1H Matrix Transformations
1H - 1
1H Matrix Transformations (1.8, 1.9)
In precalculus and calculus we study functions that have numbers as inputs and
numbers as outputs. For example, the linear function f given by the rule fx 3x
assigns to every input numb
1B Geometry of Vectors and Vector Operations
1B - 1
1B Geometry of Vectors and Vector Operations (1.3)
Matrices and Vectors are two important types of objects in linear algebra. Vectors play an important role in
geometry, physics and engineering, and in t
1D Vector Equations of Lines and Planes
1D - 1
1D Vector Equations and Parametric Equations of Lines and Planes
Vector Equation and Parametric Equations of Lines in 2 and 3
The equation of a line in 2 can be writen in the form y mx b or y y 1 mx x 1 wher
1E Inner Product Length Distance Angle Orthogonality
1E - 1
1E Inner Product, Length, Distance, Angle, Orthogonality
The inner product is a powerful tool for solving many applied problems. It can be used to express the
important geometric concepts of leng
1A Matrices - Addition, Subtraction & Scalar Multiplication
1A - 1
1A Matrices - Addition, Subtraction, Scalar Multiplication
Numerical information is often organized into a table called a matrix (plural: matrices).
Example:
A is a 4 3 matrix (matrix wit
1C Linear Combinations and Subspaces of 2 and 3
1C - 1
1C Introduction to the Concepts of Linear Combinations and Subspaces
The ideas of linear combinations and subspace are two key ideas of linear algebra.
Let v 1 , v 2 , . . , v p and v be vectors in n
Formula Sheet
1.
Existence and Uniqueness of Solutions of a System Axb
2. Facts:
a. The vectors v 1 , v 2 , . . . , v p in n are linearly independent if and only if rank v 1 v 2 . . . v p . . . . . . .
np
b. The vectors v 1 , v 2 , . . . , v p in span if
1F Applications of Orthogonnality
1F - 1
01F Applications of Orthogonality
Normal Equation and Scalar Equation of a Plane in 3
A line in 2 can be specified by giving its slope (inclination) and one of its points. Similiarly, one can specify
a plane in 3
1G Matrix Multiplication
1G - 1
1G Matrix Multiplication (2.1)
Recall that two matrices of the same size are added by adding the corresponding entries. It seems reasonable
that two matrices should be multiplied in a similiar way, that is, by multiplying t
1I Linear Transformations
1I - 1
1 I Linear Transformations (1.8, 1.9)
In linear algebra, matrix transformations are a subclass of an important type of transformations
called linear transformations.
A linear transformation is a transformation that has th
Math 232 - Midterm 2 - Spring 2016 KEY
1. Find the distance D of the point Q(-1,2,1) from the plane 2x y 3z 4 and the point R on the plane closest
to Q using the following steps:
a. (3 marks) Find a point P on the plane and find the vector PQ. Also find a
Math 232 - Final Exam Sample
1. The augmented matrix of a system Ax b in the variables x,y,z,u,v, and w has RREF
Solutionset: x 5r 3s 5t
x
y
z
u
v
r, s, t
w
yr
1 5 0 3 0 5
0
z 7 s 3t
0
0
1
1
0 3
7
us
0
0
0
0
1
4
v 4 2t
2
wt
a. (1 mark) Which variables ar
1.1 The Coordinate Plane
Introduction The coordinate plane is used to locate points, draw figures and graph equations. It helps us to
bring algebra and geometry together so that the study of one benefits the other.
Example 0 The Length of a Crease A recta
2.1
The Derivative
When was the first time you studied tangent lines?
In Geometry 11.
Lines tangent to what?
Circles.
How do you know a line is tangent to a circle?
When it intersects the circle at only one point.
Definition
A line is said to be tangent t
4.3
Derivatives of Exponential Functions
Consider f x e x .
As with every new function, we use the definition of derivative to find the derivative of f:
f x lim
h 0
f x h f x
e x h e x
exeh e x
ex eh 1
eh 1
.
lim
lim
lim
e x lim
h 0
h 0
h 0
h 0
h
h
h
h
h
Review III
Exponential Functions
2
5
x
A simple exponential function has a variable exponent like f x 3 x and g x .
In general, a simple exponential function is written as f x a x .
The domain of f is the set of all real numbers , .
The range of f is the
8.4 Derivatives of Trigonometric Functions
Consider f x sin x and g x cos x .
As with every new function, we use the definition of derivative to find the derivative of f:
f x h f x
sin x h sin x
sin x cos h cos x sin h sin x
lim
lim
h 0
h 0
h
h
h
sin x co
4.7
Curve Sketching II
Example 1
Consider f x x ln x .
a) Find all critical points of f.
b) Determine the interval(s) on which f is increasing and the interval(s) on which f is
decreasing.
c) Find all relative extrema of f.
d) Determine the interval(s) on
3.8
Differentials
Introduction
The purpose of studying functions is to understand change. In this section, we examine how the concept
of derivative helps us understand and approximate change.
Consider a differentiable function
. When the independent varia
8.6
LHospital Rule
Both f x ln x and g x x are increasing and concave downward on (0, ) and both f and g
approach infinity as x approaches infinity, which one of f and g approaches infinity faster?
lim
To answer the above question, we need to evaluate x
8.1
Radian Measure of Angles
Angle in Standard Position
Let P(x, y) be a point which moves around a circle O centred at the origin
O(0, 0) with radius r. P starts at the point A(r, 0) on the x-axis. For each
position of P, an angle = AOP is defined, which
Review I
The Laws of Exponents and Logarithms
4
Exponents are used to count the number of factors in a product. For example, 3 3 3 3 3 where the
4 times
5
exponent 4 indicates the number of 3s in this product. Similarly, a a a a a a where the exponent
1.5 Limits
Limit is the foundation of Calculus. In differential Calculus, we use limit to define the derivative
of a function. In integral calculus, we use limit to define the definite integral of a function. To
understand what limit is, let us look at th
1.7 The Intermediate Value Theorem
The Intermediate Value Theorem
Suppose that f is continuous on the closed interval [a, b] and let N be any number between f(a) and
f(b). Then there exists a number c in (a, b) such that f(c) = N.
The Intermediate Value T
3.4 Optimization
Introduction
Optimization is one of those applications in mathematics we studied since elementary
school. One only needs to know what sum and product mean (a third grader?) to be able
to solve the following problem:
The sum of two numbers
1.4
Functions
What is a function?
What does the word function mean?
Do you know what a food processor is?
Do you know what a food processor does?
Do you know what a multi-function food processor is?
Do you know what a multi-function food processor does?
H
3.7 Curve Sketching
Example 1
Sol
I
Sketch the graph of
6
f x 2
x 3
.
Gathering Info about f
The domain of f is
x |
,
.
The y-intercept is (0, 2).
There is no x-intercept because the graph of f does not intersect the x-axis.
f is an even function and t
3.5 Price Elasticity of Demand
In business, a demand equation relates the demand (the quantity demanded by consumers) for a product
and the (unit) price of the product.
Normally, when the (unit) price p of a product is decreased, the demand x for
the prod
3.1
Increasing and Decreasing Functions
Definition
A function f is increasing on (a, b) if for any x1 and x2 in (a, b),
x 2 x1 implies f x 2 f x1 .
If f is increasing on (a, b), its graph goes upward from left to right
over (a, b).
A function f is decreas
CBSY 2205
In-class assessment
Chapter 9 CRM
1.
Click on this URL: http:/www.timhortons.com List and explain the IT-supported
customer touch points and customer-touching applications (e-CRM) provided by the
company.
2.
In Terms of CIC & SFA, will Tim Horto