Calculus: Early Transcendentals

Cover
Inside Cover
Title Page
Copyright
Contents
About the Authors
Preface
Acknowledgments
Supplements
Applications Index
P Preparing for Calculus
	P.1 Functions and Their Graphs
		1 Evaluate a Function
		2 Find the Domain of a Function
		3 Identify the Graph of a Function
		4 Analyze a Piecewise-Defined Function
		5 Obtain Information from or about the Graph of a Function
		6 Use Properties of Functions
		7 Find the Average Rate of Change of a Function
		P.1 Assess Your Understanding
	P.2 Library of Functions; MathematicalModeling
		1 Develop a Library of Functions
		2 Analyze a Polynomial Function and Its Graph
		3 Find the Domain and the Intercepts of a Rational Function
		4 Construct a Mathematical Model
		P.2 Assess Your Understanding
	P.3 Operations on Functions;Graphing Techniques
		1 Form the Sum, Difference, Product, and Quotientof Two Functions
		2 Form a Composite Function
		3 Transform the Graph of a Function with Verticaland Horizontal Shifts
		4 Transform the Graph of a Function with Compressionsand Stretches
		5 Transform the Graph of a Function by ReflectingIt about the x-axis or the y-axis
		P.3 Assess Your Understanding
	P.4 Inverse Functions
		1 Determine Whether a Function Is One-to-One
		2 Determine the Inverse of a Function Definedby a Set of Ordered Pairs
		3 Obtain the Graph of the Inverse Functionfrom the Graph of a One-to-One Function
		4 Find the Inverse of a One-to-One Function Definedby an Equation
		P.4 Assess Your Understanding
	P.5 Exponential and Logarithmic Functions
		1 Analyze an Exponential Function
		2 Define the Number e
		3 Analyze a Logarithmic Function
		4 Solve Exponential Equations and Logarithmic Equations
		P.5 Assess Your Understanding
	P.6 Trigonometric Functions
		1 Work with Properties of Trigonometric Functions
		2 Graph the Trigonometric Functions
		P.6 Assess Your Understanding
	P.7 Inverse Trigonometric Functions
		1 Define the Inverse Trigonometric Functions
		2 Use the Inverse Trigonometric Functions
		3 Solve Trigonometric Equations
		P.7 Assess Your Understanding
	P.8 Technology Used in Calculus
1 Limits and Continuity
	1.1 Limits of Functions Using Numericaland Graphical Techniques
		1 Discuss the Slope of a Tangent Line to a Graph
		2 Investigate a Limit Using a Table of Numbers
		3 Investigate a Limit Using a Graph
		1.1 Assess Your Understanding
	1.2 Limits of Functions Using Properties of Limits
		1 Find the Limit of a Sum, a Difference, and a Product
		2 Find the Limit of a Power and the Limit of a Root
		3 Find the Limit of a Polynomial
		4 Find the Limit of a Quotient
		5 Find the Limit of an Average Rate of Change
		6 Find the Limit of a Difference Quotient
		1.2 Assess Your Understanding
	1.3 Continuity
		1 Determine Whether a Function Is Continuous at a Number
		2 Determine Intervals on Which a Function Is Continuous
		3 Use Properties of Continuity
		4 Use the Intermediate Value Theorem
		1.3 Assess Your Understanding
	1.4 Limits and Continuity of Trigonometric,Exponential, and Logarithmic Functions
		1 Use the Squeeze Theorem to Find a Limit
		2 Find Limits Involving Trigonometric Functions
		3 Determine Where the Trigonometric FunctionsAre Continuous
		4 Determine Where an Exponential or a LogarithmicFunction Is Continuous
		1.4 Assess Your Understanding
	1.5 Infinite Limits; Limits at Infinity; Asymptotes
		1 Investigate Infinite Limits
		2 Find the Vertical Asymptotes of a Function
		3 Investigate Limits at Infinity
		4 Find the Horizontal Asymptotes of a Function
		5 Find the Asymptotes of a Rational Function Using Limits
		1.5 Assess Your Understanding
	1.6 The  -δ Definition of a Limit
		1 Use the  -δ Definition of a Limit
		1.6 Assess Your Understanding
	Chapter Review
	CHAPTER 1 PROJECT Pollution in Clear Lake
2 The Derivative
