Calculus: Early Transcendentals, 2nd Edition
- Length: 1328 pages
- Edition: 2
- Language: English
- Publisher: WH Freeman
- Publication Date: 2019-01-22
- ISBN-10: 1319248470
- ISBN-13: 9781319248475
- Sales Rank: #0 (See Top 100 Books)
Using clear chapter objectives and review questions, as well as accessible applications, this textbook guides a student throughout a calculus course, building their confidence and engaging them with the subject. Feel like your students could do with extra support when learning calculus Then this is the textbook for you. The textbook contains a wide array of features including challenge problems and applied exercises drawn from the worlds of physical and life sciences, economics, and other disciplines. Students struggling with algebra will benefit from marginal annotations that help strengthen understanding, the many references to review material, and extensive practice exercises. In short, Sullivan and Mirandas Calculus has been written with todays instructors and students in mind.The 2nd edition of Sullivan and Miranda’s Calculus is a refreshing addition to the field of calculus textbooks. The 1st edition was awarded TAA’s Most Promising New Textbook Award in 2016.
Applications Index P Preparing for Calculus P.1 Functions and Their Graphs 1 Evaluate a Function 2 Find the Difference Quotient of a Function 3 Find the Domain of a Function 4 Identify the Graph of a Function 5 Analyze a Piecewise-Defined Function 6 Obtain Information From or About the Graph of a Function 7 Use Properties of Functions 8 Find the Average Rate of Change of a Function Assess Your Understanding P.2 Library of Functions; Mathematical Modeling 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 Assess Your Understanding P.3 Operations on Functions; Graphing Techniques 1 Form the Sum, Difference, Product, and Quotient of Two Functions 2 Form a Composite Function 3 Transform the Graph of a Function with Vertical and Horizontal Shifts 4 Transform the Graph of a Function with Compressions and Stretches 5 Transform the Graph of a Function by Reflecting It About the x-axis or the y-axis Assess Your Understanding P.4 Inverse Functions 1 Determine Whether a Function Is One-to-One 2 Determine the Inverse of a Function Defined by a Set of Ordered Pairs 3 Obtain the Graph of the Inverse Function from the Graph of a One-to-One Function 4 Find the Inverse of a One-to-One Function Defined by an Equation 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 Assess Your Understanding P.6 Trigonometric Functions 1 Work with Properties of Trigonometric Functions 2 Graph the Trigonometric Functions Assess Your Understanding P.7 Inverse Trigonometric Functions 1 Define the Inverse Trigonometric Functions 2 Use the Inverse Trigonometric Functions 3 Solve Trigonometric Equations Assess Your Understanding P.8 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 Assess Your Understanding 1 Limits and Continuity 1.1 Limits of Functions Using Numerical and Graphical Techniques The Slope of the Tangent Line to a Graph 1 Discuss the Idea of a Limit 2 Investigate a Limit Using a Table 3 Investigate a Limit Using a Graph 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 Summary 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 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 Functions Are Continuous 4 Determine Where an Exponential or a Logarithmic Function Is Continuous Summary Basic Limits Assess Your Understanding 1.5 Infinite Limits; Limits at Infinity; Asymptotes 1 Find Infinite Limits 2 Find the Vertical Asymptotes of a Graph 3 Find Limits at Infinity 4 Find the Horizontal Asymptotes of a Graph 5 Find the Asymptotes of the Graph of a Rational Function Assess Your Understanding 1.6 The ε-δ Definition of a Limit 1 Use the ε-δ definition of a limit 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 Equations for the Tangent Line and the Normal Line to the Graph of a Function 2 Find the Rate of Change of a Function 3 Find Average Velocity and Instantaneous Velocity 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 Product and 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 Trigonometric Functions 1 Differentiate Trigonometric Functions 2.4 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 1 Find a Derivative Using Implicit Differentiation 2 Find Higher-Order Derivatives Using Implicit Differentiation 3 Differentiate Functions with Rational Exponents 3.2 Assess