Calculus Tutorials

Calculus 
Numbering approximately corresponds to James Stewart, Calculus, 6th ed. (Belmont, CA: Thompson, 2008).

Chapter 1: Functions
1.1 What is a function?
1.2 Types of functions
1.3 Transforming functions

Chapter 2: Limits
2.1 The Tangent Problem (the beginning of Differential Calculus)
2.2a What is a Limit?
2.2b Right and Left Hand Limits
2.3a Limit Laws
2.3b The Squeeze Theorem
2.4 The Precise Definition of a Limit
2.5 Limits and Continuity

Chapter 3: Derivatives
3.1 What is a Derivative?
3.2 The Power Rule (also Constant Rule, Sum Rule, Difference Rule)
3.3 The Product and Quotient Rule
3.4 Derivatives of Trig Functions
3.5 The Chain Rule
3.6 Implicit Differentiation

Chapter 4: Applications of Differentiation
4.1 Maximum and Minimum Values
4.2 The Mean Value Theorem
4.3 Derivative Tests
4.4 Infinite Limits
4.5 Summary of Sketching Graphs
4.6 Optimization Problems (including First Derivative Test, business functions)
4.7 Newton's Method
4.8 Antiderivatives

Chapter 5: Integral Calculus
5.1 The Area Problem (beginning of Integral Calculus)
5.2 The Definite Integral
5.3 The Fundamental Theorem of Calculus
5.4a Indefinite Integrals
5.4b Some Basic Indefinite Integral Formulas
5.4c Net Change Theorem
5.5a The Substitution Method (for indefinite integrals)
5.5b The Substitution Method (for definite integrals)

Chapter 6: Applications of Integration
6.1 Areas Between Curves
6.2 Volumes
6.3 Volume by Cylindrical Shell
6.4 Work (see also the physics video on work)
6.5 Average Value of a Function

Chapter 7: Inverse Functions
7.1 Introduction to Inverse Functions

Approach 1 to Logarithmic and Exponential Functions (from more intuitive perspective)
7.2a Review of Exponential Functions
7.2b Natural Exponential Function e^x
7.3 Logarithmic Functions (including natural log)
7.4a Derivatives of Natural Log Functions

7.4b Derivatives of Logarithm and Exponential Functions

7.4c Logarithmic Differentiation (and e as a limit)

Approach 2 to Logarithmic and Exponential Functions (from more rigorous perspective)

7.2a* Defining the Natural Logarithmic Function

7.2b* Derivatives of Natural Logarithms

7.2c* Logarithmic Differentiation

7.3a* Defining the Natural Exponential Function (e^x)

7.3b* Differentiating the Natural Exponential Function

7.4a* General Exponential Functions

7.4b* Four Cases of Derivatives for Bases and Exponents (& proof of power rule)

7.4c* General Logarithmic Functions

7.4d* e as a Limit

7.2-4 Summary of Calculus of Logs, Natural Logs, and Exponential Functions

7.5a Exponential Growth and Decay

7.5b Population Growth

7.5c Radioactive Decay

7.5d Newton's Law of Cooling

7.5e Continuously Compounded Interest

7.6 Inverse Trigonometric Functions

7.7 Hyperbolic Functions

7.8 Indeterminate Forms and L'Hospital's Rule

Chapter 8 Techniques of Integration

8.0 Some Basic Integration Formulas

8.1a Integration by Parts (introduction)

8.1b More Examples of Integration by Parts

8.1c Integration by Parts for Definite Integral

8.1d The Reduction Formula for Powers of Sine

8.2a Trigonometric Integrals (sine, cosine to an odd power)

8.2b Trigonometric Integrals (sine, cosine to even powers)

8.2c Trigonometric Integrals (secant to even powers with tangent)

8.2d Trigonometric Integrals (tangent to odd power with secant)

8.3a Trigonometric Substitution (using a-sin-theta)

8.3b Trigonometric Substitution (using a-tangent-theta)

8.3c Trigonometric Substitution (using a-sec-theta)

8.4a Integration by Partial Fractions (numerator larger)

8.4b Integration by Partial Fractions (denominator first power)

8.4c Integration by Partial Fractions (denominator higher power)

8.4d Integration by Partial Fractions (completing the square in denominator)

8.5 Strategies for Integration

8.6 Integration using Formulas

8.7a Approximate Integration (using Left, Right Endpoints and Midpoints)

8.7b Approximate Integration (using the Trapezoidal and Simpson Rules)

8.8a Improper Integrals 

8.8b Magnetic Potential u on Axis of Coil Equation

Chapter 9 Further Applications of Integration

9.1 Arc Length (7.4 in Essentials)

9.2 Area of a Surface of Revolution

9.3 Applications to Physics and Engineering

9.4 Applications to Economics and Biology

9.5 Probability

Chapter 10: Differential Equations
10.1 Modeling with Differential Equations
10.2a Slope and Distance Fields
10.2b Euler's Method

Chapter 11: Parametric Equations and Polar Coordinates

11.1 Curves Defined by Parametric Equations

11.2 Calculus with Parametric Curves

11.3 Polar Coordinates

11.4 Areas and Lengths in Polar Coordinates

11.5 Conic Sections

11.6 Conic Sections in Polar Coordinates

Chapter 12: Infinite Sequences and Series (Chapter 8 in Essentials)

12.1 Sequences

12.2 Series

12.3 The Integral Test and Estimates of Sums

12.4 The Comparison Tests

12.5 Alternating Series

12.6 Absolute Convergence and the Ratio and Root Tests

12.7 Strategy for Testing Series

12.8 Power Series

12.9 Representations of Functions as Power Series

12.10 Taylor and Maclaurin Series

12.11 Applications of Taylor Polynomials

12.7

Chapter 13: Vectors and the Geometry of Space
13.1 Three Dimensional Coordinate Systems
13.2 Vectors
13.3 The Dot Product (see also this physics version)
13.4 The Cross Product (see also this physics version)