	2.1 Rates of Change and the Derivative
		1 Find Instantaneous Velocity
		2 Find an Equation of the Tangent Lineto the Graph of a Function
		3 Find the Rate of Change of a Function
		4 Find the Derivative of a Function at a Number
		2.1 Assess Your Understanding
	2.2 The Derivative as a Function
		1 Define the Derivative Function
		2 Graph the Derivative Function
		3 Identify Where a Function Has No Derivative
		2.2 Assess Your Understanding
	2.3 The Derivative of a Polynomial Function;The Derivative of y = ex
		1 Differentiate a Constant Function
		2 Differentiate a Power Function
		3 Differentiate the Sum and the Difference of Two Functions
		4 Differentiate the Exponential Function y = ex
		2.3 Assess Your Understanding
	2.4 Differentiating the Productand the Quotient of Two Functions;Higher-Order Derivatives
		1 Differentiate the Product of Two Functions
		2 Differentiate the Quotient of Two Functions
		3 Find Higher-Order Derivatives
		4 Work with Acceleration
		2.4 Assess Your Understanding
	2.5 The Derivative of the TrigonometricFunctions
		1 Differentiate Trigonometric Functions
		2.5 Assess Your Understanding
	Chapter Review
	CHAPTER 2 PROJECT The Lunar Module
3 More About Derivatives
	3.1 The Chain Rule
		1 Differentiate a Composite Function
		2 Differentiate y = ax, a > 0, a  = 1
		3 Use the Power Rule for Functions to Find a Derivative
		4 Use the Chain Rule for Multiple Composite Functions
		3.1 Assess Your Understanding
	3.2 Implicit Differentiation; Derivativesof the Inverse Trigonometric Functions
		1 Find a Derivative Using Implicit Differentiation
		2 Find Higher-Order Derivatives Using ImplicitDifferentiation
		3 Differentiate Functions with Rational Exponents
		4 Find the Derivative of an Inverse Function
		5 Differentiate the Inverse Trigonometric Functions
		3.2 Assess Your Understanding
	3.3 Derivatives of Logarithmic Functions
		1 Differentiate Logarithmic Functions
		2 Use Logarithmic Differentiation
		3 Express e as a Limit
		3.3 Assess Your Understanding
	3.4 Differentials; Linear Approximations;Newton’s Method
		1 Find the Differential of a Function and InterpretIt Geometrically
		2 Find the Linear Approximation to a Function
		3 Use Differentials in Applications
		4 Use Newton's Method to Approximatea Real Zero of a Function
		3.4 Assess Your Understanding
	3.5 Taylor Polynomials
		1 Find a Taylor Polynomial
		3.5 Assess Your Understanding
	3.6 Hyperbolic Functions
		1 Define the Hyperbolic Functions
		2 Establish Identities for Hyperbolic Functions
		3 Differentiate Hyperbolic Functions
		4 Differentiate Inverse Hyperbolic Functions
		3.6 Assess Your Understanding
	Chapter Review
	CHAPTER 3 PROJECT World Population
4 Applications of the Derivative
	4.1 Related Rates
		1 Solve Related Rate Problems
		4.1 Assess Your Understanding
	4.2 Maximum and Minimum Values;Critical Numbers
		1 Identify Absolute Maximum and Minimum Valuesand Local Extreme Values of a Function
		2 Find Critical Numbers
		3 Find Absolute Maximum and Absolute Minimum Values
		4.2 Assess Your Understanding
	4.3 The Mean Value Theorem
		1 Use Rolle's Theorem
		2 Work with the Mean Value Theorem
		3 Identify Where a Function Is Increasingand Decreasing
		4.3 Assess Your Understanding
	4.4 Local Extrema and Concavity
		1 Use the First Derivative Test to Find Local Extrema
		2 Use the First Derivative Test with Rectilinear Motion
		3 Determine the Concavity of a Function
		4 Find Inflection Points
		5 Use the Second Derivative Test to Find Local Extrema
		4.4 Assess Your Understanding
	4.5 Indeterminate Forms and L'Hôpital's Rule
		1 Identify Indeterminate Forms of the Type00and∞∞
		2 Use L'Hôpital's Rule to Find a Limit
		3 Find the Limit of an Indeterminate Formof the Type 0·∞,∞−∞, 00, 1∞, or∞0
		4.5 Assess Your Understanding