Your Understanding 3.3 Derivatives of the Inverse Trigonometric Functions 1 Find the Derivative of an Inverse Function 2 Find the Derivative of the Inverse Trigonometric Functions 3.3 Assess Your Understanding 3.4 Derivatives of Logarithmic Functions 1 Differentiate logarithmic functions 2 Use logarithmic differentiation 3 Express e as a Limit 3.4 Assess Your Understanding 3.5 Differentials; Linear Approximations; Newton’s Method 1 Find the Differential of a Function and Interpret It Geometrically 2 Find the Linear Approximation to a Function 3 Use Differentials in Applications 4 Use Newton’s Method to Approximate a Real Zero of a Function 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 Assess Your Understanding 4.2 Maximum and Minimum Values; Critical Numbers 1 Identify Absolute Maximum and Minimum Values and Local Extreme Values of a Function 2 Find Critical Numbers 3 Find Absolute Maximum and Absolute Minimum Values 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 Increasing and Decreasing 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 Assess Your Understanding 4.5 Indeterminate Forms and L’Hôpital’s Rule 1 Identify Indeterminate Forms of the Type 0/0 and ∞/∞ 2 Use L’Hôpital’s Rule to Find a Limit 3 Find the Limit of an Indeterminate Form of the Type 0.∞, ∞–∞, 00, 1∞, or ∞0 Assess Your Understanding 4.6 Using Calculus to Graph Functions 1 Graph a Function Using Calculus Assess Your Understanding 4.7 Optimization 1 Solve Optimization Problems Assess Your Understanding 4.8 Antiderivatives; Differential Equations 1 Find Antiderivatives 2 Solve a Differential Equation 3 Solve Applied Problems Modeled by Differential Equations Assess Your Understanding Chapter Review Chapter Project 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 Assess Your Understanding 5.2 The Definite Integral 1 Form Riemann Sums 2 Define a Definite Integral as the Limit of Riemann Sums 3 Approximate a Definite Integral Using Riemann Sums 4 Know Conditions That Guarantee a Definite Integral Exists 5 Find a Definite Integral Using the Limit of Riemann Sums 6 Form Riemann Sums from a Table 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 the Integral of a Rate of Change 4 Interpret the Integral as an Accumulation Function 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 4 Interpret Integrals Involving Rectilinear Motion Assess Your Understanding 5.5 The Indefinite Integral; Method of Substitution 1 Find Indefinite Integrals 2 Use Properties of Indefinite Integrals 3 Find an Indefinite Integral Using Substitution 4 Find a Definite Integral Using Substitution 5 Integrate Even and Odd Functions Assess Your Understanding 5.6 Separable First-Order Differential Equations; Uninhibited and Inhibited Growth and Decay Models 1 Solve a Separable First-Order Differential Equation 2 Solve Differential Equations Involving Uninhibited Growth and Decay 3 Solve Differential Equations Involving Inhibited Growth and Decay 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 Functions by Partitioning the x-Axis 2 Find the Area Between the Graphs of Two Functions by Partitioning the y-Axis 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 Solid Formed by Revolving a Region About the x-Axis 2 Use the Disk Method to Find the Volume of a Solid Formed by Revolving a Region About the y-Axis 3 Use the Washer Method to Find the Volume of a Solid Formed by Revolving a Region About the x-Axis 4 Use the Washer Method to Find the Volume of a Solid Formed by Revolving a Region About the y-Axis 5 Find the Volume of a Solid Formed by Revolving a Region About a Line Parallel to a Coordinate Axis 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 Formed by Revolving a Region About the y-Axis 2 Use the Shell Method to Find the Volume of a Solid Formed by Revolving a Region About the x-Axis 3 Use the Shell Method to Find the Volume of a Solid Formed by Revolving a Region About a Line Parallel to a Coordinate Axis Assess Your Understanding 6.4 Volume of a Solid: Slicing 1 Use Slicing to Find the Volume of a Solid Assess Your Understanding 6.5 Arc Length; Surface Area of a Solid of Revolution 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 Using a Partition of the y-Axis 3 Find the Surface Area of a Solid of Revolution 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: Gravitational Force Assess Your Understanding 6.7 Hydrostatic Pressure and Force 1 Find Hydrostatic Pressure and Force 