About this Book
Cover Page
Title Page
Copyright
What Is Calculus?
Contents
Preface
Acknowledgments
Applications Index
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 9 Project Polar Graphs and Microphones
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: Keplerโs Laws of Planetary Motion
1 Discuss Keplerโs 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
Assess Your Understanding
14.2 The Double Integral over Nonrectangular Regions
1 Use Fubiniโs Theorem for an x-simple region
2 Use Fubiniโs Theorem for a y-simple region
3 Use properties of double integrals
4 Use double integrals to find area
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
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
Assess Your Understanding
14.5 Surface Area
1 Find the area of a surface that lies above a region R
Assess Your Understanding
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
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
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
Application: Spherical coordinates in navigation
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
Assess Your Understanding
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 โซCf(x, y)dx and โซCf(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 Greenโs Theorem
1 Use Greenโs Theorem to Find a Line Integral
2 Use Greenโs Theorem to Find Area
3 Use Greenโs 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 Stokesโ Theorem
1 Find the Curl of F
2 Verify Stokesโ Theorem
3 Use Stokesโ Theorem to Find an Integral
4 Use Stokesโ 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
Assess Your Understanding
16.2 Separable and Homogeneous First-Order Differential Equations; Slope Fields; Eulerโs 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 Eulerโs method to approximate a particular solution of a first-order differential equation
Assess Your Understanding
16.3 Exact Differential Equations
1 Identify and solve an exact differential equation
Assess Your Understanding
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
Assess Your Understanding
16.5 Power Series Methods
1 Use power series to solve a linear differential equation
Assess Your Understanding
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 Taylorโs Formula with Remainder
Appendix C Technology Used in Calculus
C.1 Graphing Calculators
C.2 Computer Algebra Systems
Answers
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