, Contents
CHAPTER 1 Introduction to Calculus
1.1 Velocity and Distance
1.2 Calculus Without Limits
1.3 The Velocity at an Instant
1.4 Circular Motion
1.5 A Review of Trigonometry
1.6 A Thousand Points of Light
1.7 Computing in Calculus
CHAPTER 2 Derivatives
The Derivative of a Function
Powers and Polynomials
The Slope and the Tangent Line
Derivative of the Sine and Cosine
The Product and Quotient and Power Rules
Limits
Continuous Functions
CHAPTER 3 Applications of the Derivative
3.1 Linear Approximation
3.2 Maximum and Minimum Problems
3.3 Second Derivatives: Minimum vs. Maximum
3.4 Graphs
3.5 Ellipses, Parabolas, and Hyperbolas
3.6 ,
Iterations x,+ = F(x,)
3.7 Newton's Method and Chaos
3.8 The Mean Value Theorem and l'H8pital's Rule
, Contents
CHAPTER 4 The Chain Rule
4.1 Derivatives by the Chain Rule
4.2 Implicit Differentiation and Related Rates
4.3 Inverse Functions and Their Derivatives
4.4 Inverses of Trigonometric Functions
CHAPTER 5 Integrals
5.1 The Idea of the Integral 177
5.2 Antiderivatives 182
5.3 Summation vs. Integration 187
5.4 Indefinite Integrals and Substitutions 195
5.5 The Definite Integral 201
5.6 Properties of the Integral and the Average Value 206
5.7 The Fundamental Theorem and Its Consequences 213
5.8 Numerical Integration 220
CHAPTER 6 Exponentials and Logarithms
6.1 An Overview 228
6.2 The Exponential ex 236
6.3 Growth and Decay in Science and Economics 242
6.4 Logarithms 252
6.5 Separable Equations Including the Logistic Equation 259
6.6 Powers Instead of Exponentials 267
6.7 Hyperbolic Functions 277
CHAPTER 7 Techniques of Integration
7.1 Integration by Parts
7.2 Trigonometric Integrals
7.3 Trigonometric Substitutions
7.4 Partial Fractions
7.5 Improper Integrals
CHAPTER 8 Applications of the Integral
8.1 Areas and Volumes by Slices
8.2 Length of a Plane Curve
8.3 Area of a Surface of Revolution
8.4 Probability and Calculus
8.5 Masses and Moments
8.6 Force, Work, and Energy
, Contents
CHAPTER 9 Polar Coordinates and Complex Numbers
9.1 Polar Coordinates 348
9.2 Polar Equations and Graphs 351
9.3 Slope, Length, and Area for Polar Curves 356
9.4 Complex Numbers 360
CHAPTER 10 Infinite Series
10.1 The Geometric Series
10.2 Convergence Tests: Positive Series
10.3 Convergence Tests: All Series
10.4 The Taylor Series for ex, sin x, and cos x
10.5 Power Series
CHAPTER 11 Vectors and Matrices
11.1 Vectors and Dot Products
11.2 Planes and Projections
11.3 Cross Products and Determinants
11.4 Matrices and Linear Equations
11.5 Linear Algebra in Three Dimensions
CHAPTER 12 Motion along a Curve
12.1 The Position Vector 446
12.2 Plane Motion: Projectiles and Cycloids 453
12.3 Tangent Vector and Normal Vector 459
12.4 Polar Coordinates and Planetary Motion 464
CHAPTER 13 Partial Derivatives
13.1 Surfaces and Level Curves 472
13.2 Partial Derivatives 475
13.3 Tangent Planes and Linear Approximations 480
13.4 Directional Derivatives and Gradients 490
13.5 The Chain Rule 497
13.6 Maxima, Minima, and Saddle Points 504
13.7 Constraints and Lagrange Multipliers 514
CHAPTER 1 Introduction to Calculus
1.1 Velocity and Distance
1.2 Calculus Without Limits
1.3 The Velocity at an Instant
1.4 Circular Motion
1.5 A Review of Trigonometry
1.6 A Thousand Points of Light
1.7 Computing in Calculus
CHAPTER 2 Derivatives
The Derivative of a Function
Powers and Polynomials
The Slope and the Tangent Line
Derivative of the Sine and Cosine
The Product and Quotient and Power Rules
Limits
Continuous Functions
CHAPTER 3 Applications of the Derivative
3.1 Linear Approximation
3.2 Maximum and Minimum Problems
3.3 Second Derivatives: Minimum vs. Maximum
3.4 Graphs
3.5 Ellipses, Parabolas, and Hyperbolas
3.6 ,
Iterations x,+ = F(x,)
3.7 Newton's Method and Chaos
3.8 The Mean Value Theorem and l'H8pital's Rule
, Contents
CHAPTER 4 The Chain Rule
4.1 Derivatives by the Chain Rule
4.2 Implicit Differentiation and Related Rates
4.3 Inverse Functions and Their Derivatives
4.4 Inverses of Trigonometric Functions
CHAPTER 5 Integrals
5.1 The Idea of the Integral 177
5.2 Antiderivatives 182
5.3 Summation vs. Integration 187
5.4 Indefinite Integrals and Substitutions 195
5.5 The Definite Integral 201
5.6 Properties of the Integral and the Average Value 206
5.7 The Fundamental Theorem and Its Consequences 213
5.8 Numerical Integration 220
CHAPTER 6 Exponentials and Logarithms
6.1 An Overview 228
6.2 The Exponential ex 236
6.3 Growth and Decay in Science and Economics 242
6.4 Logarithms 252
6.5 Separable Equations Including the Logistic Equation 259
6.6 Powers Instead of Exponentials 267
6.7 Hyperbolic Functions 277
CHAPTER 7 Techniques of Integration
7.1 Integration by Parts
7.2 Trigonometric Integrals
7.3 Trigonometric Substitutions
7.4 Partial Fractions
7.5 Improper Integrals
CHAPTER 8 Applications of the Integral
8.1 Areas and Volumes by Slices
8.2 Length of a Plane Curve
8.3 Area of a Surface of Revolution
8.4 Probability and Calculus
8.5 Masses and Moments
8.6 Force, Work, and Energy
, Contents
CHAPTER 9 Polar Coordinates and Complex Numbers
9.1 Polar Coordinates 348
9.2 Polar Equations and Graphs 351
9.3 Slope, Length, and Area for Polar Curves 356
9.4 Complex Numbers 360
CHAPTER 10 Infinite Series
10.1 The Geometric Series
10.2 Convergence Tests: Positive Series
10.3 Convergence Tests: All Series
10.4 The Taylor Series for ex, sin x, and cos x
10.5 Power Series
CHAPTER 11 Vectors and Matrices
11.1 Vectors and Dot Products
11.2 Planes and Projections
11.3 Cross Products and Determinants
11.4 Matrices and Linear Equations
11.5 Linear Algebra in Three Dimensions
CHAPTER 12 Motion along a Curve
12.1 The Position Vector 446
12.2 Plane Motion: Projectiles and Cycloids 453
12.3 Tangent Vector and Normal Vector 459
12.4 Polar Coordinates and Planetary Motion 464
CHAPTER 13 Partial Derivatives
13.1 Surfaces and Level Curves 472
13.2 Partial Derivatives 475
13.3 Tangent Planes and Linear Approximations 480
13.4 Directional Derivatives and Gradients 490
13.5 The Chain Rule 497
13.6 Maxima, Minima, and Saddle Points 504
13.7 Constraints and Lagrange Multipliers 514