	4.6 Using Calculus to Graph Functions
		1 Graph a Function Using Calculus
		4.6 Assess Your Understanding
	4.7 Optimization
		1 Solve Optimization Problems
		4.7 Assess Your Understanding
	4.8 Antiderivatives; Differential Equations
		1 Find Antiderivatives
		2 Solve a Differential Equation
		3 Solve Applied Problems Modeled by Differential Equations
		4.8 Assess Your Understanding
	Chapter Review
	CHAPTER 4 PROJECT The U.S. Economy
5 The Integral
	5.1 Area
		1 Approximate the Area Under the Graph of a Function
		2 Find the Area Under the Graph of a Function
		5.1 Assess Your Understanding
	5.2 The Definite Integral
		1 Define a Definite Integral as the Limit of Riemann Sums
		2 Find a Definite Integral Using the Limit of Riemann Sums
		5.2 Assess Your Understanding
	5.3 The Fundamental Theorem of Calculus
		1 Use Part 1 of the Fundamental Theorem of Calculus
		2 Use Part 2 of the Fundamental Theorem of Calculus
		3 Interpret an Integral Using Part 2 of the FundamentalTheorem of Calculus
		5.3 Assess Your Understanding
	5.4 Properties of the Definite Integral
		1 Use Properties of the Definite Integral
		2 Work with the Mean Value Theorem for Integrals
		3 Find the Average Value of a Function
		5.4 Assess Your Understanding
	5.5 The Indefinite Integral; Growthand Decay Models
		1 Find Indefinite Integrals
		2 Use Properties of Indefinite Integrals
		3 Solve Differential Equations InvolvingGrowth and Decay
		5.5 Assess Your Understanding
	5.6 Method of Substitution;Newton's Law of Cooling
		1 Find an Indefinite Integral Using Substitution
		2 Find a Definite Integral Using Substitution
		3 Integrate Even and Odd Functions
		4 Solve Differential Equations: Newton's Law of Cooling
		5.6 Assess Your Understanding
	Chapter Review
	CHAPTER 5 PROJECT Managing the Klamath River
6 Applications of the Integral
	6.1 Area Between Graphs
		1 Find the Area Between the Graphs of Two Functionsby Partitioning the x-Axis
		2 Find the Area Between the Graphs of Two Functionsby Partitioning the y-Axis
		6.1 Assess Your Understanding
	6.2 Volume of a Solid of Revolution:Disks and Washers
		1 Use the Disk Method to Find the Volume of a SolidFormed by Revolving a Region About the x-Axis
		2 Use the Disk Method to Find the Volume of a Solid Formedby Revolving a Region About the y-Axis
		3 Use the Washer Method to Find the Volume of a SolidFormed by Revolving a Region About the x-Axis
		4 Use the Washer Method to Find the Volume of a SolidFormed by Revolving a Region About the y-Axis
		5 Find the Volume of a Solid Formed by Revolving a RegionAbout a Line Parallel to a Coordinate Axis
		6.2 Assess Your Understanding
	6.3 Volume of a Solid of Revolution:Cylindrical Shells
		1 Use the Shell Method to Find the Volume of a Solid Formedby Revolving a Region About the y-Axis
		2 Use the Shell Method to Find the Volume of a Solid Formedby Revolving a Region About the x-Axis
		3 Use the Shell Method to Find the Volume of a SolidFormed by Revolving a Region About a Line Parallelto a Coordinate Axis
		6.3 Assess Your Understanding
	6.4 Volume of a Solid: Slicing Method
		1 Use the Slicing Method to Find the Volume of a Solid
		6.4 Assess Your Understanding
	6.5 Arc Length
		1 Find the Arc Length of the Graph of a Function y = f (x)
		2 Find the Arc Length of the Graph of a Function Usinga Partition of the y-Axis
		6.5 Assess Your Understanding
	6.6 Work
		1 Find the Work Done by a Variable Force
		2 Find the Work Done by a Spring Force
		3 Find the Work Done to Pump a Liquid
		Application to Gravitational Force
		6.6 Assess Your Understanding
	6.7 Hydrostatic Pressure and Force
		1 Find Hydrostatic Pressure and Force
		6.7 Assess Your Understanding
	6.8 Center of Mass; Centroid;the Pappus Theorem
		1 Find the Center of Mass of a Finite System of Objects
		2 Find the Centroid of a Homogeneous Lamina
