Description
Book SynopsisAbout our authors
Joel Hass received his PhD from the University of California - Berkeley. He is currently a professor of mathematics at the University of California - Davis. He has coauthored 6 widely used calculus texts as well as 2 calculus study guides. He is currently on the editorial board of?Geometriae Dedicata and Media-Enhanced Mathematics. He has been a member of the Institute for Advanced Study at Princeton University and of the Mathematical Sciences Research Institute, and he was a Sloan Research Fellow. Hass's current areas of research include the geometry of proteins, 3-dimensional manifolds, applied math and computational complexity. In his free time, Hass enjoys kayaking.
Christopher Heil received his PhD from the University of Maryland. He is currently a professor of mathematics at the Georgia Institute of Technology. He is the author of a graduate text on analysis and a number of highly cited research survey article
Table of Contents
Table of Contents
-
Functions
- 1.1 Functions and Their Graphs
- 1.2 Combining Functions; Shifting and Scaling Graphs
- 1.3 Trigonometric Functions
- 1.4 Graphing with Software
- 1.5 Exponential Functions
- 1.6 Inverse Functions and Logarithms
-
Limits and Continuity
- 2.1 Rates of Change and Tangent Lines to Curves
- 2.2 Limit of a Function and Limit Laws
- 2.3 The Precise Definition of a Limit
- 2.4 One-Sided Limits
- 2.5 Continuity
- 2.6 Limits Involving Infinity; Asymptotes of Graphs
- Questions to Guide Your Review
- Practice Exercises
- Additional and Advanced Exercises
-
Derivatives
- 3.1 Tangent Lines and the Derivative at a Point
- 3.2 The Derivative as a Function
- 3.3 Differentiation Rules
- 3.4 The Derivative as a Rate of Change
- 3.5 Derivatives of Trigonometric Functions
- 3.6 The Chain Rule
- 3.7 Implicit Differentiation
- 3.8 Derivatives of Inverse Functions and Logarithms
- 3.9 Inverse Trigonometric Functions
- 3.10 Related Rates
- 3.11 Linearization and Differentials
- Questions to Guide Your Review
- Practice Exercises
- Additional and Advanced Exercises
-
Applications of Derivatives
- 4.1 Extreme Values of Functions on Closed Intervals
- 4.2 The Mean Value Theorem
- 4.3 Monotonic Functions and the First Derivative Test
- 4.4 Concavity and Curve Sketching
- 4.5 Indeterminate Forms and L’Hôpital’s Rule
- 4.6 Applied Optimization
- 4.7 Newton’s Method
- 4.8 Antiderivatives
- Questions to Guide Your Review
- Practice Exercises
- Additional and Advanced Exercises
-
Integrals
- 5.1 Area and 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 Method
- 5.6 Definite Integral Substitutions and the Area Between Curves
- Questions to Guide Your Review
- Practice Exercises
- Additional and Advanced Exercises
-
Applications of Definite Integrals
- 6.1 Volumes Using Cross-Sections
- 6.2 Volumes Using Cylindrical Shells
- 6.3 Arc Length
- 6.4 Areas of Surfaces of Revolution
- 6.5 Work
- 6.6 Moments and Centers of Mass
- Questions to Guide Your Review
- Practice Exercises
- Additional and Advanced Exercises
-
Integrals and Transcendental Functions
- 7.1 The Logarithm Defined as an Integral
- 7.2 Exponential Change and Separable Differential Equations
- 7.3 Hyperbolic Functions
- Questions to Guide Your Review
- Practice Exercises
- Additional and Advanced Exercises
-
Techniques of Integration
- 8.1 Integration by Parts
- 8.2 Trigonometric Integrals
- 8.3 Trigonometric Substitutions
- 8.4 Integration of Rational Functions by Partial Fractions
- 8.5 Integral Tables and Computer Algebra Systems
- 8.6 Numerical Integration
- 8.7 Improper Integrals
- Questions to Guide Your Review
- Practice Exercises
- Additional and Advanced Exercises
-
Infinite Sequences and Series
- 9.1 Sequences
- 9.2 Infinite Series
- 9.3 The Integral Test
