Suppose V is a vector space and S = cfw_v1 , v2 , . vn is a nite set of vectors in V over R.
Precisely explain what is meant by a linear combination of vectors in
Week 10 Version 1
Time allowed: 25 minutes.
1. (0 marks if correct, -1 mark if incorrect)
Suppose V is a vector space and cfw_v1 , v2 , . vn is a set of vectors in V .
Precisely explain what is meant if v1 v2 , . vn is a spanning set for V .
cfw_ ,
2.
MATH1251 Sample Solutions
2008 Test 2 ver. 1a
These solutions were written and typed by Ryan Xie. Thanks to Richard Chen and Allan Chen
for their help. Please ask for permission if you are using this for commercial purposes.
Steps have most likely been om
UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
MATH1251 CALCULUS S2 2007
TEST 1 VERSION 2a
This sheet must be lled in and stapled to the front of your answers
Students Family Name
Initials
Tutorial Code
Tutors Name
Note: The use of a c
MATH1251 Sample Solutions
Appendix for Calculus Test 2
In this Ill show something I think is pretty neat, and I think its more rigorous than the
other proofs to nd L for a sequence. Obviously, use this at your own peril and I accept no
responsibility for
Week 10 Version 3
Time allowed: 25 minutes.
1. (0 marks if correct, -1 mark if incorrect)
Dene the statement, The set of vectors cfw_v1 , v2 , . vn is linearly dependent.
2. (3 marks)
Prove that S = cfw_x R3 | x1 + x2 = x3 is a subspace of R3 .
3. (1
lOMoARcPSD
Summary - Notes for final exam covering all course material
Macroeconomics 1 (University of New South Wales)
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lOMoARcPSD
ECON 1102 ~ 2013 Notes
Measuring Macroeconomic Perf
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Summary - book "Principles of Macroeconomics" macroeconomics notes
Macroeconomics 1 (University of New South Wales)
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Macroeconomics Notes
Chapter 1: Measuring Ma
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Lecture notes, lectures 1-20 - Comprehensive econ1102
Macroeconomics 1 (University of New South Wales)
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ECON1102
Macroeconomics 1
MACROECONOMICS
OUTPUT AND
MATH1251 Solutions
2007 Test 1 ver. 2a
These solutions were written and typed by Ryan Xie, and checked and edited by Richard Chen.
Thanks to Allan Chen for his help. Please ask for permission if you are using this for commercial
purposes.
Steps have most
Week 7 Version 3
Time allowed: 25 minutes.
1. (4 marks)
(a) Find the general solution of the dierential equation
du
+ 2u = e2x .
dx
(b) Hence, use the substitution
dy
+ 2y
dx
to nd the general solution of the dierential equation
u=
d2 y
dy
+ 4 + 4y = e2x
Week 7 Version 2
Time allowed: 25 minutes.
1. (2 marks)
Find the general solution of the dierential equation
d2 y
dy
+ 4 + 4y = 8x.
2
dx
dx
2. (3 marks)
dy
Find all functions f (x) such that the dierential equation y 2 sin x + yf (x)
= 0 is exact.
dx
Henc
Week 7 Version 1
Time allowed: 25 minutes.
1. (3 marks)
Solve the dierential equation 2ydx + xdy = 0 by multiplying through by a function of the
form xa y b to make the equation exact.
2. (2 marks)
x
Solve the integral equation y(x) = 1 +
y(t) dt.
0
3. (2
Math1251 Solutions
Algebra
2007 Test 2 ver. 1a
These solutions were written and typed by Ryan Xie. Thanks to Allan Chen and Richard Chen
for their help. Please ask for permission if you are using this for commercial purposes.
Steps have most likely been o
MATH1251 Solutions
Algebra
2007 Test 1 ver. 1b
These solutions were written and typed by Ryan Xie, and checked and edited by Richard Chen.
Thanks to Allan Chen for his help. Please ask for permission if you are using this for commercial
purposes.
Steps ha
Math1251 Solutions
Algebra
2007 Test 3 ver. 2b
These solutions were written and typed by Ryan Xie. Thanks to Allan Chen and Richard Chen
for their help. Please ask for permission if you are using this for commercial purposes.
Steps have most likely been o
MATH1251 Solutions
Algebra
2009 Test 1 ver. 2a
These solutions were written and typed by Ryan Xie, and checked and edited by Richard Chen.
Thanks to Allan Chen for his help. Please ask for permission if you are using this for commercial
purposes.
Steps ha
Math1251 Solutions
Algebra
2009 Test 2 ver. 1a
These solutions were written and typed by Ryan Xie. Thanks to Allan Chen and Richard Chen
for their help. Please ask for permission if you are using this for commercial purposes.
Steps have most likely been o
Math1251 Solutions
Algebra
2008 Test 2 ver. 3b
These solutions were written and typed by Ryan Xie. Thanks to Allan Chen and Richard Chen
for their help. Please ask for permission if you are using this for commercial purposes.
Steps have most likely been o
Math1251 Solutions
Algebra
2008 Test 1 ver. 2a
These solutions were written and typed by Ryan Xie, and checked and edited by Richard Chen.
Thanks to Allan Chen for his help. Please ask for permission if you are using this for commercial
purposes.
Steps ha
UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
MATH1251 MATHEMATICS FOR ACTUARIAL STUDIES
AND FINANCE 1B Algebra S2 2007
TEST 2 VERSION 1A
This sheet must be lled in and stapled to the front of your answers
Students Family Name
Initial
Week 3 Version 1
Time allowed: 25 minutes.
1. (3 marks)
1
(1 + x2 )n dx.
Let In =
0
(a) Show that In =
2n
2n
+
In1 for n 1.
1 + 2n 2n + 1
1
(1 + x2 )2 dx.
(b) Use the reduction formula to evaluate
0
2. (2 marks)
Find
1
dx.
x+ 4x
3. (2 marks)
Solve the di
Week 3 Version 2
Time allowed: 25 minutes.
1. (3 marks)
e
(ln x)n dx.
Let In =
1
(a) Show that In = e nIn1 for n 1.
e
(ln x)4 dx.
(b) Use your result to nd
1
2. (2 marks)
cos4 x dx.
Find
0
3. (2 marks)
Solve the dierential equation (1 + x2 )
dy
= 12xy 2 ,
Week 4 Version 1
Time allowed: 25 minutes.
1. (3 marks)
Show that if z +
1
is a real number, then either Im(z) = 0, or |z| = 1.
z
2. (2 marks)
d2 x
dx
Are the solutions of the continuous time system 2 2 2 + x = 0 stable or unstable?
dt
dt
Give reasons.
3.