Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics
AM 1411

Fall 2010
The University of Western Ontario
Applied Mathematics 1411b
First Tutorial Test: 29 January 2013
Time: 45 minutes
Use this examination booklet to answer questions.
You are allowed to use calculators and your textbook during this test.
Name:
STUDENT NUMBER
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics
AM 1411

Fall 2010
Western University
Applied Mathematics 1411A
Test 1 Friday February 5 2016
Time: 60 minutes
Version A
o This exam booklet has 3 problems on 7 pages, for a total of 50 points.
a Please answer all questions in the spaces provided.
o Answers without justicat
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics
AM 1411

Fall 2010
Exercises for AM025
D.J. Jeffrey
Department of Applied Mathematics
U. Western Ontario
November 8, 2006
1
Determinants
1. For an n n matrix A,
(a) if A is singular then det(A) = 0.
(b) if A is singular then det(A) 6= 0.
(c) if A is singular then det(A) can
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics
AM 1411

Fall 2010
_I_;,<:z.:_ .5) 7 I;
7, on, 7 , + i: no. .129.U+\OH.I
5.
0x ._ _
. 'H'mrc
, SQ.
.Z_._6 "1.26.
HIM1311
omLLb = . V.
J.H_L_%6_ ._ . 4. Consider the transformation given by
(3;) Z G 31) (:2) (1)
USe the lefthand axes to plot vectors x = [$1, $2]. Use
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics
AM 1411

Fall 2010
Exercises for AM1411
D.J. Jeffrey
Department of Applied Mathematics
U. Western Ontario
March 24, 2015
1
Row Echelon Form
1. If A is an n n matrix and x is an n 1 vector of unknowns, the equation Ax = 0 is called
(a) homogeneous,
(b) inhomogeneous,
(c) inc
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics
AM 1411

Fall 2010
Solutions of equations  AM1411
D.J. Jeffrey
Department of Applied Mathematics
U. Western Ontario
March 23, 2015
1
Equations to solve
1. A system of 3 equations in 3 unknowns
(a) is never inconsistent.
(b) is always consistent.
(c) never has a unique solu
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics
AM 1411

Fall 2010
MultipleChoice Questions on Transformations
D.J. Jeffrey
Department of Applied Mathematics
U. Western Ontario
March 13, 2015
x2
y1
1
1. The matrix equation
=
y2
0
(1, 1)
0
x1
transforms the square
1
x2
x1
into
(a) None of the below.
y2
y2
y1
(b)
(1,
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics
AM 1411

Fall 2014
THE UNIVERSITY OF WESTERN ONTARIO
DEPARTMENT OF APPLIED MATHEMATICS
Applied Mathematics 1411a
Second Tutorial Test, version 1, October 2223, 2013
Time: 60 min
The test is closed book! You are allowed to use only simple nonprogrammable
calculators. Justi
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics
AM 1411

Fall 2014
THE UNIVERSITY OF WESTERN ONTARIO
DEPARTMENT OF APPLIED MATHEMATICS
Applied Mathematics 1411a
First Tutorial Test, version 1, October 1 and 2, 2013
Time: 60 min
The test is closed book! You are allowed to use only simple nonprogrammable
calculators. Just
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics
AM 1411

Fall 2014
THE UNIVERSITY OF WESTERN ONTARIO
DEPARTMENT OF APPLIED MATHEMATICS
Applied Mathematics 1411a
Third Tutorial Test, version 1, November 1213, 2013
Time: 60 min
The test is closed book! You are allowed to use only simple nonprogrammable
calculators. Justi
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics
AM 1411

Fall 2014
THE UNIVERSITY OF WESTERN ONTARIO
DEPARTMENT OF APPLIED MATHEMATICS
Applied Mathematics 1411a
First Tutorial Test, version 1, October 1 and 2, 2013
Time: 60 min
The test is closed book! You are allowed to use only simple nonprogrammable
calculators. Just
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics
AM 1411

Fall 2014
THE UNIVERSITY OF WESTERN ONTARIO
DEPARTMENT OF APPLIED MATHEMATICS
Applied Mathematics 1411a
Second Tutorial Test, version 1, October 2223, 2013
Time: 60 min
The test is closed book! You are allowed to use only simple nonprogrammable
calculators. Justi
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics
AM 1411

Fall 2014
THE UNIVERSITY OF WESTERN ONTARIO
DEPARTMENT OF APPLIED MATHEMATICS
Applied Mathematics 1411a
Third Tutorial Test, version 1, November 1213, 2013
Time: 60 min
The test is closed book! You are allowed to use only simple nonprogrammable
calculators. Justi
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics
AM 1411

