1
CARLETON UNIVERSITY
FINAL EXAMINATION SOLUTIONS
MATH 2004 A, B, C, D
Fall 2013
DURATION: 3 HOURS
Department Name and Course Number: School of Mathematics and Statistics, MATH 2004 A, B, C, D.
Course Instructor(s): Dr. A.B. Mingarelli (Sect. A), Dr. R. C
Multivariable Calculus for Engineering or Physics
School of Mathematics and Statistics, Carleton University
October 20, 2010
(Carleton University)
Multivariable Calculus for Engineering or Physics
October 20, 2010
1 / 20
Outline
1
Review
2
Vector Function
MATH2004 Notes - By Eric Hua
Contents
1.11.6. Vectors in the Plane and in 3-D
3
1.7-1.10. The Dot Product, Cross Product, and Applications
5
2.1-2.5. Lines and Planes
7
2.6 Rotations in the Plane
9
2.7-2.8. Parametric Curves
9
2.9 Applications to Area Pro
Preface
This Complete Solutions Manual contains solutions to all of the exercises in chapters 1015 of Calculus, First Edition, by Soo T.
Tan. These correspond to chapters 914 of Calculus, Early Transcendentals, First Edition. A Student Solutions Manual is
MATH2004E/BIT2005
Multivariable Calculus for Engineering or Physics (Multivariable Calculus for photonics)
Winter 2015
Instructor:
Mathieu Lemire
Office: 5250 Herzberg Building
Tel.: 613-520-2600 ext. 1983
E-mail: [email protected] or through cuLe
Part A: Multiple Choice Questions (2 marks each)
A1. In polar coordinates, the equation of the circle (x 1)2 + y 2 = 1 is
(a) r = , (b) r = 1, (c) r = 1 + sin , (d) r = 2 cos , (e) r2 = 1 + 2r sin .
A2. The area enclosed by one loop of the lemniscate r2 =
Carleton U, School of Math & Stats
Multivariable Calculus
for Engineering or Physics
Math 2004-C
Lecturer: Dr. <J Cova
Fall 2014
. OFFICE:
HP 5250.
:
TELEPHONE:
520-2600 ext 1983.
:
. E-MAIL:
[email protected]
:
. TEXTBOOK:
Multivariable Calculus, by
"!#%$'&)(*$,+-.!/1032*+54768/19;:</1=>!?(*+@!BAC+5!?([email protected]
DEGFHIKJ)LMNLPOQRIOQGSTOQ*UVOHQ*WNXN*Y
ZL[ONXN\O]SG^X_
`3^X_a`QbNX
ced
dhgidkj
dmlnd
d>lqp3r
WfO
WfO
*^o_
ced
cxw
y w
c w
y w
c w
y w
]'[
M
g
P
\
*^X_
dsgidkj
g
WfOcfw_z|WfOcfw_~|
b^X_z|*^o_~|
1
Practice problems Week 4
MATH2004E
Winter 2015
Practicing is very important in order to do well in this course. Practice as many
problems as possible from the following list. Make sure to try at least a few questions
from each possible type of questions
1
Practice problems Week 2
MATH2004E
Winter 2015
Practicing is very important in order to do well in this course. Practice as many
problems as possible from the following list. Make sure to try at least a few questions
from each possible type of questions
1
Practice problems Week 3
MATH2004E
Winter 2015
Practicing is very important in order to do well in this course. Practice as many
problems as possible from the following list. Make sure to try at least a few questions
from each possible type of questions
1
Practice problems List Week 6
MATH2004E
Practicing is very important in order to do well in this course. Practice as many problems as possible until you feel confortable with the material. Solutions to odd-value
problems of the textbook can be found at
1
Practice problems List Week 5
MATH2004E
Practicing is very important in order to do well in this course. Practice as many problems as possible until you feel confortable with the material. Solutions to odd-value
problems of the textbook can be found at
1
Practice problems Week 8
MATH2004E
Winter 2015
Practicing is very important in order to do well in this course. Practice as many
problems as possible from the following list. Make sure to try at least a few questions
from each possible type of questions
1
Practice problems Week 9
MATH2004E
Winter 2015
Practicing is very important in order to do well in this course. Practice as many
problems as possible from the following list. Make sure to try at least a few questions
from each possible type of questions
Math 2004 B
Tutorial # 7
Date: Dec 6, 2010. In Tutorial
I
Tutorial Questions
Question I.1. Evaluate the line integral:
1)
2)
3)
4)
5)
6)
ydx + (x + y 2 )dy where C is the arc of the parabola x = 1 y 2 from (0, 1) to (0, 1).
C 2yz cos xds where C : x = t,
Math 2004 B
Test 4
Last Name
Nov 22, 2010
First Name
Student Number
Question 1. [17] Let
f (x, y, z) = 2x + z, g(x, y, z) = x2 + y 2 + z 2
1) [3 Points] Find f (x, y, z) and g(x, y, z).
