Calculus 1 for Honours Mathematics

Instructor: Barbara Forrest

 

Access to the course notes and online lectures are available according to the following terms of use. The course notes are available as a PDF file. The lectures are available as MP4 files. These files are being made available so that students may view the course notes and lectures on their mobile devices. No technical support is available.


Terms of Use


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All rights, including copyright, images, slides, audio, and video components, of the content of this course are owned by the course authors Barbara Forrest and Brian Forrest.

 

By accessing these web pages, you agree that you may only download the content for your own personal, non-commercial use.

 

You are not permitted to copy, broadcast, download, store (in any medium), transmit, show or play in public, adapt, or change in any way the content of these web pages for any other purpose whatsoever without the prior written permission of the course authors.

 

Author Contact Information:

 

Barbara Forrest (baforres@uwaterloo.ca)

Brian Forrest (beforres@uwaterloo.ca)

 

 

 

 


Course Notes


 

Click on the following link to access the course notes.

 

Calculus 1 for Honours Mathematics: Course Notes

 

 

 


Lectures


 

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 your device in order to view the files.

 

 

 

Chapter 1: Sequences and Convergence

 

A.    Absolute Value and the Triangle Inequality

B.    Introduction to Sequences

C.    Examples of Recursively Defined Sequences

D.    Heron's Algorithm

E.    Subsequences and Tails of Sequences

F.     Limits of Sequences (part 1)

G.    Limits of Sequences (part 2)

H.    Limits of Sequences (part 3)

I.       Divergence to Infinity

J.      Arithmetic for Limits of Sequences(part 1)

K.    Arithmetic for Limits of Sequences (part 2)

L.      Arithmetic for Limits of Sequences (part 3)

M.   Squeeze Theorem for Sequences

N.    Least Upper Bound Property

O.    Monotone Convergence Theorem

P.    Introduction to Series

Q.    Geometric Series

R.     Divergence Test

 

Chapter 2: Limits and Continuity

 

A.    Limits of Functions (part 1)

B.    Limits of Functions (part 2)

C.    Uniqueness of Limits

D.    Sequential Characterization of the Limit

E.    Arithmetic Rules for Limits of Functions

F.     One-Sided Limits

G.    Squeeze Theorem for Limits of Functions

H.    Fundamental Trig Limit

I.       Horizontal Asymptotes and Limits at Infinity (part 1)

J.      Horizontal Asymptotes and Limits at Infinity (part 2)

K.    Fundamental Log Limit

L.      Vertical Asymptotes and Infinite Limits

M.   Continuity

N.    Types of Discontinuities

O.    Continuity of Polynomials, Trigonometric Functions, and Exponentials

P.    Arithmetic Rules for Continuity

Q.    Continuity of an Interval

R.     Intermediate Value Theorem

S.    Approximating Roots of a Polynomial

T.      Bisection Algorithm for Approximating Zeros

U.    Extreme Value Theorem

V.    Curve Sketching (part 1)

 

Chapter 3: Derivatives

 

A.    Instantaneous Velocity

B.    Derivatives

C.    Derivatives and Continuity

D.    The Derivative Function

E.    Derivatives of the Sine and Cosine Functions

F.     Derivatives of Exponential Functions

G.    Linear Approximation

H.    The Error in Linear Approximation

I.       Linear Approximation Applications

J.      Newton's Method

K.    Arithmetic Rules for Differentiation

L.      Chain Rule

M.   More Trigonometric Derivatives

N.    Inverse Function Theorem

O.    Derivatives of Inverse Trigonometric Functions

P.    Implicit Differentiation

Q.    Extrema

R.     Exponential Growth and Decay (optional)

 

 

Chapter 4: The Mean Value Theorem

 

A.    Mean Value Theorem

B.    Applications of the MVT: Antiderivatives

C.    Applications of the MVT: Increasing Function Theorem

D.    Applications of the MVT: Functions with Bounded Derivatives

E.    Applications of the MVT: Comparing Functions through their Derivatives

F.    Applications of the MVT: Concavity

G.    Applications of the MVT: First Derivative Test

H.    Applications of the MVT: Second Derivative Test

I.     L'Hospital's Rule (part 1)

J.    L'Hospital's Rule (part 2)

K. Curve Sketching (part 2)

 

 

Chapter 5: Taylor Polynomials and Taylors Theorem

 

 

A.    Taylor Polynomials

B.    Taylor Polynomials: Examples : Part 1

C.    Taylor Polynomials: Examples : Part 2

D.    Taylor's Theorem

E.    Taylor's Approximation Theorem

F.    Big-O Notation

G.    Arithmetic with Big-O Notation: Part 1

H.    Arithmetic with Big-O Notation: Part 2

I.     Big-O Notation: Examples

J.    Big-O Notation: Calculating Taylor Polynomials

 


This page is maintained by Barbara Forrest.

Users are encouraged to contact the authors to report any errors.