MAT137Y1

U of T

Course Overview

Lessons & Practice

I. Welcome

1. Functions/Pre-Calculus (Review)

4hr

1.1.1. Interval Notation 4 min

1.1.2. Set Notation3 min

1.1.3. Example4 min

1.1.4. Practice2 min

1.1.5. Practice52 sec

1.1.6. Practice44 sec

1.1.7. Cheatsheet

1.2.1. Domain2 min

1.2.2. Range2 min

1.2.3. Example2 min

1.2.4. Practice 40 sec

1.2.5. Practice 3 min

1.3.1. Operations with Functions3 min

1.3.2. Example2 min

1.3.3. Practice2 min

1.3.4. Practice1 min

1.4.1. Even and Odd Functions3 min

1.4.2. Example2 min

1.5.1. Transformation of Graphs2 min

1.5.2. Example1 min

1.5.3. Practice2 min

1.6.1. Basic of Polynomials4 min

1.6.2. Example2 min

1.6.3. Practice3 min

1.6.4. Polynomial Graphs & Characteristics4 min

1.6.5. Example2 min

1.6.6. Practice2 min

1.7.1. Factoring 5 min

1.7.2. Factoring a Difference & Sum of Cubes3 min

1.7.3. Example2 min

1.7.4. Practice46 sec

1.7.5. Practice2 min

1.7.6. Practice2 min

1.8.1. Dividing Polynomials - Long Division6 min

1.8.2. Example3 min

1.8.3. Dividing Polynomials - Synthetic Division8 min

1.8.4. Example2 min

1.8.5. Practice2 min

1.8.6. Practice3 min

1.9.1. Rational Functions4 min

1.9.2. Solving Rational Equations3 min

1.9.3. Example5 min

1.9.4. Example3 min

1.9.5. Practice6 min

1.9.6. Practice2 min

1.10.1. Radical Functions59 sec

1.10.2. Solving Radical Equations5 min

1.10.3. Example5 min

1.10.4. Practice8 min

1.11.1. Piecewise Functions4 min

1.11.2. Example5 min

1.11.3. Practice3 min

1.12.1. Absolute Value Function1 min

1.12.2. Example3 min

1.13.1. Inverse Functions3 min

1.13.2. Example3 min

1.13.3. Practice2 min

1.14.1. Exponential Functions3 min

1.14.2. Example1 min

1.14.3. Example2 min

1.15.1. Logarithmic Functions4 min

1.15.2. Example3 min

1.15.3. Example:3 min

1.15.4. Practice2 min

1.16.1. Trigonometric Functions6 min

1.16.2. Example4 min

1.16.3. Trig Identities2 min

1.16.4. Practice3 min

1.16.5. Practice4 min

1.17.1. Inverse Trigonometric Functions4 min

1.17.2. Example3 min

1.17.3. Practice2 min

1.17.4. Practice2 min

2. Logic, Quantifiers, and Proofs

59min

3. Limits

1.5hr

3.1.1. Limit Basics2 min

3.1.2. Example1 min

3.2.1. Limit Laws 2 min

3.2.2. Practice1 min

3.3.1. One-Sided Limits2 min

3.3.2. Example: One-Sided Limit2 min

3.3.3. Practice: One-Sided Limits3 min

3.4.1. Infinite Limits2 min

3.4.2. Example1 min

3.4.3. Example3 min

3.5.1. Limits at Infinity3 min

3.5.2. Example60 sec

3.5.3. Example3 min

3.5.4. Example4 min

3.5.5. Practice 3 min

3.5.6. Practice3 min

3.6.1. Indeterminate Forms50 sec

3.7.1. Computing Limits by Factoring1 min

3.7.2. Example1 min

3.7.3. Practice2 min

3.8.1. Computing Limits by Multiplying by the Conjugate3 min

3.8.2. Example3 min

3.8.3. Practice3 min

3.9.1. Computing Limits by Getting a Common Denominator1 min

3.9.2. Example2 min

3.9.3. Practice4 min

3.10.1. The Fundamental Trig Limit2 min

3.10.2. Example3 min

3.11.1. The Squeeze Theorem2 min

3.11.2. Example2 min

3.11.3. Practice2 min

3.12.1. Continuity2 min

3.12.2. Example2 min

3.12.3. Practice4 min

3.13.1. The Intermediate Value Theorem2 min

3.13.2. Extreme Value Theorem1 min

3.13.3. Example2 min

3.13.4. Practice2 min

3.14.1. The Epsilon Delta Definition of a Limit

3.14.2. Example3 min

3.14.3. Practice

4. Derivatives

1.7hr

4.1.1. Derivatives as a Rate of Change2 min

4.1.2. Average Rate of Change46 sec

4.1.3. Example2 min

4.1.4. Practice1 min

4.2.1. The Limit Definition of a Derivative2 min

4.2.2. Example3 min