		3 Find the Volume of a Solid of Revolution Usingthe Pappus Theorem
		6.8 Assess Your Understanding
	Chapter Review
	CHAPTER 6 PROJECT Determining the Amount of Concrete Needed for a Cooling Tower
7 Techniques of Integration
	7.1 Integration by Parts
		1 Integrate by Parts
		2 Derive a Formula Using Integration by Parts
		7.1 Assess Your Understanding
	7.2 Integrals Containing Trigonometric Functions
		1 Find Integrals of the form sinn x dx or cosn x dx, n ≥ 2an Integer
		2 Find Integrals of the Form sinm x cosn x dx
		3 Find Integrals of the Form tanm x secn x dxor cotm x cscn x dx
		4 Find Integrals of the Form sin(ax) sin(bx) dx, sin(ax) cos(bx) dx, or cos(ax) cos(bx) dx
		7.2 Assess Your Understanding
	7.3 Integration Using TrigonometricSubstitution: Integrands Containing  a2 − x2, x2 +a2, or x2 −a2, a > 0
		1 Find Integrals Containing a2 − x2
		2 Find Integrals Containing x2 +a2
		3 Find Integrals Containing x2 −a2
		4 Use Trigonometric Substitution to Find Definite Integrals
		7.3 Assess Your Understanding
	7.4 Substitution: IntegrandsContaining ax2 +bx +c
		1 Find an Integral That Contains a Quadratic Expression
		7.4 Assess Your Understanding
	7.5 Integration of Rational Functions UsingPartial Fractions
		1 Integrate a Rational Function Whose DenominatorContains Only Distinct Linear Factors
		2 Integrate a Rational Function Whose DenominatorContains a Repeated Linear Factor
		3 Integrate a Rational Function Whose DenominatorContains a Distinct Irreducible Quadratic Factor
		4 Integrate a Rational Function Whose DenominatorContains a Repeated Irreducible Quadratic Factor
		7.5 Assess Your Understanding
	7.6 Integration Using Numerical Techniques
		1 Approximate an Integral Using the Trapezoidal Rule
		2 Approximate an Integral Using Simpson's Rule
		7.6 Assess Your Understanding
	7.7 Integration Using Tables and ComputerAlgebra Systems
		1 Use a Table of Integrals
		2 Use a Computer Algebra System
		7.7 Assess Your Understanding
	7.8 Improper Integrals
		1 Find Integrals with an Infinite Limit of Integration
		2 Interpret an Improper Integral Geometrically
		3 Integrate Functions over [a, b] That Are Not Definedat an Endpoint
		4 Use the Comparison Test for Improper Integrals
		7.8 Assess Your Understanding
	Chapter Review
	CHAPTER 7 PROJECT The Birds of Rügen Island
8 Infinite Series
	8.1 Sequences
		1 Write the Terms of a Sequence
		2 Find the nth Term of a Sequence
		3 Use Properties of Convergent Sequences
		4 Use a Related Function or the Squeeze Theoremto Show a Sequence Converges
		5 Determine Whether a Sequence Converges or Diverges
		8.1 Assess Your Understanding
	8.2 Infinite Series
		1 Determine Whether a Series Has a Sum
		2 Analyze a Geometric Series
		3 Analyze the Harmonic Series
		Using a Geometric Series in a Biology Application∗
		8.2 Assess Your Understanding
	8.3 Properties of Series; the Integral Test
		1 Use the Test for Divergence
		2 Work with Properties of Series
		3 Use the Integral Test
		4 Analyze a p-Series
		8.3 Assess Your Understanding
	8.4 Comparison Tests
		1 Use Comparison Tests for Convergence and Divergence
		2 Use the Limit Comparison Test
		8.4 Assess Your Understanding
	8.5 Alternating Series; Absolute Convergence
		1 Determine Whether an Alternating Series Converges
		2 Approximate the Sum of a Convergent Alternating Series
		3 Determine Whether a Series Converges
		8.5 Assess Your Understanding
	8.6 Ratio Test; Root Test
		1 Use the Ratio Test
		2 Use the Root Test
		8.6 Assess Your Understanding
	8.7 Summary of Tests
		1 Choose an Appropriate Test to Determine Whethera Series Converges
		8.7 Assess Your Understanding
	8.8 Power Series
		1 Determine Whether a Power Series Converges
		2 Find the Interval of Convergence of a Power Series
		3 Define a Function Using a Power Series