2) [3 Points] Find the equation of the tangent plane to the surface x
Math 2004 B
Test 3
Last Name
First Name
Nov 8, 2010
Student Number
Question 1. [15] Consider the curve r(t) = 4 cos(t) i +3t j +4 sin(t) k .
1)
2)
3)
4)
5)
[3
[3
[3
[3
[3
Points]
Points]
Points]
Points]
Points]
Find
Find
Find
Find
Find
0
r(t)dt.
the unit
Multivariable Calculus for Engineering or Physics
School of Mathematics and Statistics, Carleton University
October 18, 2010
(Carleton University)
Multivariable Calculus for Engineering or Physics
October 18, 2010
1 / 30
Outline
1
Review
2
Vector Equation
Multivariable Calculus for Engineering or Physics
School of Mathematics and Statistics, Carleton University
October 25, 2010
(Carleton University)
Multivariable Calculus for Engineering or Physics
October 25, 2010
1 / 23
Outline
1
Review
2
Functions of Se
Multivariable Calculus for Engineering or Physics
School of Mathematics and Statistics, Carleton University
October 25, 2010
(Carleton University)
Multivariable Calculus for Engineering or Physics
October 25, 2010
1 / 19
Outline
1
Review
2
ARC length and
Multivariable Calculus for Engineering or Physics
School of Mathematics and Statistics, Carleton University
October 13, 2010
(Carleton University)
Multivariable Calculus for Engineering or Physics
October 13, 2010
1 / 32
Outline
1
Review
2
Dot Product
Den
Math 2004 B
Tutorial # 1
Date: Sept 20, 2010. In Tutorial
Question 1. Let
< x < 0,
1,
f (x) =
1,
0 x < ,
and
f (x + 2) = f (x), for all x R.
Find the Fourier series of f .
Solution: The period of f is 2L = 2 so L = . Since f (x) is an odd function, an =
FT
RA
Lecture Notes on Fourier Series (Math 2004 B)
V. 1
Dr. Gang Li
School of Mathematics and Statistics
D
Carleton University
Fourier Series
Notes by G. Li
D
RA
FT
Math 2004 B
Page 1 of 17
FT
Contents
Introduction
1.1 Denition of Fourier series . . . .
Math 2004 B
Tutorial # 2
Date: Sept 27, 2010. In Tutorial
Question 1. Let
x
, 0<x<
4
2
(i) Find the Fourier series of f , (ii) Find the Fourier cosine series of f , (iii) Find the Fourier sine series of
f (x) =
f.
Solution: (i) f can be extended to a pe
Multivariable Calculus for Engineering or Physics
School of Mathematics and Statistics, Carleton University
November 4, 2010
(Carleton University)
Multivariable Calculus for Engineering or Physics
November 4, 2010
1 / 16
Outline
1
Review
(Carleton Univers
Multivariable Calculus for Engineering or Physics
School of Mathematics and Statistics, Carleton University
November 3, 2010
(Carleton University)
Multivariable Calculus for Engineering or Physics
November 3, 2010
1 / 35
Outline
1
Review
(Carleton Univers
Math 2004 B
Tutorial # 3
Date: Oct 18, 2010. In Tutorial
I
Tutorial Questions
Question I.1. Consider the parametric curve given by x = et + t, y = et t.
1) Find
dy
dx .
2) Find the equation of the tangent line to the curve at t = 1.
3) Find the point on t
Math 2004 B
Tutorial # 5
Date: Nov 15, 2010. In Tutorial
I
Tutorial Questions
Question I.1. Find the equation of the tangent plane to the given surface at the specied point
z2
x2 y 2
+
+
= 1, (1, 2, 11)
4
9
36
Solution: The tangent plane is:
22
2 11
21
(x
Functions and their properties
9
=
y
z
y
=
=
x
1.2
z
z
+y
x
y
=
x2
x2 + y 2
1
1
y ( 2
) (2y)
2 x + y2
y2
x2 + y 2
1
Directional Derivatives
Let z = f (x, y) dene a surface S in R3 and P0 (x0 , y0 ) a point on S.
The gradient of z (or f ) at P0 is a vector
1
CARLETON UNIVERSITY
SPECIAL FINAL
EXAMINATION
December 2010
DURATION: 3 HOURS
Department Name and Course Number: School of Mathematics and Statistics, MATH 1004 A, B, C, D.
Course Instructor(s): Dr. A.B. Mingarelli (Sect. A), Dr. M. Sadeghi (Sect. B), M
mg . I I mm + Na
U ad I 85:35 Op BUM 28:0 9? 8D #33 | a. van pm 0mm 1 H Swag ta|
3%
.Amm Hm "HIVnN 8m m3 + mam H N H3 swim 833m 0H? 0p 9H: EHHHHOHH 2%. mo HHOEHWHHUo mg UHHE Am WMHNS m+
H 3:3: peg 3
wummoom 0Q 53 wwwmm Mmaomvm OZ
.mmmvv 95m Em 95: 03 mm 3