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 Using the Pappus Theorem Assess Your Understanding Chapter Review Chapter 6 Project 7 Techniques of Integration 7.1 Integration by Parts 1 Integrate by Parts 2 Find a Definite Integral Using Integration by Parts 3 Derive a General Formula Using Integration by Parts Assess Your Understanding 7.2 Integrals Containing Trigonometric Functions 1 Find Integrals of the Form ∫ sinnxdx or ∫ cosnxdx, n≥2 an Integer 2 Find Integrals of the Form ∫sinmx cosnxdx 3 Find Integrals of the Form ∫tanmx secnxdx or ∫cotmx cscnxdx 4 Find Integrals of the Form ∫sin(ax) sin(bx)dx, ∫sin(ax)cos(bx)dx, or ∫cos(ax)cos(bx)dx Assess Your Understanding 7.3 Integration Using Trigonometric Substitution: Integrands Containing a2-x2, x2+a2, or x2-a2,a>0 1 Integrate a Function Containing a2-x2 2 Integrate a Function Containing x2+a2 3 Integrate a Function Containing x2-a2 4 Use Trigonometric Substitution to Find Definite Integrals Assess Your Understanding 7.4 Integrands Containing ax+bx+c, a≠0 1 Integrate a Function That Contains a Quadratic Expression Assess Your Understanding 7.5 Integration of Rational Functions Using Partial Fractions; the Logistic Model 1 Integrate a Proper Rational Function Whose Denominator Contains Only Distinct Linear Factors 2 Integrate a Proper Rational Function Whose Denominator Contains a Repeated Linear Factor 3 Integrate a Proper Rational Function Whose Denominator Contains a Distinct Irreducible Quadratic Factor 4 Integrate a Proper Rational Function Whose Denominator Contains a Repeated Irreducible Quadratic Factor 5 Work with the Logistic Model Assess Your Understanding 7.6 Approximating Integrals: The Trapezoidal Rule, the Midpoint Rule, Simpson’s Rule 1 Approximate an Integral Using the Trapezoidal Rule 2 Approximate an Integral Using the Midpoint Rule 3 Approximate an Integral Using Simpson’s Rule Assess Your Understanding 7.7 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 Defined at an Endpoint 4 Use the Comparison Test for Improper Integrals Assess Your Understanding 7.8 Integration Using Tables and Computer Algebra Systems 1 Use a Table of Integrals 2 Use a Computer Algebra System Assess Your Understanding 7.9 Mixed Practice 1 Recognize the Form of an Integrand and Find Its Integral 2 Assess Your Understanding Chapter Review Chapter 7 Project The Birds of Rügen Island 8 Infinite Series 8.1 Sequences 1 Write Several 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 Theorem to Show a Sequence Converges 5 Determine Whether a Sequence Converges or Diverges 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 Application: Using a Geometric Series in Biology Assess Your Understanding 8.3 Properties of Series; Series with Positive Terms; 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 5 Approximate the Sum of a Convergent Series Assess Your Understanding 8.4 Comparison Tests 1 Use Comparison Tests for Convergence and Divergence 2 Use the Limit Comparison Test Assess Your Understanding 8.5 Alternating Series; Absolute Convergence 1 Determine Whether an Alternating Series Converges or Diverges 2 Approximate the Sum of a Convergent Alternating Series 3 Use the Absolute Convergence Test Assess Your Understanding 8.6 Ratio Test; Root Test 1 Use the Ratio Test 2 Use the Root Test Assess Your Understanding 8.7 Summary of Tests 1 Choose an Appropriate Test to Determine Whether a Series Converges 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 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 Assess Your Understanding 8.10 Approximations Using Taylor/Maclaurin Expansions 1 Approximate a Function and Its Graph Using a Taylor Polynomial 2 Approximate the Number e; Approximate Logarithms 3 Approximate Definite Integrals 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 Curve Represented Parametrically 3 Use Time as the Parameter in Parametric Equations 4 Convert a Rectangular Equation to Parametric Equations Assess Your Understanding 9.2 Tangent Lines 1 Find an Equation of the Tangent Line at a Point on a Plane Curve Assess Your Understanding 9.3 Arc Length; Surface Area of a Solid of Revolution 1 Find the Arc Length of a Plane Curve 2 Find the Surface Area of a Solid of Revolution Obtained from Parametric Equations Assess Your Understanding 9.4 Polar Coordinates 1 Plot Points Using Polar Coordinates 2 Convert Between Rectangular Coordinates and Polar Coordinates 3 Identify and Graph Polar Equations 