		4 Use Properties of Power Series
		8.8 Assess Your Understanding
	8.9 Taylor Series; Maclaurin Series
		1 Express a Function as a Taylor Series or a Maclaurin Series
		2 Determine the Convergence of a Taylor/Maclaurin Series
		3 Find Taylor/Maclaurin Expansions
		4 Work with a Binomial Series
		8.9 Assess Your Understanding
	8.10 Approximations Using Taylor/MaclaurinExpansions
		1 Approximate Functions and Their Graphs
		2 Approximate the Number e; Approximate Logarithms
		3 Approximate Definite Integrals
		8.10 Assess Your Understanding
	Chapter Review
	CHAPTER 8 PROJECT How Calculators Calculate
9 Parametric Equations;Polar Equations
	9.1 Parametric Equations
		1 Graph Parametric Equations
		2 Find a Rectangular Equation for a CurveRepresented Parametrically
		3 Use Time as the Parameter in Parametric Equations
		4 Convert a Rectangular Equation to Parametric Equations
		9.1 Assess Your Understanding
	9.2 Tangent Lines; Arc Length
		1 Find an Equation of the Tangent Line at a Pointon a Plane Curve
		2 Find the Arc Length of a Plane Curve
		9.2 Assess Your Understanding
	9.3 Surface Area of a Solid of Revolution
		1 Find the Surface Area of a Solid of Revolution Obtainedfrom Parametric Equations
		2 Find the Surface Area of a Solid of Revolution Obtainedfrom a Rectangular Equation
		9.3 Assess Your Understanding
	9.4 Polar Coordinates
		1 Plot Points Using Polar Coordinates
		2 Convert Between Rectangular Coordinatesand Polar Coordinates
		3 Identify and Graph Polar Equations
		9.4 Assess Your Understanding
	9.5 Polar Equations;Parametric Equations of Polar Equations;Arc Length of Polar Equations
		1 Graph a Polar Equation; Find Parametric Equations
		2 Find the Arc Length of a Curve Representedby a Polar Equation
		9.5 Assess Your Understanding
	9.6 Area in Polar Coordinates
		1 Find the Area of a Region Enclosed by the Graphof a Polar Equation
		2 Find the Area of a Region Enclosed by the Graphsof Two Polar Equations
		3 Find the Surface Area of a Solid of Revolution Obtainedfrom a Polar Equation
		9.6 Assess Your Understanding
	9.7 The Polar Equation of a Conic
		1 Express a Conic as a Polar Equation
		9.7 Assess Your Understanding
	Chapter Review
	CHAPTER 9 PROJECT Polar Graphs and Microphones
10 Vectors; Lines, Planes, andQuadric Surfaces in Space
	10.1 Rectangular Coordinates in Space
		1 Locate Points in Space
		2 Find the Distance Between Two Points in Space
		3 Find the Equation of a Sphere
		10.1 Assess Your Understanding
	10.2 Introduction to Vectors
		1 Represent Vectors Geometrically
		2 Use Properties of Vectors
		10.2 Assess Your Understanding
	10.3 Vectors in the Plane and in Space
		1 Represent a Vector Algebraically
		2 Add, Subtract, and Find Scalar Multiples of Vectors
		3 Find the Magnitude of a Vector
		4 Find a Unit Vector
		5 Find a Vector in the Plane from Its Directionand Magnitude
		10.3 Assess Your Understanding
	10.4 The Dot Product
		1 Find the Dot Product of Two Vectors
		2 Find the Angle Between Two Vectors
		3 Determine Whether Two Vectors Are Orthogonal
		4 Find a Vector in Space from Its Magnitude and Direction
		5 Find the Projection of a Vector
		6 Compute Work
		10.4 Assess Your Understanding
	10.5 The Cross Product
		1 Find the Cross Product of Two Vectors
		2 Prove Algebraic Properties of the Cross Product
		3 Apply Geometric Properties of the Cross Product
		10.5 Assess Your Understanding
	10.6 Equations of Lines and Planes in Space
		1 Find a Vector Equation of a Line in Space
		2 Find Parametric Equations of a Line in Space
		3 Find Symmetric Equations of a Line in Space
		4 Determine Whether Two Distinct Lines Are Skew,Parallel, or Intersecting
		5 Find an Equation of a Plane
		6 Determine Whether Two Distinct PlanesAre Parallel or Intersecting
		7 Find the Distance from a Point to a Plane
