Topics to study

 

Volume 1

 

Functions and Graphs

1.1 Review of Functions

1.2 Basic Classes of Functions

1.3 Trigonometric Functions

1.4 Inverse Functions

1.5 Exponential and Logarithmic Functions

 

Limits

2.1 A Preview of Calculus

2.2 The Limit of a Function

2.3 The Limit Laws

2.4 Continuity

2.5 The Precise Definition of a Limit

 

Derivatives

3.1 Defining the Derivative

3.2 The Derivative as a Function

3.3 Differentiation Rules

3.4 Derivatives as Rates of Change

3.5 Derivatives of Trigonometric Functions

3.6 The Chain Rule

3.7 Derivatives of Inverse Functions

3.8 Implicit Differentiation

3.9 Derivatives of Exponential and Logarithmic Functions

 

Applications of Derivatives

4.1 Related Rates

4.2 Linear Approximations and Differentials

4.3 Maxima and Minima

4.4 The Mean Value Theorem

4.5 Derivatives and the Shape of a Graph

4.6 Limits at Infinity and Asymptotes

4.7 Applied Optimization Problems

4.8 L’Hôpital’s Rule

4.9 Newton’s Method

4.10 Antiderivatives

Integration

5.1 Approximating Areas

5.2 The Definite Integral

5.3 The Fundamental Theorem of Calculus

5.4 Integration Formulas and the Net Change Theorem

5.5 Substitution

5.6 Integrals Involving Exponential and Logarithmic Functions

5.7 Integrals Resulting in Inverse Trigonometric Functions

 

Applications of Integration

6.1 Areas between Curves

6.2 Determining Volumes by Slicing

6.3 Volumes of Revolution: Cylindrical Shells

6.4 Arc Length of a Curve and Surface Area

6.5 Physical Applications

6.6 Moments and Centers of Mass

6.7 Integrals, Exponential Functions, and Logarithms

6.8 Exponential Growth and Decay

6.9 Calculus of the Hyperbolic Functions

Volume 2

 

Integration

1.1 Approximating Areas

1.2 The Definite Integral

1.3 The Fundamental Theorem of Calculus

1.4 Integration Formulas and the Net Change Theorem

1.5 Substitution

1.6 Integrals Involving Exponential and Logarithmic Functions

1.7 Integrals Resulting in Inverse Trigonometric Functions

 

Applications of Integration

2.1 Areas between Curves

2.2 Determining Volumes by Slicing

2.3 Volumes of Revolution: Cylindrical Shells

2.4 Arc Length of a Curve and Surface Area

2.5 Physical Applications

2.6 Moments and Centers of Mass

2.7 Integrals, Exponential Functions, and Logarithms

2.8 Exponential Growth and Decay

2.9 Calculus of the Hyperbolic Functions

 

Techniques of Integration

3.1 Integration by Parts

3.2 Trigonometric Integrals

3.3 Trigonometric Substitution

3.4 Partial Fractions

3.5 Other Strategies for Integration

3.6 Numerical Integration

3.7 Improper Integrals

 

Introduction to Differential Equations

4.1 Basics of Differential Equations

4.2 Direction Fields and Numerical Methods

4.3 Separable Equations

4.4 The Logistic Equation

4.5 First-order Linear Equations

 

Sequences and Series

5.1 Sequences

5.2 Infinite Series

5.3 The Divergence and Integral Tests

5.4 Comparison Tests

5.5 Alternating Series

5.6 Ratio and Root Tests

 

Power Series

6.1 Power Series and Functions

6.2 Properties of Power Series

6.3 Taylor and Maclaurin Series

6.4 Working with Taylor Series

 

Parametric Equations and Polar Coordinates

7.1 Parametric Equations

7.2 Calculus of Parametric Curves

7.3 Polar Coordinates

7.4 Area and Arc Length in Polar Coordinates

7.5 Conic Sections

 

Volume 3

 

Parametric Equations and Polar Coordinates

1.1 Parametric Equations

1.2 Calculus of Parametric Curves

1.3 Polar Coordinates

1.4 Area and Arc Length in Polar Coordinates

1.5 Conic Sections

 

Vectors in Space

2.1 Vectors in the Plane

2.2 Vectors in Three Dimensions

2.3 The Dot Product

2.4 The Cross Product

2.5 Equations of Lines and Planes in Space

2.6 Quadric Surfaces

2.7 Cylindrical and Spherical Coordinates

 

Vector-Valued Functions

3.1 Vector-Valued Functions and Space Curves

3.2 Calculus of Vector-Valued Functions

3.3 Arc Length and Curvature

3.4 Motion in Space

 

Differentiation of Functions of Several Variables

4.1 Functions of Several Variables

4.2 Limits and Continuity

4.3 Partial Derivatives

4.4 Tangent Planes and Linear Approximations

4.5 The Chain Rule

4.6 Directional Derivatives and the Gradient

4.7 Maxima/Minima Problems

4.8 Lagrange Multipliers

 

Multiple Integration

5.1 Double Integrals over Rectangular Regions

5.2 Double Integrals over General Regions

5.3 Double Integrals in Polar Coordinates

5.4 Triple Integrals

5.5 Triple Integrals in Cylindrical and Spherical Coordinates

5.6 Calculating Centers of Mass and Moments of Inertia

5.7 Change of Variables in Multiple Integrals

 

Vector Calculus

6.1 Vector Fields

6.2 Line Integrals

6.3 Conservative Vector Fields

6.4 Green’s Theorem

6.5 Divergence and Curl

6.6 Surface Integrals

6.7 Stokes’ Theorem

6.8 The Divergence Theorem

 

Second-Order Differential Equations

7.1 Second-Order Linear Equations

7.2 Nonhomogeneous Linear Equations

7.3 Applications

7.4 Series Solutions of Differential Equations