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 Represented by a Polar Equation Assess Your Understanding 9.6 Area in Polar Coordinates 1 Find the Area of a Region Enclosed by the Graph of a Polar Equation 2 Find the Area of a Region Enclosed by the Graphs of Two Polar Equations 3 Find the Surface Area of a Solid of Revolution Obtained from the Graph of a Polar Equation Assess Your Understanding 9.7 The Polar Equation of a Conic 1 Express a Conic as a Polar Equation Assess Your Understanding Chapter Review Chapter Project 10 Vectors; Lines, Planes, and Quadric 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 Assess Your Understanding 10.2 Introduction to Vectors 1 Represent vectors geometrically 2 Use properties of vectors 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 direction and magnitude 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 Express a vector in space using its magnitude and direction 5 Find the Projection of a Vector 6 Compute Work 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 Use geometric properties of the cross product 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 planes are parallel or intersecting 7 Find the distance from a point to a plane and from a point to a line 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 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 continuity of a vector function 4 Find the derivative of a vector function 5 Find the derivative of a vector function using derivative rules Assess Your Understanding 11.2 Unit Tangent and Principal Unit Normal Vectors; Arc Length 1 Interpret the derivative of a vector function geometrically 2 Find the unit tangent vector and the principal unit normal vector of a smooth curve 3 Find the arc length of a curve traced out by a vector function Assess Your Understanding 11.3 Arc Length as Parameter; Curvature 1 Determine whether the parameter used in a vector function 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 Assess Your Understanding 11.4 Motion Along a Curve 1 Find the velocity, acceleration, and speed of a moving particle 2 Express the acceleration vector using tangential and normal components Assess Your Understanding 11.5 Integrals of Vector Functions; Projectile Motion 1 Integrate vector functions 2 Solve projectile motion problems Assess Your Understanding 11.6 Application: Kepler’s Laws of Planetary Motion 1 Discuss Kepler’s Laws of Planetary Motion 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 Variables and 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 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 where a function is continuous Assess Your Understanding 12.3 Partial Derivatives 1 Find the partial derivatives of a function of 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 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 as an approximating tool 4 Find the differential of a function of three or more variables Assess Your Understanding 12.5 Chain Rules 1 Differentiate functions of several variables where each variable is a function of a single variable 2 Differentiate functions of several variables where each variable is a function of two or more variables 3 Differentiate an implicitly defined function of several variables 4 Use a Chain Rule in a proof 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 Function of Two Variables 2 Find the Gradient of a Function of Two Variables 3 Find the Gradient of a Function of Two Variables 4 Find the Directional Derivative and Gradient of a Function of Three Variables Assess Your Understanding 13.2 Tangent Planes 1 Find an Equation of a Tangent Plane to a Surface 2 Find an Equation of a Normal Line to a Tangent Plane 3 Find an Equation of a Tangent Plane to a Surface Defined Explicitly 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 Assess Your Understanding 13.4 Lagrange Multipliers 1 Use Lagrange Multipliers for an Optimization Problem with One Constraint 2 Use Lagrange Multipliers for an Optimization Problem with Two Constraints Assess Your Understanding Chapter Review Chapter 13 Project Measuring Ice Thickness on Crystal Lake 14 Multiple Integrals 14.1 The Double Integral over a Rectangular Region 1 Find double Riemann sums of z=f(x,y) over a closed rectangular region 2 Find the value of a double integral defined on a closed rectangular region 3 Find the volume under a surface and over a rectangular region 14.2 The Double Integral over Nonrectangular Regions 