Calculus With Analytic Geometry
Table of Contents

Interested in seeing the entire table of contents?

Program Details

CHAPTER 1: Numbers, Functions, and Graphs

1-1 Introduction

1-2 The Real Line and Coordinate Plane: Pythagoras

1-3 Slopes and Equations of Straight Lines

1-4 Circles and Parabolas: Descartes and Fermat

1-5 The Concept of a Function

1-6 Graphs of Functions

1-7 Introductory Trigonometry

1-8 The Functions Sin O and Cos O

CHAPTER 2: The Derivative of a Function

2-0 What is Calculus ?

2-1 The Problems of Tangents

2-2 How to Calculate the Slope of the Tangent

2-3 The Definition of the Derivative

2-4 Velocity and Rates of Change: Newton and Leibriz

2-5 The Concept of a Limit: Two Trigonometric Limits

2-6 Continuous Functions: The Mean Value Theorem and Other Theorem

CHAPTER 3: The Computation of Derivatives

3-1 Derivatives of Polynomials

3-2 The Product and Quotient Rules

3-3 Composite Functions and the Chain Rule

3-4 Some Trigonometric Derivatives

3-5 Implicit Functions and Fractional Exponents

3-6 Derivatives of Higher Order

CHAPTER 4: Applications of Derivatives

4-1 Increasing and Decreasing Functions: Maxima and Minima

4-2 Concavity and Points of Inflection

4-3 Applied Maximum and Minimum Problems

4-4 More Maximum-Minimum Problems

4-5 Related Rates

4-6 Newtons Method for Solving Equations

4-7 Applications to Economics: Marginal Analysis

CHAPTER 5: Indefinite Integrals and Differential Equations

5-1 Introduction

5-2 Differentials and Tangent Line Approximations

5-3 Indefinite Integrals: Integration by Substitution

5-4 Differential Equations: Separation of Variables

5-5 Motion Under Gravity: Escape Velocity and Black Holes

CHAPTER 6: Definite Integrals

6-1 Introduction

6-2 The Problem of Areas

6-3 The Sigma Notation and Certain Special Sums

6-4 The Area Under a Curve: Definite Integrals

6-5 The Computation of Areas as Limits

6-6 The Fundamental Theorem of Calculus

6-7 Properties of Definite Integrals

CHAPTER 7: Applications of Integration

7-1 Introduction: The Intuitive Meaning of Integration

7-2 The Area between Two Curves

7-3 Volumes: The Disk Method

7-4 Volumes: The Method of Cylindrical Shells

7-5 Arc Length

7-6 The Area of a Surface of Revolution

7-7 Work and Energy

7-8 Hydrostatic Force


CHAPTER 8: Exponential and Logarithm Functions

8-1 Introduction

8-2 Review of Exponents and Logarithms

8-3 The Number e and the Function y = e <^>x

8-4 The Natural Logarithm Function y = ln x

8-5 Applications

Population Growth and Radioactive Decay

8-6 More Applications

CHAPTER 9: Trigonometric Functions

9-1 Review of Trigonometry

9-2 The Derivatives of the Sine and Cosine

9-3 The Integrals of the Sine and Cosine

9-4 The Derivatives of the Other Four Functions

9-5 The Inverse Trigonometric Functions

9-6 Simple Harmonic Motion

9-7 Hyperbolic Functions

CHAPTER 10 : Methods of Integration

10-1 Introduction

10-2 The Method of Substitution

10-3 Certain Trigonometric Integrals

10-4 Trigonometric Substitutions

10-5 Completing the Square

10-6 The Method of Partial Fractions

10-7 Integration by Parts

10-8 A Mixed Bag

10-9 Numerical Integration

CHAPTER 11: Further Applications of Integration

11-1 The Center of Mass of a Discrete System

11-2 Centroids

11-3 The Theorems of Pappus

11-4 Moment of Inertia

CHAPTER 12: Indeterminate Forms and Improper Integrals

12-1 Introduction. The Mean Value Theorem Revisited

12-2 The Interminate Form 0/0. L'Hospital's Rule

12-3 Other Interminate Forms

12-4 Improper Integrals

12-5 The Normal Distribution

CHAPTER 13: Infinite Series of Constants

13-1 What is an Infinite Series ?