		10.6 Assess Your Understanding
	10.7 Quadric Surfaces
		1 Identify Quadric Surfaces Based on an Ellipse
		2 Identify Quadric Surfaces Based on a Hyperbola
		3 Identify Cylinders
		4 Graph Quadric Surfaces
		10.7 Assess Your Understanding
	Chapter Review
	CHAPTER 10 PROJECT The Hall Effect
11 Vector Functions
	11.1 Vector Functions and Their Derivatives
		1 Find the Domain of a Vector Function
		2 Graph a Vector Function
		3 Find the Limit and Determine the Continuityof a Vector Function
		4 Find the Derivative of a Vector Function
		5 Find the Derivative of a Vector Function UsingDerivative Formulas
		11.1 Assess Your Understanding
	11.2 Unit Tangent and Principal Unit NormalVectors; Arc Length
		1 Interpret the Derivative of a Vector Function Geometrically
		2 Find the Unit Tangent Vector and the Principal UnitNormal Vector of a Smooth Curve
		3 Find the Arc Length of a Curve Traced Outby a Vector Function
		11.2 Assess Your Understanding
	11.3 Arc Length as Parameter; Curvature
		1 Determine Whether the Parameter Used in a VectorFunction Is Arc Length
		2 Find the Curvature of a Curve
		3 Find the Curvature of a Space Curve
		4 Find the Curvature of a Plane Curve Given by y = f (x)
		5 Find an Osculating Circle
		11.3 Assess Your Understanding
	11.4 Motion Along a Curve
		1 Find the Velocity, Acceleration, and Speedof a Moving Particle
		2 Express the Acceleration Vector Using Tangentialand Normal Components
		11.4 Assess Your Understanding
	11.5 Integrals of Vector Functions; ProjectileMotion
		1 Integrate Vector Functions
		2 Solve Projectile Motion Problems
		11.5 Assess Your Understanding
	11.6 Application: Kepler's Laws of PlanetaryMotion
		1 Discuss Kepler's Laws of Planetary Motion
		11.6 Assess Your Understanding
	Chapter Review
	CHAPTER 11 PROJECT How to Design a Safe Road
12 Functions of Several Variables
	12.1 Functions of Two or More Variablesand Their Graphs
		1 Work with Functions of Two or Three Variables
		2 Graph Functions of Two Variables
		3 Graph Level Curves
		4 Describe Level Surfaces
		12.1 Assess Your Understanding
	12.2 Limits and Continuity
		1 Define the Limit of a Function of Several Variables
		2 Find a Limit Using Properties of Limits
		3 Examine When Limits Exist
		4 Determine Whether a Function Is Continuous
		12.2 Assess Your Understanding
	12.3 Partial Derivatives
		1 Find the Partial Derivatives of a Functionof Two Variables
		2 Interpret Partial Derivatives as the Slope of a Tangent Line
		3 Interpret Partial Derivatives as a Rate of Change
		4 Find Second-Order Partial Derivatives
		5 Find the Partial Derivatives of a Function of n Variables
		12.3 Assess Your Understanding
	12.4 Differentiability and the Differential
		1 Find the Change in z = f (x, y)
		2 Show That a Function of Two Variables Is Differentiable
		3 Use the Differential dz to Approximate a Change in z
		4 Find the Differential of a Function of Threeor More Variables
		12.4 Assess Your Understanding
	12.5 Chain Rules
		1 Differentiate Functions of Two or More VariablesWhere Each Variable Is a Function of a Single Variable
		2 Differentiate Functions of Two or More Variables WhereEach Variable is a Function of Two or More Variables
		3 Differentiate an Implicitly-Defined Functionof Two or More Variables
		4 Use a Chain Rule in a Proof
		12.5 Assess Your Understanding
	Chapter Review
	CHAPTER 12 PROJECT Searching for Exoplanets
13 Directional Derivatives,Gradients, and Extrema
	13.1 Directional Derivatives; Gradients
		1 Find the Directional Derivative of a Functionof Two Variables
		2 Find the Gradient of a Function of Two Variables
		3 Use Properties of the Gradient
		4 Find the Directional Derivative and Gradient of a Functionof Three Variables
		13.1 Assess Your Understanding
	13.2 Tangent Planes
		1 Find a Tangent Plane to a Surface