4.2.3. Practice5 min

4.3.1. Differentiation Laws51 sec

4.4.1. The Power Rule1 min

4.4.2. Example2 min

4.4.3. Practice51 sec

4.5.1. Tangent Lines1 min

4.5.2. Example3 min

4.5.3. Practice2 min

4.6.1. The Product Rule49 sec

4.6.2. Example2 min

4.6.3. Example2 min

4.6.4. Practice2 min

4.7.1. The Quotient Rule1 min

4.7.2. Example2 min

4.7.3. Practice4 min

4.8.1. The Chain Rule1 min

4.8.2. Example2 min

4.8.3. Practice3 min

4.9.1. Derivatives of Trigonometric Functions1 min

4.9.2. Example39 sec

4.9.3. Example3 min

4.9.4. Example1 min

4.9.5. Practice3 min

4.9.6. Practice2 min

4.10.1. Derivatives of Exponential Functions1 min

4.10.2. Example52 sec

4.10.3. Example3 min

4.10.4. Practice1 min

4.11.1. Higher Order Derivatives2 min

4.11.2. Example3 min

4.11.3. Practice2 min

4.11.4. Practice

4.11.5. Practice2 min

4.12.1. Derivatives of Inverse Functions52 sec

4.12.2. Example2 min

4.12.3. Practice3 min

4.13.1. Derivatives of Logarithmic Functions1 min

4.13.2. Example56 sec

4.13.3. Example2 min

4.13.4. Practice2 min

4.13.5. Practice: Log and Ln2 min

4.14.1. Inverse Trig Derivatives1 min

4.14.2. Example1 min

4.14.3. Practice3 min

4.15.1. Logarithmic Differentiation2 min

4.15.2. Example2 min

4.15.3. Practice3 min

5. Applications of Differentiation

2.4hr

5.1.1. Implicit Differentiation2 min

5.1.2. Example3 min

5.1.3. Example3 min

5.1.4. Practice3 min

5.2.1. Related Rates1 min

5.2.2. Example5 min

5.2.3. Example4 min

5.2.4. Practice2 min

5.2.5. Practice4 min

5.2.6. Practice4 min

5.3.1. Rolle's Theorem1 min

5.3.2. Example2 min

5.3.3. Practice1 min

5.4.1. The Mean Value Theorem1 min

5.4.2. Example2 min

5.4.3. Practice2 min

5.5.1. Intervals of Increase and Decrease1 min

5.5.2. Example2 min

5.5.3. Practice3 min

5.6.1. Critical Points and Extrema2 min

5.6.3. Practice3 min

5.7.1. Maximum and Minimum on Closed Intervals2 min

5.7.2. Example4 min

5.7.3. Practice3 min

5.7.4. Practice3 min

5.8.1. 1st and 2nd Derivative Tests3 min

5.8.2. Example7 min

5.8.4. Practice3 min

5.9.1. Optimization1 min

5.9.2. Example4 min

5.9.3. Example4 min

5.9.4. Practice6 min

5.9.5. Practice4 min

5.10.1. L’Hopital’s Rule2 min

5.10.2. Example2 min

5.10.3. Practice5 min

5.11.1. Solving General Indeterminate Forms2 min

5.11.2. Example2 min

5.11.3. Practice5 min

5.11.4. Practice2 min

5.12.1. Concavity and Inflection Points56 sec

5.12.2. Example6 min

5.12.3. Practice6 min

5.13.1. Curve Sketching2 min

5.13.2. Example7 min

5.13.3. Practice9 min

5.13.4. Practice

6. Integration

2hr

6.1.1. Finite Sums & Summation Notation3 min

6.1.2. Example5 min

6.1.3. Practice4 min

6.1.4. Practice4 min

6.2.1. Approximating Areas & Riemann Sums 3 min

6.2.2. Example4 min

6.2.3. Practice3 min

6.3.1. The Definite Integrals & Area Under a Curve2 min

6.3.2. Example7 min

6.3.3. Practice2 min

6.3.4. Practice12 min

6.4.1. Properties of the Definite Integral2 min

6.4.2. Practice2 min

6.4.3. Practice2 min

6.5.1. Antiderivatives (Indefinite Integral)3 min

6.5.2. Example2 min

6.5.3. Practice1 min

6.5.4. Practice1 min

6.5.5. Practice1 min

6.6.1. Antiderivatives (Indefinite Integral) of Trig Functions4 min

6.6.2. Example1 min

6.6.3. Practice1 min

6.6.4. Practice1 min

6.7.1. Antiderivatives (Indefinite Integrals) of Inverse Trig Functions2 min

6.7.2. Example1 min

6.7.3. Practice1 min

6.8.1. Antiderivatives (Indefinite Integral) of Exponential Functions1 min

6.8.2. Example42 sec

6.8.3. Practice1 min

6.8.4. Practice2 min

