asher 557
Calculus Section 1.4 Continuity and One-Sided Limits
Determine continuity at a point and continuity on an open interval.
- Determine one=sided limits and continuity on a closed intervai.
Homework: pages 78-79 #5 2, 4,
6, 7-10, 15, 25, 26, 33, 3
Name:
Teacher:
Score:
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PREVIEW
o 2.2: SEPARABLE EQUATIONS
What it means for an ODE to be separable 2.1, and recognizing if a
given differential equation is separable or not;
How to solve a separable rst order ODE 1.
o 2.3: LINEAR EQUATIONS
How to recognize linear rstorder di
PREVIEW
1.4: PREDICATES and QUANTIFIERS
Predicates
The universal and existential quantifiers
universe of discourse, scope of quantifiers
quantifiers with restricted domains
precedence of quantifiers
binding variables
logical equivalence with quant
PREVIEW
1.7: Introduction to Proofs
Terminology
How to read a Theorem
Direct proof
Contrapositive proof
Proof by contradiction
Disproving by counterexample
Definitions: odd, even integers; rationals
HONOUR HOMEWORK: To be done before next weeks tu
PREVIEW
1.6: RULES OF INFERENCE
valid argument
rules of inference for propositional logic
using rules of inference to build arguments
4 important rules of inference for quantifiers
fallacies (your reading)
HONOUR HOMEWORK: To be done before next wee
PREVIEW
1.3: PROPOSITIONAL EQUIVALENCES
Tautology/Contradiction
Establishing logical equivalences using truth tables
Using established identities to simplify logical formulas, and to prove
logical equivalences
HONOUR HOMEWORK: To be done before next w
PREVIEW
1.1: PROPOSITIONAL LOGIC
Propositions
Logical connectives
Truth tables
Compound propositions
Translating English sentences to logical formulas
Bit strings and bit operations
Truth tables for compound propositions
HONOUR HOMEWORK: To be don
PREVIEW
2.1: Sets
Sets and Elements of Sets
Standard Sets of Numbers: , , , , . . .
Understanding Set Builder Notation
Subsets
Cardinality of a Set
Power Set
Cartesian Product
Truth Set of a Predicate
2.2: Set Operations
Union, Intersection, Se
PREVIEW
1.8: Proof Methods and Strategy
Exhaustive proofs; proof by cases
Without loss of generality (wlog)
Existence proofs:
Constructive proof
Non-constructive proof
Uniqueness proofs
Backwards/Forwards Reasoning
HONOUR HOMEWORK: To be done befo
PREVIEW
3.1: Algorithms
Understanding what an algorithm is
Writing pseudocode for an algorithm
Working through an algorithm example for a specific input
3.2: Growth of Functions
Big-O notation - upper bound of growth; finding witnesses, bounds
Some
PREVIEW
2.4: Sequences and Summations
common sequences: harmonic, arithmetic, geometric, powers, factorials, etc.
inferring formulas for sequences from terms
summation notation: limits, index variables
double sums, summation over specified index sets
PREVIEW
2.3: Functions Continued
Definitions: Function, Domain, Codomain, Range
One-to-one (Injective) and Onto (Surjective) Functions
Inverse Functions; Finding the Inverse
Composition of Functions
Graph of a Function
Important Functions: Floor, C
Scalar Autonomous ODEs: The Phase Line
Definition. An ODE x = f (x, t) with f : Rn R Rn is said to be
autonomous if it does not depend explicitly on the independent variable, t in
this case. We can write x = f (x).
Example. Consider the ODE x = ax with sa
Existence and uniqueness of solutions
Given a differential equation IVP:
dx
= f (x, t),
dt
x(t0 ) = x0 .
Consider the following questions:
1. Do all equations such as (1) have a solution?
(1)
2. If a solution exists, is it unique?
Example. Consider the so
Differential Equations: MATH 2060U
Chapter 0: Review of Prerequisites
1. The Derivative of a Function: The derivative of f (x) with respect to
df
x is the function f 0 (x)
and is defined as,
dx
f (x + h) f (x)
h0
h
f 0 (x) = lim
1
2. Differentiation Tech
Part 1: First Order Ordinary Differential Equations
Consider the following examples that will provide some context to how ODEs
fit in with other types of equations that you have seen, and will see later.
1. Algebraic Equations
(a) Scalar algebraic equatio
P-L. Buono
1
1.1
MATH 2055U Adv. Linear Algebra and Appl.
1 of 8
Chapter 1 - Sections 6.4-6.5 and 7
Isomorphisms
Definition. An invertible transformation A : V W is called an isomorphism. Two vectors spaces V and W are called isomorphic (denoted
V ' W ) i
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MATH 2055U Adv. Linear Algebra and Appl.
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Chapter 1 - Section 3
I will now discuss the main definitions and results of the section.
Definition. A transformation T from a set X to a set Y is a rule that
for each argument x X assigns a va
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MATH 2055U Adv. Linear Algebra and App.
Kernel and Range
Definition. Let A : V W be a linear transformation, then
ker A := cfw_v V | Av = 0 V
is the kernel of A, and
Ran A := cfw_w W | w = Av
is the range of A.
for some v V W
1 of 9
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MATH 2055U Adv. Linear Algebra and Appl.
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Chapter 1 - Section 6
Invertible transformations
Definition (Identity Transformation). Let V be a vector space, the
identity transformation is the linear transformation I = IV : V V
defined by I
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MATH 2055U Adv. Linear Algebra and Appl.
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Chapter 1 - Sections 4 and 5
Linear Transformations as a Vector Space
Definition. Let V, W be two vector spaces. We denote by L(V, W ) the
set of all linear transformations T : V W .
Proposition
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MATH 2055U Adv. Linear Algebra and Appl.
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Reading Analysis: Chapter 1 - Section 2
I will now discuss the main definitions and results of the section.
Definition. A system of vectors v1 , v2 , . . . , vn V is called a basis of
the vector
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1
MATH 2055U Adv. Linear Algebra and Appl.
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Reading Analysis: Chapter 1 - Section 1
(a) Definitions:
(a) Vector space
(b) Real vs complex vector space
(c) Matrix notation: aj,k
(d) Transpose of a matrix.
(b) Results:
(a) Many examples of
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MATH 2055U Adv. Linear Algebra and App.
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Finite Dimensional Spaces
Definition. The dimension dim V of a vector space V is the number of
vectors in a basis.
(a) Finite dimensional spaces
(b) Infinite dimensional spaces
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MATH 20
PREVIEW
o 4.2: REDUCTION OF ORDER
The technique reduction of order" and how to use it to construct a
second linearly independent solution given one solution to a linear
2nd order equation 1,1.
0 4.3: HOMOGENEOUS LINEAR EQUATIONS WITH CONSTANT COEFFICIENT
PREVIEW
o 4.4: UNDETERMINED COEFFICIENTS. SUPERPOSITION APPROACH (covered
on Assign #2 but not in lecture .will only be tested on this section in
short answer or multiple choice questions)
The form of the particular solution to be assumed.
o 4.6: VARIATI
PREVIEW
o 2.4: EXACT EQUATIONS
Recognizing if a rstorder differential equation is exact 2.2;
Solving exact differential equations 1;
To nd (where possible) an integrating factor which makes a rstorder
ode exact 2, and to subsequently solve the equation
PREVIEW
o 4.1: LINEAR DIFFERENTIAL EQUATIONS: BASIC THEORY
What it means for a 2nd order equation to be linear; what it means to
be homogeneous 4.3;
Familiarity with operator notation 4.4, and how to use this to represent
differential equations 4;
Find