Student Solutions Manual Thomas' Calculus, Single Variable

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 widely used calculus texts as well as calculus study guides. He is currently on the editorial board of several publications, including the Notices of the American Mathematical Society. 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, three 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 articles.  He serves on the editorial boards of Applied and Computational Harmonic Analysis and The Journal of Fourier Analysis and Its Applications.  Heil's current areas of research include redundant representations, operator theory, and applied harmonic analysis.  In his spare time, Heil pursues his hobby of astronomy.

Maurice D. Weir (late) of the the Naval Postgraduate School in Monterey, California was Professor Emeritus as a member of the Department of Applied Mathematics. He held a DA and MS from Carnegie-Mellon University and received his BS at Whitman College. Weir was awarded the Outstanding Civilian Service Medal, the Superior Civilian Service Award, and the Schieffelin Award for Excellence in Teaching. He co-authored eight books, including University Calculus and Thomas’ Calculus.

Table of Contents

1. Functions
   • 1.1 Functions and Their Graphs
   • 1.2 Combining Functions; Shifting and Scaling Graphs
   • 1.3 Trigonometric Functions
   • 1.4 Graphing with Software
2. 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
3. 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 Related Rates
   • 3.9 Linearization and Differentials
4. 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 Applied Optimization
   • 4.6 Newton’S Method
   • 4.7 Antiderivatives
5. 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
6. 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 and Fluid Forces
   • 6.6 Moments and Centers of Mass
7. Transcendental Functions
   • 7.1 Inverse Functions and Their Derivatives
   • 7.2 Natural Logarithms
   • 7.3 Exponential Functions
   • 7.4 Exponential Change and Separable Differential Equations
   • 7.5 Indeterminate Forms and L’Hôpital's Rule
   • 7.6 Inverse Trigonometric Functions
   • 7.7 Hyperbolic Functions
   • 7.8 Relative Rates of Growth
8. Techniques of Integration
   • 8.1 Using Basic Integration Formulas
   • 8.2 Integration by Parts
   • 8.3 Trigonometric Integrals
   • 8.4 Trigonometric Substitutions
   • 8.5 Integration of Rational Functions by Partial Fractions
   • 8.6 Integral Tables and Computer Algebra Systems
   • 8.7 Numerical Integration
   • 8.8 Improper Integrals
   • 8.9 Probability
9. First-Order Differential Equations
   • 9.1 Solutions, Slope Fields, and Euler’s Method
   • 9.2 First-Order Linear Equations
   • 9.3 Applications
   • 9.4 Graphical Solutions of Autonomous Equations
   • 9.5 Systems of Equations and Phase Planes
10. Infinite Sequences and Series
    • 10.1 Sequences
    • 10.2 Infinite Series
    • 10.3 The Integral Test
    • 10.4 Comparison Tests
    • 10.5 Absolute Convergence; The Ratio and Root Tests
    • 10.6 Alternating Series and Conditional Convergence
    • 10.7 Power Series
    • 10.8 Taylor and Maclaurin Series
    • 10.9 Convergence of Taylor Series
    • 10.10 Applications of Taylor Series
11. Parametric Equations and Polar Coordinates
    • 11.1 Parametrizations of Plane Curves
    • 11.2 Calculus with Parametric Curves
    • 11.3 Polar Coordinates
    • 11.4 Graphing Polar Coordinate Equations
    • 11.5 Areas and Lengths in Polar Coordinates
    • 11.6 Conic Sections
    • 11.7 Conics in Polar Coordinates
12. Vectors and the Geometry of Space
    • 12.1 Three-Dimensional Coordinate Systems
    • 12.2 Vectors
    • 12.3 The Dot Product
    • 12.4 The Cross Product
    • 12.5 Lines and Planes in Space
    • 12.6 Cylinders and Quadric Surfaces
13. Vector-Valued Functions and Motion in Space
    • 13.1 Curves in Space and Their Tangents
    • 13.2 Integrals of Vector Functions; Projectile Motion
    • 13.3 Arc Length in Space
    • 13.4 Curvature and Normal Vectors of a Curve
    • 13.5 Tangential and Normal Components of Acceleration
    • 13.6 Velocity and Acceleration in Polar Coordinates
14. Partial Derivatives
    • 14.1 Functions of Several Variables
    • 14.2 Limits and Continuity in Higher Dimensions
    • 14.3 Partial Derivatives
    • 14.4 The Chain Rule
    • 14.5 Directional Derivatives and Gradient Vectors
    • 14.6 Tangent Planes and Differentials
    • 14.7 Extreme Values and Saddle Points
    • 14.8 Lagrange Multipliers
    • 14.9 Taylor’s Formula for Two Variables
    • 14.10 Partial Derivatives with Constrained Variables
15. Multiple Integrals
    • 15.1 Double and Iterated Integrals over Rectangles
    • 15.2 Double Integrals over General Regions
    • 15.3 Area by Double Integration
    • 15.4 Double Integrals in Polar Form
    • 15.5 Triple Integrals in Rectangular Coordinates
    • 15.6 Applications
    • 15.7 Triple Integrals in Cylindrical and Spherical Coordinates
    • 15.8 Substitutions in Multiple Integrals
16. Integrals and Vector Fields
    • 16.1 Line Integrals of Scalar Functions
    • 16.2 Vector Fields and Line Integrals: Work, Circulation, and Flux
    • 16.3 Path Independence, Conservative Fields, and Potential Functions
    • 16.4 Green’s Theorem in the Plane
    • 16.5 Surfaces and Area
    • 16.6 Surface Integrals
    • 16.7 Stokes' Theorem
    • 16.8 The Divergence Theorem and a Unified Theory
17. Second-Order Differential Equations (Online at www.goo.gl/MgDXPY)
    • 17.1 Second-Order Linear Equations
    • 17.2 Nonhomogeneous Linear Equations
    • 17.3 Applications
    • 17.4 Euler Equations
    • 17.5 Power-Series Solutions

Appendices

1. Real Numbers and the Real Line
2. Mathematical Induction
3. Lines, Circles, and Parabolas
4. Proofs of Limit Theorems
5. Commonly Occurring Limits
6. Theory of the Real Numbers
7. Complex Numbers
8. The Distributive Law for Vector Cross Products
9. The Mixed Derivative Theorem and the Increment Theorem

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