Calculus

Chapter 1: Precalculus Review
1.1 Real Numbers, Functions, and Graphs
1.2 Linear and Quadratic Functions
1.3 The Basic Classes of Functions
1.4 Trigonometric Functions
1.5 Technology: Calculators and Computers
Chapter Review Exercises

Chapter 2: Limits
2.1 The Limit Idea: Instantaneous Velocity and Tangent Lines
2.2 Investigating Limits
2.3 Basic Limit Laws
2.4 Limits and Continuity
2.5 Indeterminate Forms
2.6 The Squeeze Theorem and Trigonometric Limits
2.7 Limits at Infinity
2.8 The Intermediate Value Theorem
2.9 The Formal Definition of a Limit
Chapter Review Exercises

Chapter 3: Differentiation
3.1 Definition of the Derivative
3.2 The Derivative as a Function
3.3 Product and Quotient Rules
3.4 Rates of Change
3.5 Higher Derivatives
3.6 Trigonometric Functions
3.7 The Chain Rule
3.8 Implicit Differentiation
3.9 Related Rates
Chapter Review Exercises

Chapter 4: Applications of the Derivative
4.1 Linear Approximation and Applications
4.2 Extreme Values
4.3 The Mean Value Theorem and Monotonicity
4.4 The Second Derivative and Concavity
4.5 Analyzing and Sketching Graphs of Functions
4.6 Applied Optimization
4.7 Newton’s Method
Chapter Review Exercises

Chapter 5: Integration
5.1 Approximating and Computing Area
5.2 The Definite Integral
5.3 The Indefinite Integral
5.4 The Fundamental Theorem of Calculus, Part I
5.5 The Fundamental Theorem of Calculus, Part II
5.6 Net Change as the Integral of a Rate of Change
5.7 The Substitution Method
Chapter Review Exercises

Chapter 6: Applications of the Integral
6.1 Area Between Two Curves
6.2 Setting Up Integrals: Volume, Density, Average Value
6.3 Volumes of Revolution: Disks and Washers
6.4 Volumes of Revolution: Cylindrical Shells
6.5 Work and Energy
Chapter Review Exercises

Chapter 7: Exponential and Logarithmic Functions
7.1 The Derivative of f (x) = bx and the Number e
7.2 Inverse Functions
7.3 Logarithmic Functions and Their Derivatives
7.4 Applications of Exponential and Logarithmic Functions
7.5 L’Hopital’s Rule
7.6 Inverse Trigonometric Functions
7.7 Hyperbolic Functions
Chapter Review Exercises

Chapter 8: Techniques of Integration
8.1 Integration by Parts
8.2 Trigonometric Integrals
8.3 Trigonometric Substitution
8.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions
8.5 The Method of Partial Fractions
8.6 Strategies for Integration
8.7 Improper Integrals
8.8 Numerical Integration
Chapter Review Exercises

Chapter 9: Further Applications of the Integral
9.1 Probability and Integration
9.2 Arc Length and Surface Area
9.3 Fluid Pressure and Force
9.4 Center of Mass
Chapter Review Exercises

Chapter 10: Introduction to Differential Equations
10.1 Solving Differential Equations
10.2 Models Involving y'=k(y-b)
10.3 Graphical and Numerical Methods
10.4 The Logistic Equation
10.5 First-Order Linear Equations
Chapter Review Exercises

Chapter 11: Infinite Series
11.1 Sequences
11.2 Summing an Infinite Series
11.3 Convergence of Series with Positive Terms
11.4 Absolute and Conditional Convergence
11.5 The Ratio and Root Tests and Strategies for Choosing Tests
11.6 Power Series
11.7 Taylor Polynomials
11.8 Taylor Series
Chapter Review Exercises

Chapter 12: Parametric Equations, Polar Coordinates, and Conic Sections
12.1 Parametric Equations
12.2 Arc Length and Speed
12.3 Polar Coordinates
12.4 Area and Arc Length in Polar Coordinates
12.5 Conic Sections
Chapter Review Exercises

Chapter 13: Vector Geometry
13.1 Vectors in the Plane
13.2 Three-Dimensional Space: Surfaces, Vectors, and Curves
13.3 Dot Product and the Angle Between Two Vectors
13.4 The Cross Product
13.5 Planes in 3-Space
13.6 A Survey of Quadric Surfaces
13.7 Cylindrical and Spherical Coordinates
Chapter Review Exercises

Chapter 14: Calculus of Vector-Valued Functions
14.1 Vector-Valued Functions
14.2 Calculus of Vector-Valued Functions
14.3 Arc Length and Speed
14.4 Curvature
14.5 Motion in 3-Space
14.6 Planetary Motion According to Kepler and Newton
Chapter Review Exercises

Chapter 15: Differentiation in Several Variables
15.1 Functions of Two or More Variables
15.2 Limits and Continuity in Several Variables
15.3 Partial Derivatives
15.4 Differentiability, Tangent Planes, and Linear Approximation
15.5 The Gradient and Directional Derivatives
15.6 Multivariable Calculus Chain Rules
15.7 Optimization in Several Variables
15.8 Lagrange Multipliers: Optimizing with a Constraint
Chapter Review Exercises

Chapter 16: Multiple Integration
16.1 Integration in Two Variables
16.2 Double Integrals over More General Regions
16.3 Triple Integrals
16.4 Integration in Polar, Cylindrical, and Spherical Coordinates
16.5 Applications of Multiple Integrals
16.6 Change of Variables
Chapter Review Exercises

Chapter 17: Line and Surface Integrals
17.1 Vector Fields
17.2 Line Integrals
17.3 Conservative Vector Fields
17.4 Parametrized Surfaces and Surface Integrals
17.5 Surface Integrals of Vector Fields
Chapter Review Exercises

Chapter 18: Fundamental Theorems of Vector Analysis
18.1 Green’s Theorem
18.2 Stokes’ Theorem
18.3 Divergence Theorem
Chapter Review Exercises

Appendices
A. The Language of Mathematics
B. Properties of Real Numbers
C. Induction and the Binomial Theorem
D. Additional Proofs

ANSWERS TO ODD-NUMBERED EXERCISES

REFERENCES

INDEX

Additional content can be accessed online at www.macmillanlearning.com/calculuset4e:

Additional Proofs:
L’Hôpital’s Rule
Error Bounds for Numerical
Integration
Comparison Test for Improper
Integrals

Additional Content:
Second-Order Differential
Equations
Complex Numbers