- 9.4 Comparison Tests
- 9.5 Absolute Convergence; The Ratio and Root Tests
- 9.6 Alternating Series and Conditional Convergence
- 9.7 Power Series
- 9.8 Taylor and Maclaurin Series
- 9.9 Convergence of Taylor Series
- 9.10 Applications of Taylor Series
- Questions to Guide Your Review
- Practice Exercises
- Additional and Advanced Exercises
-
Parametric Equations and Polar Coordinates
- 10.1 Parametrizations of Plane Curves
- 10.2 Calculus with Parametric Curves
- 10.3 Polar Coordinates
- 10.4 Graphing Polar Coordinate Equations
- 10.5 Areas and Lengths in Polar Coordinates
- Questions to Guide Your Review
- Practice Exercises
- Additional and Advanced Exercises
-
Vectors and the Geometry of Space
- 11.1 Three-Dimensional Coordinate Systems
- 11.2 Vectors
- 11.3 The Dot Product
- 11.4 The Cross Product
- 11.5 Lines and Planes in Space
- 11.6 Cylinders and Quadric Surfaces
- Questions to Guide Your Review
- Practice Exercises
- Additional and Advanced Exercises
-
Vector-Valued Functions and Motion in Space
- 12.1 Curves in Space and Their Tangents
- 12.2 Integrals of Vector Functions; Projectile Motion
- 12.3 Arc Length in Space
- 12.4 Curvature and Normal Vectors of a Curve
- 12.5 Tangential and Normal Components of Acceleration
- 12.6 Velocity and Acceleration in Polar Coordinates
- Questions to Guide Your Review
- Practice Exercises
- Additional and Advanced Exercises
-
Partial Derivatives
- 13.1 Functions of Several Variables
- 13.2 Limits and Continuity in Higher Dimensions
- 13.3 Partial Derivatives
- 13.4 The Chain Rule
- 13.5 Directional Derivatives and Gradient Vectors
- 13.6 Tangent Planes and Differentials
- 13.7 Extreme Values and Saddle Points
- 13.8 Lagrange Multiplier
- Questions to Guide Your Review
- Practice Exercises
- Additional and Advanced Exercises
-
Multiple Integrals
- 14.1 Double and Iterated Integrals over Rectangles
- 14.2 Double Integrals over General Regions
- 14.3 Area by Double Integration
- 14.4 Double Integrals in Polar Form
- 14.5 Triple Integrals in Rectangular Coordinates
- 14.6 Applications
- 14.7 Triple Integrals in Cylindrical and Spherical Coordinates
- 14.8 Substitutions in Multiple Integrals
- Questions to Guide Your Review
- Practice Exercises
- Additional and Advanced Exercises
-
Integrals and Vector Fields
- 15.1 Line Integrals of Scalar Functions
- 15.2 Vector Fields and Line Integrals: Work, Circulation, and Flux
- 15.3 Path Independence, Conservative Fields, and Potential Functions
- 15.4 Green’s Theorem in the Plane
- 15.5 Surfaces and Area
- 15.6 Surface Integrals
- 15.7 Stokes’ Theorem
- 15.8 The Divergence Theorem and a Unified Theory
- Questions to Guide Your Review
- Practice Exercises
- Additional and Advanced Exercises
-
First-Order Differential Equations (online at bit.ly/2pzYlEq)
- 16.1 Solutions, Slope Fields, and Euler’s Method
- 16.2 First-Order Linear Equations
- 16.3 Applications
- 16.4 Graphical Solutions of Autonomous Equations
- 16.5 Systems of Equations and Phase Planes
-
Second-Order Differential Equations (online at bit.ly/2IHCJyE)
- 17.1 Second-Order Linear Equations
- 17.2 Non-homogeneous Linear Equations
- 17.3 Applications
- 17.4 Euler Equations
- 17.5 Power-Series Solutions
Appendix
- A.1 Real Numbers and the Real Line
- A.2 Mathematical Induction
- A.3 Lines and Circles
- A.4 Conic Sections
- A.5 Proofs of Limit Theorems
- A.6 Commonly Occurring Limits
- A.7 Theory of the Real Numbers
- A.8 Complex Numbers
- A.9 The Distributive Law for Vector Cross Products
- A.10 The Mixed Derivative Theorem and the increment Theorem
Additional Topics (online at bit.ly/2IDDl8w)
- B.1 Relative Rates of Growth
- B.2 Probability
- B.3 Conics in Polar Coordinates
- B.4 Taylor’s Formula for Two Variables
- B.5 Partial Derivatives with Constrained Variables
Odd Answers
