Elementary Calculus

Chapter 1. Functions and Limits

1.3 New Functions from Old Functions

1.4 The Tangent and Velocity Problems

1.5 The Limit of a Function

1.6 Calculating Limits Using the Limit Laws

1.7 The Precise Definition of a Limit

1.8 Continuity

Chapter 2. Derivatives

2.1 Derivatives and Rates of Change

2.2 The Derivatives as a Function

2.3 Differentiation Formulas

2.4 Derivatives of Trigonometric Functions

2.5 The Chain Rule

2.6 Implicit Differentiation

2.8 Related Rates

2.9 Linear Approximations and Differentials

Chapter 3. Applications of Differentiation

3.1 Maximum and Minimum Values

3.2 The Mean Value Theorem

3.3 What Derivatives Tells Us about the Shape of a Graph

3.4 Limits at Infinity; Horizontal Asymptotes

3.5 Summary of Curve Sketching

3.7 Optimization Problems

3.8 Newton’s Method

3.9 Antiderivatives

Chapter 4. Integrals

4.1 The Area and Distance Problems

4.2 The Definite Integral

4.3 The Fundamental Theorem of Calculus

4.4 Indefinite Integrals and the Net Change Theorem

4.5 The Substitution Rule

Chapter 5. Applications of Integration

5.1 Areas Between Curves

5.2 Volumes

5.3 Volumes by Cylindrical Shells

5.4 Work

5.5 Average Value of a Function

Chapter 6. Inverse Functions: Exponential, Logarithmic, and Inverse Trigonometric Functions

6.1 Inverse Functions and Their Derivatives

6.2 Exponential Functions and Their Derivatives

6.3 Logarithmic Functions

6.4 Derivatives of Logarithmic Functions

6.6 Inverse Trigonometric Functions

6.7 Hyperbolic Functions

6.8 Indeterminate Forms and l’Hospital’s Rule

Chapter 7. Techniques of Integration

7.1 Integration by Parts

7.2 Trigonometric Integrals

7.3 Trigonometric Substitution

7.4 Integration of Rational Functions by Partial Fractions

7.5 Strategy for Integration

7.8 Improper Integrals

Chapter 8. Further Applications of Integration

8.1 Arc Length

8.2 Area of Surface of Revolution

Calculus 1

Chapter 1. Series

1.1 Sequence and Series
1.2 Geometric Series

1.3 Comparison Test
1.4 Root Test

1.5 Ratio Test
1.6 Integral Test
1.7 Alternating Series and Absolute Convergence

Chapter 2. Power Series

2.1 Power Series
2.2 Radius of Convergence

2.3 Power Series of The Exponential Function

2.4 Power Series of Trigonometric Functions

2.5 Hyperbolic Functions
2.6 Inverse Trigonometric Functions

Chapter 3. Taylor Expansion

3.1 Cauchy’s Mean Value Theorem

3.2 L’Hôpital’s Rule

3.3 Linear and Polynomial Approximations

3.4 Taylor’s Theorem
3.5 Taylor Series

3.6 Taylor Expansion at an Arbitrary Point

Chapter 4. Space and Coordinate System

4.1 Cartesian Spaces
4.2 Polar Coordinates

4.3 Cylindrical and Spherical Coordinates

Chapter 5. Vectors

5.1 Parallel Translation
5.2 Directed Line Segments and Vectors

5.3 The Inner Product

5.4 Equations of Lines and Planes

5.5 Linear Dependence and Independence

Chapter 6. Matrices and Linear Transformations

6.1 Matrices

6.2 Linear Transformations

Chapter 7. Square matrices and Determinants

7.1 Inverse Matrices
7.2 Permutations

7.3 Determinants

7.4.2 Properties of Determinants

Chapter 8. Three-Dimensional Space and Cross Product

8.1 The Cross Product

Chapter 9. Curves

9.1 Parametrized Curves

9.2 Acceleration Vectors

9.3 Reparametrizations

9.4 Lengths of Curves

9.5 Polar Coordinates and Areas

9.6 Arc Length and Reparametrization

9.7 Line Integrals