		2 Find a Normal Line to a Tangent Plane
		13.2 Assess Your Understanding
	13.3 Extrema of Functions of Two Variables
		1 Find Critical Points
		2 Use the Second Partial Derivative Test
		3 Find the Absolute Extrema of a Function of Two Variables
		4 Solve Optimization Problems
		13.3 Assess Your Understanding
	13.4 Lagrange Multipliers
		1 Use Lagrange Multipliers for an Optimization Problemwith One Constraint
		2 Use Lagrange Multipliers for an Optimization Problemwith Two Constraints
		13.4 Assess Your Understanding
	Chapter Review
	CHAPTER 13 PROJECT Measuring Ice Thickness on Crystal Lake
14 Multiple Integrals
	14.1 The Double Integral overa Rectangular Region
		1 Find Riemann Sums of z = f (x, y) over a ClosedRectangular Region
		2 Find the Value of a Double Integral Definedon a Closed Rectangular Region
		3 Find the Volume Under a Surface and overa Rectangular Region
		14.1 Assess Your Understanding
	14.2 The Double Integral overNonrectangular Regions
		1 Use Fubini's Theorem for an x-Simple Region
		2 Use Fubini's Theorem for a y-Simple Region
		3 Work with Properties of Double Integrals
		4 Use Double Integrals to Find Area and Volume
		14.2 Assess Your Understanding
	14.3 Double Integrals Using Polar Coordinates
		1 Find a Double Integral Using Polar Coordinates
		2 Find Area and Volume Using Polar Coordinates
		14.3 Assess Your Understanding
	14.4 Center of Mass; Moment of Inertia
		1 Find the Mass and the Center of Mass of a Lamina
		2 Find Moments of Inertia
		14.4 Assess Your Understanding
	14.5 Surface Area
		1 Find the Surface Area That Lies Above a Region R
		14.5 Assess Your Understanding
	14.6 The Triple Integral
		1 Find a Triple Integral Defined over a Closed Box
		2 Find a Triple Integral Defined over a More General Solid
		3 Find the Volume of a Solid
		4 Find the Mass, Center of Mass, and Momentsof Inertia of a Solid
		5 Find a Triple Integral Defined over xz-Simpleand yz-Simple Solids
		14.6 Assess Your Understanding
	14.7 Triple Integrals Using Cylindrical Coordinates
		1 Convert Rectangular Coordinates to Cylindrical Coordinates
		2 Find a Triple Integral Using Cylindrical Coordinates
		14.7 Assess Your Understanding
	14.8 Triple Integrals Using Spherical Coordinates
		1 Convert Rectangular Coordinates to Spherical Coordinates
		2 Find a Triple Integral Using Spherical Coordinates
		14.8 Assess Your Understanding
	14.9 Change of Variables Using Jacobians
		1 Find a Jacobian in Two Variables
		2 Change the Variables of a Double Integral Using a Jacobian
		3 Change the Variables of a Triple Integral Using a Jacobian
		14.9 Assess Your Understanding
	Chapter Review
	CHAPTER 14 PROJECT The Mass of Stars
15 Vector Calculus
	15.1 Vector Fields
		1 Describe a Vector Field
		15.1 Assess Your Understanding
	15.2 Line Integrals
		1 Define a Line Integral in the Plane
		2 Find the Value of a Line Integral Along a Smooth Curve
		3 Find Line Integrals of the Form C f (x, y) dxand C f (x, y) dy
		4 Find Line Integrals Along a Piecewise-Smooth Curve
		5 Find the Value of a Line Integral in Space
		15.2 Assess Your Understanding
	15.3 Fundamental Theorem of Line Integrals
		1 Identify a Conservative Vector Fieldand Its Potential Function
		2 Use the Fundamental Theorem of Line Integrals
		3 Reconstruct a Function from Its Gradient: finding thePotential Function for a Conservative Vector Field
		4 Determine Whether a Vector Field Is Conservative
		15.3 Assess Your Understanding
	15.4 An Application of Line Integrals: Work
		1 Compute Work
		15.4 Assess Your Understanding
	15.5 Green's Theorem
		1 Use Green's Theorem to Find a Line Integral
		2 Use Green's Theorem to Find Area
		3 Use Green's Theorem with Multiply-Connected Regions
		15.5 Assess Your Understanding
	15.6 Parametric Surfaces
		1 Describe Surfaces Defined Parametrically
