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
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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
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics for Engineers I
AM 1413

Fall 2013
Chapter 5: Integration
5.1: Sums and sigma Notation
is used to denote the sum of
sequence values.
Ex:
m is the lower limit and n is the upper limit of the summing values, Ie start at m, m+1, m+2,. n1, n
i, is the index of the sequence and has no effect
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics for Engineers I
AM 1413

Fall 2013
Chapter 8: Conics, Parametric Curves and Polar Curves
8.2: Parametric Curves
Breaking down the curve, into functions of x=f(r), and y=g(r)
Allows for curves and vertical lines to be possible.
The parameter r, is independent from the axis,
Parameter of a s
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics for Engineers I
AM 1413

Fall 2013
Chapter 7: Applications of Integration
7.1: Volumes by slicingSolids of Revolution
By taking slices of a cylinder, the volume of a shape can be determined.
V=A(x)x
V=A(x)dx
Rotation of a function around an axis:
Determine the area of a cross section, det
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics for Engineers I
AM 1413

Fall 2013
Chapter 6: Techniques of integration
6.1: Integration by parts
Suppose that U(x) and V(x) are tow differentiable functions: than according to chain rule:
Than the inverse of this will be given by:
Which simplifies to
Ex: xexdx, let U=x, than dU=dx, V=ex,
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics for Engineers I
AM 1413

Fall 2013
Chapter 3: Transcendental Functions
3.1: Inverse functions:
The function must be a onetoone function. This means that for any f(x) there is only a single x value that will give that f(x) y. Value. If it does not meet this the
function must be broken apa
Western University (Ontario)  Also known as University of Western Ontario
Applied Mathematics for Engineers I
AM 1413

Fall 2013
Chapter 2: Differentiation
Methods for detonating differential equations:
f `(x ) = Df(x)=Dxf(x)=Dx1f(x)= d/dx*f(x)
f `(x)=D2f(x)=D2xf(x)=d2/dx*f(x)
*Note, go back to first principles for any derivative
2.1 Tangent lines and their slopes:
Definition of a