1 Use Fubini’s Theorem for an x-simple region 2 Use Fubini’s Theorem for a y-simple region 3 Use properties of double integrals 4 Use double integrals to find area 14.3 Double Integrals Using Polar Coordinates 1 Find a double integral using polar coordinates 2 Find area and volume using polar coordinates 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 14.5 Surface Area 1 Find the area of a surface that lies above a region R 14.6 The Triple Integral 1 Find a triple integral of a function defined in a closed box 2 Find a triple integral of a function defined in a more general solid 3 Find the volume of a solid 4 Find the mass, center of mass, and moments of inertia of a solid 5 Find a triple integral of a function defined in an xz-simple or a yz-simple solid 14.7 Triple Integrals Using Cylindrical Coordinates 1 Convert rectangular coordinates to cylindrical coordinates 2 Find a triple integral using cylindrical coordinates 14.8 Triple Integrals Using Spherical Coordinates 1 Convert rectangular coordinates to spherical coordinates 2 Find a triple integral using spherical coordinates Application: Spherical coordinates in navigation 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 Chapter Review Chapter 14 Project The Mass of Stars 15 Vector Calculus 15.1 Vector Fields 1 Describe a Vector Field Assess Your Understanding 15.2 Line Integrals of Scalar Functions 1 Define a Line Integral in the Plane 2 Find the Value of a Line Integral Along a Smooth Curve Application: Finding theMass of a Wire of Variable Density Application: Finding the Lateral Surface Area of a Cylinder 3 Find Line Integrals of the Form c f(x, y)dx and c f(x, y)dy 4 Find Line Integrals Along a Piecewise-Smooth Curve 5 Find the Value of a Line Integral in Space Assess Your Understanding 15.3 Line Integrals of Vector Fields; Work 1 Find the Line Integral of a Vector Field 2 Compute Work Assess Your Understanding 15.4 Fundamental Theorem of Line Integrals 1 Identify a Conservative Vector Field and Its Potential Function 2 Use the Fundamental Theorem of Line Integrals 3 Reconstruct a Function from Its Gradient: Finding the Potential Function for a Conservative Vector Field 4 Determine Whether a Vector Field Is Conservative 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 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 Smooth Surface 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 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 Application: Electric Force Fields 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 Use Stokes’ Theorem with Conservative Vector Fields 5 Interpret the Curl of F Assess Your Understanding Chapter Review Chapter 15 Project Modeling a Tornado 16 Differential Equations 16.1 Classification of Ordinary Differential Equations 1 Classify ordinary differential equations 2 Verify the solution of an ordinary differential equation 16.2 Separable and Homogeneous First-Order Differential Equations; Slope Fields; Euler’s Method 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 homogeneous first-order differential equation 4 Find orthogonal trajectories 5 Use a slope field to represent the solution of a first-order differential equation 6 Use Euler’s method to approximate a particular solution of a first-order differential equation 16.3 Exact Differential Equations 1 Identify and solve an exact differential equation 16.4 First-Order Linear Differential Equations; Bernoulli Differential Equations 1 Solve a first-order linear differential equation Application: Free fall with air resistance Application: Flow rate in mixtures 2 Find the general solution of a Bernoulli equation Application: Logistic functions 16.5 Power Series Methods 1 Use power series to solve a linear differential equation Chapter Review Chapter 16 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 range of 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 Sines and the Law of Cosines Appendix B Theorems and Proofs B.1 Limit Theorems and Proofs B.2 Theorems and Proofs Involving Inverse Functions B.3 Derivative Theorems and Proofs B.4 Integral Theorems and Proofs B.5 A Bounded Monotonic Sequence Converges B.6 Taylor’s Formula with Remainder Appendix C Technology Used in Calculus C.1 Graphing Calculators C.2 Computer Algebra Systems 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 Index Table of Derivatives Table of Integrals Back Cover
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