6.9.1. Antiderivatives (Indefinite Integral) of

\frac{1}{x}

6.9.2. Example2 min

6.9.3. Practice1 min

6.10.1. Fundamental Theorem of Calculus - Part I2 min

6.10.2. Example1 min

6.10.3. Example48 sec

6.10.4. Practice1 min

6.10.5. Practice2 min

6.11.1. Fundamental Theorem of Calculus - Part 21 min

6.11.2. Example2 min

6.11.3. Example

6.11.4. Practice 4 min

6.11.5. Practice3 min

6.11.6. Practice1 min

6.12.1. Area between curves1 min

6.12.2. Example5 min

6.12.3. Example5 min

6.12.4. Practice

6.12.5. Practice

6.13.1. Computing Definite Integrals Geometrically44 sec

6.13.2. Example1 min

6.13.3. Practice1 min

7. Integration Methods

1.3hr

7.1.1. Integration by Substitution2 min

7.1.2. Example4 min

7.1.3. Example3 min

7.1.4. Example (w/ Back-Substitution)4 min

7.1.5. Practice

7.1.6. Practice

7.1.7. Practice

7.1.8. Practice (w/ Back-Substitution)

7.2.1. Integrating Even and Odd Functions2 min

7.2.2. Example2 min

7.2.3. Practice2 min

7.3.1. Integration by Parts3 min

7.3.2. Example5 min

7.3.3. Example3 min

7.3.4. Example6 min

7.3.5. Practice

7.3.6. Practice

7.4.1. Trigonometric Integrals5 min

7.4.2. Example3 min

7.4.3. Example: Trigonometric Integrals (Case 1-3)3 min

7.4.4. Example4 min

7.4.5. Example1 min

7.4.6. Example2 min

7.4.7. Practice

7.4.8. Practice

7.4.9. Practice

7.4.10. Practice

7.5.1. Trigonometric Substitution3 min

7.5.2. Example7 min

7.5.3. Practice

7.5.4. Practice

7.6.1. New Theory

7.7.1. Partial Fractions Decomposition4 min

7.7.2. Example: Long Division with Partial Fractions8 min

7.7.3. Example: Rationalizing Substiution5 min

7.7.4. Example: Repeated factors

7.7.5. Practice Question: Partial Fraction

7.7.6. Practice Question: Partial Fraction

8. Applications of Integration

13min

8.I. Chapter Intro23 sec

8.1.1. Volumes of Revolution by Cross Sections (Washers)3 min

8.1.2. Example4 min

8.1.3. Example3 min

8.1.4. Practice

8.1.5. Practice

8.2.1. Volumes of Revolution by Cylindrical Shells2 min

8.2.2. Practice

8.2.3. Practice

8.2.4. Practice

9. Sequences

20min

9.1.1. Sequences4 min

9.1.2. Example1 min

9.1.3. Practice3 min

9.2.1. Sequence Convergence7 min

9.2.2. Example5 min

9.2.3. Practice

10. Improper Integrals

26min

10.1.1. Improper Integrals5 min

10.1.2. Example4 min

10.1.3. Practice7 min

10.2.1. Comparison Theorem for Improper Integrals3 min

10.2.2. Practice (type 2 - converges)4 min

10.2.3. Practice (type 2 - diverges)

10.2.4. Practice (type 1 - converges)3 min

11. Series and Series Tests

57min

11.1.1. Series8 min

11.1.2. Example4 min

11.2.1. Theory: Integral Test4 min

11.2.2. Example: Convergence of a Series using the Integral Test4 min

11.3.1. Theory: Comparison and Limit Comparison Tests4 min

11.3.2. Example: Comparison Test Problem3 min

11.3.3. Practice Question: Limit Comparison Test Problem4 min

11.4.1. Theory: Alternating Series Test2 min

11.4.2. Practice Question: Using the Alternating Series Test6 min

11.5.1. Theory: Absolute Convergence3 min

11.5.2. Practice Question: Absolutely and Conditionally Convergent Series5 min

11.6.1. Theory: Ratio and Root Tests3 min

11.6.2. Example: Using the Ratio Test6 min

12. Power series and Taylor series

24min

12.1.1. Theory: Power Series4 min

12.1.2. Theory: Properties of Power Series3 min

12.1.3. Example: Finding the Radius of Convergence5 min

12.2.1. Theory: Taylor and Maclaurin Series4 min

12.2.2. Example: Expressing a Function as a Power Series8 min

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