University Calculus

Front Cover
Title Page
Copyright Page
Contents
Preface
1 Functions
1.1 Functions and Their Graphs
1.2 Combining Functions; Shifting and Scaling Graphs
1.3 Trigonometric Functions
1.4 Exponential Functions
1.5 Inverse Functions and Logarithms
1.6 Graphing with Calculators and Computers
2 Limits and Continuity
2.1 Rates of Change and Tangents to Curves
2.2 Limit of a Function and Limit Laws
2.3 The Precise Definition of a Limit
2.4 One-Sided Limits and Limits at Infinity
2.5 Infinite Limits and Vertical Asymptotes
2.6 Continuity
2.7 Tangents and Derivatives at a Point
2 Questions to Guide Your Review
2 Practice Exercises
2 Additional and Advanced Exercises
3 Differentiation
3.1 The Derivative as a Function
3.2 Differentiation Rules for Polynomials, Exponentials, Products, and Quotients
3.3 The Derivative as a Rate of Change
3.4 Derivatives of Trigonometric Functions
3.5 The Chain Rule and Parametric Equations
3.6 Implicit Differentiation
3.7 Derivatives of Inverse Functions and Logarithms
3.8 Inverse Trigonometric Functions
3.9 Related Rates
3.10 Linearization and Differentials
3.11 Hyperbolic Functions
3 Questions to Guide Your Review
3 Practice Exercises
3 Additional and Advanced Exercises
4 Applications of Derivatives
4.1 Extreme Values of Functions
4.2 The Mean Value Theorem
4.3 Monotonic Functions and the First Derivative Test
4.4 Concavity and Curve Sketching
4.5 Applied Optimization
4.6 Indeterminate Forms and L'Hôpital's Rule
4.7 Newton's Method
4.8 Antiderivatives
4 Questions to Guide Your Review
4 Practice Exercises
4 Additional and Advanced Exercises
5 Integration
5.1 Estimating with Finite Sums
5.2 Sigma Notation and Limits of Finite Sums
5.3 The Definite Integral
5.4 The Fundamental Theorem of Calculus
5.5 Indefinite Integrals and the Substitution Rule
5.6 Substitution and Area Between Curves
5.7 The Logarithm Defined as an Integral
5 Questions to Guide Your Review
5 Practice Exercises
5 Additional and Advanced Exercises
6 Applications of Definite Integrals
6.1 Volumes by Slicing and Rotation About an Axis
6.2 Volumes by Cylindrical Shells
6.3 Lengths of Plane Curves
6.4 Areas of Surfaces of Revolution
6.5 Exponential Change and Separable Differential Equations
6.6 Work
6.7 Moments and Centers of Mass
6 Questions to Guide Your Review
6 Practice Exercises
6 Additional and Advanced Exercises
7 Techniques of Integration
7.1 Integration by Parts
7.2 Trigonometric Integrals
7.3 Trigonometric Substitutions
7.4 Integration of Rational Functions by Partial Fractions
7.5 Integral Tables and Computer Algebra Systems
7.6 Numerical Integration
7.7 Improper Integrals
7 Questions to Guide Your Review
7 Practice Exercises
7 Additional and Advanced Exercises
8 Infinite Sequences and Series
8.1 Sequences
8.2 Infinite Series
8.3 The Integral Test
8.4 Comparison Tests
8.5 The Ratio and Root Tests
8.6 Alternating Series, Absolute and Conditional Convergence
8.7 Power Series
8.8 Taylor and Maclaurin Series
8.9 Convergence of Taylor Series
8.10 The Binomial Series
8 Questions to Guide Your Review
8 Practice Exercises
8 Additional and Advanced Exercises
9 Polar Coordinates and Conics
9.1 Polar Coordinates
9.2 Graphing in Polar Coordinates
9.3 Areas and Lengths in Polar Coordinates
9.4 Conic Sections
9.5 Conics in Polar Coordinates
9.6 Conics and Parametric Equations; The Cycloid
9 Questions to Guide Your Review
9 Practice Exercises
9 Additional and Advanced Exercises
10 Vectors and the Geometry of Space
10.1 Three-Dimensional Coordinate Systems
10.2 Vectors
10.3 The Dot Product
10.4 The Cross Product
10.5 Lines and Planes in Space
10.6 Cylinders and Quadric Surfaces
10 Questions to Guide Your Review
10 Practice Exercises
10 Additional and Advanced Exercises
11 Vector-Valued Functions and Motion in Space
11.1 Vector Functions and Their Derivatives
11.2 Integrals of Vector Functions
11.3 Arc Length in Space
11.4 Curvature of a Curve
11.5 Tangential and Normal Components of Acceleration
11.6 Velocity and Acceleration in Polar Coordinates
11 Questions to Guide Your Review
11 Practice Exercises
11 Additional and Advanced Exercises
12 Partial Derivatives
12.1 Functions of Several Variables
12.2 Limits and Continuity in Higher Dimensions
12.3 Partial Derivatives
12.4 The Chain Rule
12.5 Directional Derivatives and Gradient Vectors
12.6 Tangent Planes and Differentials
12.7 Extreme Values and Saddle Points
12.8 Lagrange Multipliers
12.9 Taylor's Formula for Two Variables
12 Questions to Guide Your Review
12 Practice Exercises
12 Additional and Advanced Exercises
13 Multiple Integrals
13.1 Double and Iterated Integrals over Rectangles
13.2 Double Integrals over General Regions
13.3 Area by Double Integration
13.4 Double Integrals in Polar Form
13.5 Triple Integrals in Rectangular Coordinates
13.6 Moments and Centers of Mass
13.7 Triple Integrals in Cylindrical and Spherical Coordinates
13.8 Substitutions in Multiple Integrals
13 Questions to Guide Your Review
13 Practice Exercises
13 Additional and Advanced Exercises
14 Integration in Vector Fields
14.1 Line Integrals
14.2 Vector Fields, Work, Circulation, and Flux
14.3 Path Independence, Potential Functions, and Conservative Fields
14.4 Green's Theorem in the Plane
14.5 Surfaces and Area
14.6 Surface Integrals and Flux
14.7 Stokes' Theorem
14.8 The Divergence Theorem and a Unified Theory
14 Questions to Guide Your Review
14 Practice Exercises
14 Additional and Advanced Exercises
Appendices
A.1 Real Numbers and the Real Line
A.2 Mathematical Induction
A.3 Lines, Circles, and Parabolas
A.4 Trigonometry Formulas
A.5 Proofs of Limit Theorems
A.6 Commonly Occurring Limits
A.7 Theory of the Real Numbers
A.8 The Distributive Law for Vector Cross Products
A.9 The Mixed Derivative Theorem and the Increment Theorem
Answers
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Appendices
Index
A Brief Table of Integrals
Credits
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