		2 Find a Parametric Representation of a Surface
		3 Find Equations for a Tangent Plane and a Normal Line
		4 Find the Surface Area of a Parametrized Surface
		15.6 Assess Your Understanding
	15.7 Surface and Flux Integrals
		1 Find a Surface Integral Using a Double Integral
		2 Determine the Orientation of a Surface
		3 Find the Flux of a Vector Field Across a Surface
		Application: Electric Flux
		15.7 Assess Your Understanding
	15.8 The Divergence Theorem
		1 Find the Divergence of a Vector Field
		2 Use the Divergence Theorem
		3 Interpret the Divergence of F
		15.8 Assess Your Understanding
	15.9 Stokes' Theorem
		1 Find the Curl of F
		2 Verify Stokes' Theorem
		3 Use Stokes' Theorem to Find an Integral
		4 Apply Stokes' Theorem to Conservative Vector Fields
		5 Interpret the Curl of F
		15.9 Assess Your Understanding
	Chapter Review
	CHAPTER 15 PROJECT Modeling a Tornado
16 Differential Equations
	16.1 Classification of Ordinary DifferentialEquations
		1 Classify Ordinary Differential Equations
		2 Verify the Solution of an Ordinary Differential Equation
		16.1 Assess Your Understanding
	16.2 Separation of Variables in First-OrderDifferential Equations
		1 Solve a Separable First-Order Differential Equation
		2 Identify a Homogeneous Function of Degree k
		3 Use a Change of Variables to Solve a HomogeneousFirst-Order Differential Equation
		4 Solve Applied Problems
		16.2 Assess Your Understanding
	16.3 Exact Differential Equations
		1 Identify and Solve an Exact Differential Equation
		16.3 Assess Your Understanding
	16.4 First-Order Linear Differential Equations;Bernoulli Differential Equations
		1 Solve a First-Order Linear Differential Equation
		2 Solve Applied Problems Involving First-Order LinearDifferential Equations
		3 Find the General Solution of a Bernoulli Equation
		4 Solve Applied Problems
		16.4 Assess Your Understanding
	16.5 Power Series Methods
		1 Use Power Series to Solve a Linear Differential Equation
		16.5 Assess Your Understanding
	Chapter Review
	CHAPTER 16 PROJECT The Melting Arctic Ice Cap
Appendix A Precalculus Used in Calculus
	A.1 Algebra Used in Calculus
		1 Factor and Simplify Algebraic Expressions
		2 Complete the Square
		3 Solve Equations
		4 Solve Inequalities
		5 Work with Exponents
		6 Work with Logarithms
	A.2 Geometry Used in Calculus
		1 Use Properties of Triangles and the Pythagorean Theorem
		2 Work with Congruent Triangles and Similar Triangles
		3 Use Geometry Formulas
	A.3 Analytic Geometry Used in Calculus
		1 Use the Distance Formula
		2 Graph Equations, Find Intercepts, and Test for Symmetry
		3 Work with Equations of a Line
		4 Work with the Equation of a Circle
		5 Graph Parabolas, Ellipses, and Hyperbolas
	A.4 Trigonometry Used in Calculus
		1 Work with Angles, Arc Length of a Circle,and Circular Motion
		2 Define and Evaluate Trigonometric Functions
		3 Determine the Domain and the Rangeof the Trigonometric Functions
		4 Use Basic Trigonometry Identities
		5 Use Sum and Difference, Double-Angle and Half-Angle,and Sum-to-Product and Product-to-Sum Formulas
		6 Solve Triangles Using the Law of Sinesand the Law of Cosines
	A.5 Sequences; Summation Notation;the Binomial Theorem
		1 Write the First Several Terms of a Sequence
		2 Write the Terms of a Recursively Defined Sequence
		3 Use Summation Notation
		4 Find the Sum of the First n Terms of a Sequence
		5 Use the Binomial Theorem
Appendix B Theorems and Proofs
	B.1 Limit Theorems and Proofs
	B.2 Theorems and Proofs InvolvingInverse Functions
	B.3 Derivative Theorems and Proofs
	B.4 Integral Theorems and Proofs
	B.5 A Bounded Monotonic SequenceConverges
	B.6 Taylor's Formula with Remainder
Answers
	Chapter P
	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
	Chapter 15
	Chapter 16
Photo Credits
Index