2.2 2.3
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 a
10.1 Curves Defined by Parametric Equations
If a particle moves on the plane along a path shown in
the figure on the right, it is rather difficult to describe
the path by a function of the form y f x because the
path is not the graph of a function. But if
Calculus I: MATH 1130
Kwantlen University
Sample Problems
Continuity
September 28, 2016
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More Examples:
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More Examples:
3 / 20
Continuity:
4 / 20
Continuity Very Informally: Informally speaking, a function is continuous if
its graph has no bre
MATH 1130
Recursions
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October 5, 2016
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Sequences: A sequence is an ordered collection of numbers.
Example:
1,
1 1 1
2, 4, 8, . . .
Sometimes, we might be able to find a general (rule) for the
terms of our sequences.
For example, t
Calculus I: MATH 1130
Kwantlen University
Infinite Limits and Vertical Asymptotes
Limit Laws & Calculating Limits
Indeterminate Forms
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Infinite limits: The values of y increase or decrease without bound as
x approaches to a.
Defin
Calculus I: MATH 1130
Kwantlen University
Tangent and Velocity
Limits Intro
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Tangent (from Latin word tangens, touching) line:
A line that touches a curve in exactly one point.
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Tangent (from Latin word tangens, touching) li
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Rate of Change
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Average Rate of Change:
Example: What is the average growth rate of the baby during the first
24 months of life.
Example: What is the average growth rate of the baby during th
Calculus I: MATH 1130
Kwantlen University
Limits and Infinity
Horizontal Asymptote
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Limits and Infinity:
A numerical example:
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Formal Definition:
3/5
Example:
Horizontal Asymptote:
Informally: If one of the limits at infinity is a
The test is scheduled for Friday, Nov. 21.
There are four questions worth a total of 19 marks.
The test covers Sections 4.1 and 4.2.
Can you draw a possible graph of a function given information about its local and absolute
extrema (or lack thereof)?
Do
Math 1120-S10 Test #3 Information
Your test has four questions worth a total of 30 marks.
The test covers Sections 3.1-3.5* (*not including inverse trigonometric functions and their
derivatives)
One question requires you to calculate derivatives of variou
Math 1120-S10 Test #4 Information
The test is scheduled for Friday, Nov. 7.
There are four questions worth a total of 23 marks.
The test covers the Inverse Trig. Supplement and Sections 3.6, 3.7, 3.9 and 3.10 (including the
supplement on Quadratic Approxi
Math 1120-S10 Test #1 Information
Your test has four questions worth a total of 25 marks.
All of the questions are on the topic of limits.
Dont forget that the absolute value function is a hidden piecewise function.
Make sure you remember that vertical as
Math 1120-S10 Test #2 Information
Your test has five questions worth a total of 25 marks.
You must know the definition of continuity and be able to apply it.
Know the intuitive idea of continuity.
Be able to identify point, jump and infinite discontinuiti
Math 1152 Spring 2014
1.4
23
Gaussian Elimination
Definition 1.4.1. Gauss-Jordan elimination is the systematic procedure to bring a
matrix to reduced row echelon form by starting at the upper left and working down to the
lower right using only elementary
Chapter 1
Matrices and Vectors
1.1
Basic Definitions
Definition 1.1.1. A (real) scalar is a real number. We denote the set of real numbers by
R.
Definition 1.1.2. A matrix A is a rectangular array of scalars.
1 0 53
Example 1.1.3. A =
and u = 4.32 1 e are
Math 1152 Spring 2014
1.3
15
Systems of Linear Equations
Definition 1.3.1. A linear equation in the variables x1 , x2 , . . . , xn is an equation which
can be written in the form
a1 x 1 + a2 x 2 + + an x n = b
Each xi is called a
Each ai is called a
Final
Chapter 1
Matrices and Vectors
1.1
Basic Definitions
1.2
Linear Combinations, Matrix-Vector Products
Definition 1.2.1. If cfw_u1 , u2 , . . . , uk is a set of vectors in Rn , then a linear combination
of these vectors is any sum of the form c1 u1 + c2 u2
Math 1152 Spring 2014
1.7
35
Linear Dependence and Linear Independence
Definition 1.7.1. A set S = cfw_v1 , v2 , . . . , vk in Rn is called linearly independent if
c1 v1 + c2 v2 + + ck vk = 0 if and only if c1 = 0, . . . ck = 0
If it is not linearly inde
Math 1152 Spring 2014
1.6
31
Span of a set of Vectors
Definition 1.6.1. The span of a set of vectors S = cfw_u1 , u2 , . . . , uk in Rn is
Span S = cfw_v Rn |v = c1 u1 + c2 u2 + + ck uk , ci R
that is, the set of all linear combinations of vectors in S.
Math 1152 Spring 2014
2.3
49
Invertibility, Elementary Matrices
Definition 2.3.1. Let A and B be n n matrices. We say that B is the inverse of A if
AB = In and BA = In . We write A1 = B.
Example
2.3.2.
Show that B is the
inverse of A, where:
3 7
5 7
A=
,
3.4
The Chain Rule
1
1
The Reciprocal Rule
g x lends itself as an intuitive lead to the Chain Rule.
g x 2
g x
1
1
1
Consider f x , f x 2 . So the Reciprocal Rule tells us that the derivative of f g x
g x
x
x
is equal to the derivative of f evaluate
4.9
Newtons Method
In Algebra, we spend a considerable amount of time to find zeros of functions. When algebraic
manipulation fails to produce the desired results easily and rapidly, numerical approximation is often the
logical substitute. Newtons method
3.3 Derivative of Trigonometric Functions
1.
The Radian Measure of Angles
Definition The radian measure of angle =
Special case
arclength
arc PU
=
radius
OU
When OU = 1, that is, when circle O is a unit circle, is the
length of arc PU (numerically).
Fact
4.3
How Derivatives Affects the Shape of a Graph
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
3.5
Example 1
Explicitly
Implicit Differentiation
Consider the circle x 2 y 2 25 . Find the slope of the tangent line to the circle at (4, 3).
y 2 25 x 2 , y 25 x 2 (4, 3) is on the top half of the circle.)
dy
1
x
2 x
(which is in terms of x alone)
dx 2
4.7 Optimization
Introduction
Show that among all triangles with a given perimeter, the equilateral triangle has the
largest area.
This is a typical optimization problem that only seems easy. After playing with a lope of
string long enough, we need to fig
3.6 Derivatives of Logarithmic Functions
Consider g x ln x .
g x lim
h 0
ln x h ln x
1 xh
1 ln 1 h x 1
ln 1 k 1
lim ln
lim
lim
g 1 where
h
0
h
0
k
0
h
h
x
x
h x
x
k
x
k h x .
The derivative of ln x is equal to
1
times the derivative of ln x at x 1 .
x
g
4.4
LHospital Rule
Both f x ln x and g x x are increasing
and concave downward on (0, ). Both f and g
approach infinity as x approaches infinity. Which
one of f and g approaches infinity faster?
To answer the above question, we need to evaluate
lim
x
ln
4.1
Maximum and Minimum Values
Definition
A function f has an absolute maximum (or global maximum) at c if f c f x for all x
in D, where D is the domain of f. The number f c is called the maximum value of f on
D. Similarly, a function f has an absolute mi