13-2 Convergent Sequences

13-3 Convergent and Divergent Series

13-4 General Properties of Convergent Series

13-5 Series on Non-negative Terms: Comparison Tests

13-6 The Integral Test

13-7 The Ratio Test and Root Test

13-8 The Alternating Series Test

CHAPTER 14: Power Series

14-1 Introduction

14-2 The Interval of Convergence

14-3 Differentiation and Integration of Power Series

14-4 Taylor Series and Taylor's Formula

14-5 Computations Using Taylor's Formula

14-6 Applications to Differential Equations

14. 7 (optional) Operations on Power Series

14. 8 (optional) Complex Numbers and Euler's Formula


CHAPTER 15: Conic Sections

15-1 Introduction

15-2 Another Look at Circles and Parabolas

15-3 Ellipses

15-4 Hyperbolas

15-5 The Focus-Directrix-Eccentricity Definitions

15-6 (optional) Second Degree Equations

CHAPTER 16: Polar Coordinates

16-1 The Polar Coordinate System

16-2 More Graphs of Polar Equations

16-3 Polar Equations of Circles, Conics, and Spirals

16-4 Arc Length and Tangent Lines

16-5 Areas in Polar Coordinates

CHAPTER 17: Parametric Equations

17-1 Parametric Equations of Curves

17-2 The Cycloid and Other Similar Curves

17-3 Vector Algebra

17-4 Derivatives of Vector Function

17-5 Curvature and the Unit Normal Vector

17-6 Tangential and Normal Components of Acceleration

17-7 Kepler's Laws and Newton's Laws of Gravitation

CHAPTER 18: Vectors in Three-Dimensional Space

18-1 Coordinates and Vectors in Three-Dimensional Space

18-2 The Dot Product of Two Vectors

18-3 The Cross Product of Two Vectors

18-4 Lines and Planes

18-5 Cylinders and Surfaces of Revolution

18-6 Quadric Surfaces

18-7 Cylindrical and Spherical Coordinates

CHAPTER 19: Partial Derivatives

19-1 Functions of Several Variables

19-2 Partial Derivatives

19-3 The Tangent Plane to a Surface

19-4 Increments and Differentials

19-5 Directional Derivatives and the Gradient

19-6 The Chain Rule for Partial Derivatives

19-7 Maximum and Minimum Problems

19-8 Constrained Maxima and Minima

19-9 Laplace's Equation, the Heat Equation, and the Wave Equation

19-10 (optional) Implicit Functions

CHAPTER 20: Multiple Integrals

20-1 Volumes as Iterated Integrals

20-2 Double Integrals and Iterated Integrals

20-3 Physical Applications of Double Integrals

20-4 Double Integrals in Polar Coordinates

20-5 Triple Integrals

20-6 Cylindrical Coordinates

20-7 Spherical Coordinates

20-8 Areas of curved Surfaces

CHAPTER 21: Line and Surface Integrals

21-1 Green's Theorem, Gauss's Theorem, and Stokes' Theorem

21-2 Line Integrals in the Plane

21-3 Independence of Path

21-4 Green's Theorem

21-5 Surface Integrals and Gauss's Theorem

21-6 Maxwell's Equations : A Final Thought


A: The Theory of Calculus

A-1 The Real Number System

A-2 Theorems About Limits

A-3 Some Deeper Properties of Continuous Functions

A-4 The Mean Value theorem

A-5 The Integrability of Continuous Functions

A-6 Another Proof of the Fundamental Theorem of Calculus

A-7 Continuous Curves With No Length

A-8 The Existence of e = lim h->0 (1 + h) <^>1/h

A-9 Functions That Cannot Be Integrated

A-10 The Validity of Integration by Inverse Substitution

A-11 Proof of the Partial fractions Theorem

A-12 The Extended Ratio Tests of Raabe and Gauss

A-13 Absolute vs Conditional Convergence

A-14 Dirichlet's Test

A-15 Uniform Convergence for Power Series

A-16 Division of Power Series

A-17 The Equality of Mixed Partial Derivatives

A-18 Differentiation Under the Integral Sign

A-19 A Proof of the Fundamental Lemma

A-20 A Proof of the Implicit Function Theorem

A-21 Change of Variables in Multiple Integrals

B: A Few Review Topics

B-1 The Binomial Theorem

B-2 Mathematical Induction