Fall 2014
1
AM 1411a, Fall 2014
Lectures 4, September 12
2
Gaussian and GaussJordan Elimination
2.1
Elementary row operations:
1. Scaling: cRi ! Ri (c is a nonzero number)
2. Interchange: Ri $ Rj
3. Elimination: Ri + Rj ! Ri
2.2
RREF (reduced row echelon form)
1.
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics
AM 1411

Fall 2014
Ex. Three systems of linear equations
1.
x y = 1
2x + y = 6
a) Working with equations: eq.2  2 eq.1 ) 3y = 4;
solve for y : y = 4 : Substitute y into eq.1: x = y + 1 =
3
4 = 7:
1+3
3
b) Working with the augmented matrix:
"
1
2
1 1
1 6
1R ! R
2
3 2
!
#
2R
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics
AM 1411

Fall 2014
1
AM 1411a, Fall 2014
Lectures 3, September 10
2
Gaussian and GaussJordan Elimination
2.1
Elementary row operations:
1. Scaling: cRi ! Ri (c is a nonzero number)
2. Interchange: Ri $ Rj
3. Elimination: Ri + Rj ! Ri
2.2
RREF (reduced row echelon form)
1.
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics
AM 1411

Winter 2014
Section 1.1 23
I
Every elementary row operation is reversible. TRUE You can
reverse multiplying by a constant by multiplying by its inverse.
If you add row one to row two and replace row two, then you
can subtract row one from row two to get it back.
I
A
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics
AM 1411

Winter 2014
Chapter 1:
Introduction to
linear equations
and functions
1.1
x1=x, x2=y, x3=z. etc
y=ax1+bx2+cx3+.mxn
Ex: y=3x14x2+7x3
number of solutions for
functions in:
R2
parallel lines, no solution
intercept in a single point, one solution
coincident, infinitely
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics
AM 1411

Winter 2014
Chapter 3: Euclidean Vector Space
3.1: Vectors in R2, R3 and Rn
Parallelogram Rule for vector Addition:
when adding or subtracting vectors it doesn't matter which order the addition or subtraction occurs in.
any two vectors can be used to describe a third
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics
AM 1411

Winter 2014
Chapter 5: Eigenvalues and eigenvectors
5.1: eigenvalues and eigenvectors
If A is a nxn matrix, then a nonzero vector x in Rn is
valid an eigenvector of A (or of the matrix operator TA)
if Ax is a scalar multiple of x; that is Ax=x
, is the eiganvalue of
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics
AM 1411

Winter 2014
Chapter 2: Determinants
2.1: Determinants by cofactor
expansion
Determinant: abbc if
det(A) is cross product.
A1=1/det(A )[ ]
Minors and cofactors:
the minor entry of aij is denoted by Mij and is the remainder after the i row and the j column is
removed
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics
AM 1411

Winter 2014
Chapter 6:
6.1: Inner Products
Unit circles and spheres in inner product space:
there are points in V, that satisfy u=1
Weights:
if wn represents the significance, in a real positive number of the vectors: u, and v than:
Ex:
unit circles and spheres in
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics for Engineers I
AM 1413

Fall 2013
. r
NSNA '
I I I
II>HVH0 ENVH: cuc§>m :3 x l
33 H591: NEH H5383:
$53530an 2850 $0135 mecmmam 2030 $0153
mnsmmbm mwmnwncna magma 2:302
Ham CZH<memHH< Om, gammwz OZH>WHO
b02602 O>Z>U>
>30me 33:. one \ gunrmgwnm on. mmnosm ganmugxwamswio: .
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics for Engineers I
AM 1413

Fall 2013
MIDERM REVIEW
L`Hopital's Rule:
Suppose the functions f and g are differentiable on the open interval (a,b) and
g`(x)0 (but can be at a) there. suppose also that:
Must write down the lim of g(x) and f(x), and then determine the form of these limits. see c
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics for Engineers I
AM 1413

Fall 2013
TRIG REVIEW
the trig ration can be
thought of a circle of
radius of one. This has
a right angle triangle
within it. it is this
triangle that the trig
ratios come from and
can be solved from.
The acute angle is labelled "t" and is measured in radians (), 2
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics for Engineers I
AM 1413

Fall 2013
Chapter 4: More
Applications of
Differentiation
4.1: Related Rates
When two or more quantities that change with time are linked by an
equation, that equation can be differentiated with respect to time to
produce an equation linking the rates of change of