Preface: The course textbook (Stewart, Clegg and Watson, Calculus, Ninth Edition, Cengage Publishing Co., 2020) is phenomenal. Reading the respective textbook chapters will serve you well in this course. However, many of us learn better through different mediums. Thus, I have created this study plan which consists of YouTube videos which will also take you through the respective course content. I suggest watching the videos and then doing the suggested problems from the course textbook. The videos are primarily taken from MAT137: Calculus with Proofs. It is appropriate as ESC194 and ESC195 provide a transfer credit for that course. The videos are made by Alfonso Gracia-Saz who unfortunately passed away in 2021. See here to learn more about his legacy: ‘Passionate, principled and caring’: U of T remembers math professor Alfonso Gracia-Saz
Note: The course textbook is not nearly as rigorous as the content taught in the lectures, especially in the portion taught by Professor Stangeby. There is another textbook called Spivak Calculus, however it may be far too rigorous. The MAT137 videos are appropriately rigorous, however, the videos are decently more rigorous than the portion of the course taught by Professor Davis.
| Lecture | Topic | Resources |
|---|---|---|
| 1 | Introduction (1.4 The Tangent and Velocity Problems) | See Course Textbook for 1.4 |
| 2 | The Real Number System | See Course Supplement Section 2 |
| 3 | Overview of Basic Mathematical Concepts (Part 1) |
Sets and Notation (1.1, 1.2) Quantifiers (1.3, 1.4, 1.5, 1.6) Conditionals (1.7, 1.8) Distance and Absolute Values (2.4) |
| 4 | Overview of Basic Mathematical Concepts (Part 2) |
Definitions and Proofs (1.9, 1.10, 1.11, 1.12, 1.13) Induction (1.14, 1.15) |
| 5 | 1.5 The Limit of a Function | Limits Geometrically (2.1, 2.2, 2.3) |
| 6 | 1.7 The Precise Definition of a Limit |
The Formal Definition of Limit (2.5) Proofs From the Definition of Limit (2.7, 2.8, 2.9) |
| 7 | 1.6 Calculating Limits Using the Limit Laws |
Limit Laws (2.10) Proof of the Limit Law for Sums (2.11) The Squeeze Theorem (2.12, 2.13) Computations (2.19, 2.20) |
| 8 | 1.8 Continuity |
Continuity (2.14, 2.15, 2.16, 2.17) The Intermediate Value Theorem (2.22) |
| 9 | 2.1 & 2.2 Derivatives and Rates of Change |
Definition of Derivative (3.1, 3.2) Derivative as a Rate of Change (3.3) Differentiable Implies Continuous (3.5) |
| 10 |
2.3 Differentiation Formulas 2.7 Rates of Change in the Natural Sciences |
Continuous But Not Differentiable Functions (3.9) Differentiation Rules (3.4) Proof of the Differentiation Rules (3.6, 3.7) See Course Textbook for 2.7 |
| 11 |
2.4 Derivatives of Trigonometric Functions 2.5 The Chain Rule 2.6 Implicit Differentiation |
A Geometric Proof for a Trigonometric Limit (2.18) Derivatives of Trigonometric Functions (3.12) The Chain Rule (3.10, 3.11) Implicit Differentiation (3.13) Higher Order Derivatives (3.8) |
| 12 |
2.8 Related Rates 3.1 Maximum and Minimum Values |
Related Rates (6.1, 6.2)
The Extreme Value Theorem (2.21) The Local Extreme Value Theorem (5.2, 5.3) Finding the Maximum and Minimum of a Function (5.4) |
| 13 |
3.2 The Mean Value Theorem 2.9 Linear Approximations and Differentials |
The MVT…Who Cares? (5.1) Rolle’s Theorem (5.5) How Many Zeros does a Function Have? (5.6) The Mean Value Theorem (5.7, 5.8) See Course Textbook for 2.9 |
| 14 |
3.3 How Derivatives Affect the Shape of a Graph 3.4 Limits at Infinity, Horizontal Asymptotes |
Zero Derivative Implies Constant (5.9) Why Integration is Possible (5.10) Monotonicity of Functions (5.11) Finding the Intervals Where a Function is Increasing or Decreasing (5.12) The Definition(s) of Concavity (6.13) Example: Monotonicity and Concavity of a Function (6.14) Limits at Infinity (2.6) Asymptotes (6.15, 6.16, 6.17, 6.18) |
| 15 | 3.5 Summary of Curve Sketching | See Course Textbook for 3.5 |
| 16 |
3.7 Optimization Problems 3.9 Antiderivatives Appendix E, Sigma Notation |
Applied Optimization (6.3, 6.4) Antiderivatives (8.1) “Sigma Notation” for Sums (7.2) Suprema and Infima (7.3, 7.4) |
| 17 | 4.1 Areas and Distances |
Preview of the Definition of Integral (7.1) The Definition of Integral (7.5) Properties of Lower and Upper Sums (7.6) Examples of Integrable and Non-Integrable Functions (7.7, 7.8) Integrals as Limits (7.9) Riemann Sums (7.10) |
| 18 | 4.2 The Definite Integral | Five Properties of the Definite Integral (7.11) |
| 19 | 4.3 The Fundamental Theorem of Calculus |
Functions Defined as Integrals (8.2) The Fundamental Theorem of Calculus (8.3, 8.4, 8.5, 8.6) |
| 20 |
4.4 Indefinite Integrals and the Net Change Theorem 4.5 The Substitution Rule |
Summary: The Three “Notations of Integral” (8.7) Integration by Substitution (9.1, 9.2, 9.3) |
| 21 | 5.1 Areas Between Curves | See Course Textbook for 5.1 |
| 22 | 5.2 Volumes | Volumes by Discs and Washers (10.1) |
| 23 |
5.3 Volumes by Cylindrical Shells 5.5 Average Value of a Function |
Volumes by Cylindrical Shells (10.2) See Course Textbook for 5.5 |
| 24 | 6.1 Inverse Functions | Inverse Functions (4.1, 4.2, 4.3, 4.4) |
| 25 | 6.2* The Natural Logarithmic Function |
Exponentials and Logarithms (4.5, 4.6, 4.7, 4.8) Logarithmic Differentiation (4.9) Proof of the Power Rule (For All Exponents) (4.10) LN or LOG? The Controversy (4.11) |
| 26 | 6.3* The Natural Exponential Function | |
| 27 | 6.4* General Logarithmic and Exponential Functions | |
| 28 | 6.6 Inverse Trigonometric Functions | Inverse Trigonometric Functions (4.12, 4.13, 4.14) |
| 29 |
9.1 Modelling with Differential Equations 9.3 Separable Equations |
See Course Textbook for 9.1-9.5 |
| 30 |
6.5 Exponential Growth and Decay 9.4 Models for Population Growth |
|
| 31 | 9.5 Linear Equations | |
| 32 | Complex Numbers | Stewart on Complex Numbers |
| 33 | 17.1 (online chapter) Second Order Linear Equations | LibreTexts on Second-Order Linear Equations |
| 34 | 17.2 (online chapter) Nonhomogeneous Linear Equations | LibreTexts on Nonhomogeneous Linear Equations |
Last Updated on December 18, 2024
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