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Author
Contact Information:
Barbara
Forrest (baforres@uwaterloo.ca)
Brian
Forrest (beforres@uwaterloo.ca)
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Foundations of Calculus 1 for Teachers: Course Notes
Click on any of the following links to access the lectures that accompany the course notes for this course.
All
lectures are available as MP4 files. You must have an MP4 player installed on
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Module 1:
Foundations
3.
Functions Part I: Basic Properties
4.
Functions Part II: One-to-one and Onto Functions
6.
Induction Part I: The Principle of Mathematical Induction
7.
Induction Part II: Applications
8.
Enrichment: Sets and Boolean Arithmetic
Module 2:
Real Numbers
Module 3:
Sequences
2.
Examples of Recursively Defined Sequences
4.
Subsequences and Tails of Sequences
5.
Limits of Sequences (part 1)
6.
Limits of Sequences (part 2)
7.
Limits of Sequences (part 3)
9.
Arithmetic for Limits of Sequences (part 1): Introduction
10.
Arithmetic of Sequences (part 2): Products
11.
Arithmetic of Sequences (part 3): Quotients
12.
Arithmetic of Sequences (part 4): Examples
13.
Arithmetic of Sequences (part 5): Examples II
14.
Squeeze Theorem
15.
Monotone Convergence Theorem
17.
Geometric Series
18.
Divergence Test
19.
Bolzano-Weierstrass Theorem
19b.
Limit Points
20.
Cauchy Sequences
Module 4:
Limits and Continuity
1.
Definition of Limits Part 1: The Formal Definition
2.
Definition of Limits Part 2: Examples
4.
Sequential Characterization of Limits Part I: The Characterization
5.
Sequential Characterization of Limits Part II: Two More Strange Functions
6.
Arithmetic for Limits of Functions
8.
Squeeze Theorem for Limits of Functions
10.
Horizontal Asymptotes and Limits at Infinity (part 1)
11.
Horizontal Asymptotes and Limits at Infinity (part 2)
13.
Vertical Asymptotes and Infinite Limits
14.
Continuity
15.
Continuity of Polynomials, Trigonometric Functions, and Exponentials
17.
Arithmetic Rules for Continuity
19.
Intermediate Value Theorem Part I: Introduction
20.
Intermediate Value Theorem Part II: Proof
21.
Approximating Roots of a Polynomial
22.
Bisection Algorithm for Approximating Zeros
23.
Extreme Value Theorem Part I: Introduction
24.
Extreme Value Theorem Part II: Proof
25.
Uniform Continuity Part I: Introduction
26.
Uniform Continuity Part II: Continuous Functions on [a,b]
27.
Curve Sketching
Module 5:
Derivatives
2.
Derivatives
5.
Derivatives of the Sine and Cosine Functions
6.
Derivatives of Exponential Functions
7.
Linear Approximation Part I: Basics
8.
Linear Approximation Part II: The Error
9.
Linear Approximation Part III: Applications
9b.
Linear Approximation Part IV: Newton's Method
10.
Arithmetic Rules for Differentiation
11.
Chain Rule
11b.
Chain Rule: Proof
12.
More Trigonometric Derivatives
13.
Inverse Function Theorem Part I: Introduction
14.
Inverse Function Theorem Part II: Proof
15.
Derivatives of Inverse Trigonometric Functions
17.
Local Extrema
Module 6:
The Mean Value Theorem
2.
Applications of the MVT: Antiderivatives
3.
Applications of the MVT: Increasing Function Theorem
4.
Applications of the MVT: Functions with Bounded Derivatives
5.
Applications of the MVT: Comparing Functions through their Derivatives
6.
Applications of the MVT: Concavity
7.
Applications of the MVT: First Derivative Test
8.
Applications of the MVT: Second Derivative Test
9.
L'Hopital's Rule Part I: Introduction
10.
L'Hopitals' Rule Part II: More Examples
12.
L'Hopital's Rule Part III: Proof
13.
Curve Sketching (revisited)
Module 7: Taylor
Polynomials and Taylor's Theorem
2.
Taylor Polynomials: Examples : Part 1
3.
Taylor Polynomials: Examples : Part 2
5.
Taylor's Approximation Theorem
7.
Arithmetic with Big-O Notation: Part 1
8.
Arithmetic with Big-O Notation: Part 2
10.
Big-O Notation: Calculating Taylor Polynomials
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