University Calculus: Early Transcendentals 4th Edition by Joel Hass, ISBN-13: 978-0134995540

Description

University Calculus: Early Transcendentals 4th Edition by Joel Hass, ISBN-13: 978-0134995540

[PDF eBook eTextbook]

• Publisher: ‎ Pearson; 4th edition (January 1, 2019)
• Language: ‎ English
• 1104 pages
• ISBN-10: ‎ 0134995546
• ISBN-13: ‎ 978-0134995540

For 3-semester or 4-quarter¿ courses covering single¿ variable and multivariable calculus, taken by students of mathematics, engineering, natural sciences, or economics.

Clear, precise, concise.

University Calculus: Early Transcendentals helps students generalize and apply the key ideas of calculus through clear and precise explanations, thoughtfully chosen examples, meticulously crafted figures, and superior exercise sets. This text offers the right mix of basic, conceptual, and challenging exercises, along with meaningful applications. In the 4th Edition, new co-authors Chris Heil (Georgia Institute of Technology) and Przemyslaw Bogacki (Old Dominion University) partner with author Joel Hass to preserve the text’s time-tested features while revisiting every word, figure, and MyLab™ question with today’s students in mind.

Table of Contents:

Preface

New to This Edition

Accuracy Checkers

Reviewers for the Fourth Edition

1 Functions

1.1 Functions and Their Graphs

Functions; Domain and Range

Graphs of Functions

Representing a Function Numerically

The Vertical Line Test for a Function

Piecewise-Defined Functions

Increasing and Decreasing Functions

Even Functions and Odd Functions: Symmetry

Common Functions

Linear Functions

Power Functions

Polynomials

Rational Functions

Algebraic Functions

Trigonometric Functions

Exponential Functions

Logarithmic Functions

Transcendental Functions

Exercises 1.1

Functions

Finding Formulas for Functions

Functions and Graphs

Piecewise-Defined Functions

The Greatest and Least Integer Functions

Increasing and Decreasing Functions

Even and Odd Functions

Theory and Examples

1.2 Combining Functions; Shifting and Scaling Graphs

Sums, Differences, Products, and Quotients

Composing Functions

Shifting a Graph of a Function

Scaling and Reflecting a Graph of a Function

For c > 1 the graph is scaled:

For c = −1, the graph is reflected:

Exercises 1.2

Algebraic Combinations

Compositions of Functions

Shifting Graphs

Vertical and Horizontal Scaling

Graphing

Combining Functions

1.3 Trigonometric Functions

Angles

Angle Convention: Use Radians

The Six Basic Trigonometric Functions

Periodicity and Graphs of the Trigonometric Functions

Trigonometric Identities

The Law of Cosines

Two Special Inequalities

Transformations of Trigonometric Graphs

Exercises 1.3

Radians and Degrees

Evaluating Trigonometric Functions

Graphing Trigonometric Functions

Using the Addition Formulas

Using the Half-Angle Formulas

Solving Trigonometric Equations

Theory and Examples

General Sine Curves

1.4 Graphing with Software

Graphing Windows

Obtaining a Complete Graph

Exercises 1.4

Choosing a Viewing Window

Finding a Viewing Window

1.5 Exponential Functions

Exponential Behavior

The Natural Exponential Function ex

Exponential Growth and Decay

Exercises 1.5

Sketching Exponential Curves

Applying the Laws of Exponents

Compositions Involving Exponential Functions

Applications

1.6 Inverse Functions and Logarithms

One-to-One Functions

Inverse Functions

Finding Inverses

Logarithmic Functions

Properties of Logarithms

Applications

Inverse Trigonometric Functions

The Arcsine and Arccosine Functions

Identities Involving Arcsine and Arccosine

Exercises 1.6

Identifying One-to-One Functions Graphically

Graphing Inverse Functions

Formulas for Inverse Functions

Inverses of Lines

Logarithms and Exponentials

Arcsine and Arccosine

Theory and Examples

2 Limits and Continuity

2.1 Rates of Change and Tangent Lines to Curves

Average and Instantaneous Speed

Average Rates of Change and Secant Lines

Defining the Slope of a Curve

Rates of Change and Tangent Lines

Exercises 2.1

Average Rates of Change

Slope of a Curve at a Point

Instantaneous Rates of Change

2.2 Limit of a Function and Limit Laws

Limits of Function Values

An Informal Description of the Limit of a Function

The Limit Laws

Evaluating Limits of Polynomials and Rational Functions

Eliminating Common Factors from Zero Denominators

Using Calculators and Computers to Estimate Limits

The Sandwich Theorem

Exercises 2.2

Limits from Graphs

Existence of Limits

Calculating Limits

Using Limit Rules

Limits of Average Rates of Change

Using the Sandwich Theorem

Estimating Limits

Theory and Examples

Graphical Estimates of Limits

2.3 The Precise Definition of a Limit

Definition of Limit

Examples: Testing the Definition

Finding Deltas Algebraically for Given Epsilons

Using the Definition to Prove Theorems

Exercises 2.3

Centering Intervals About a Point

Finding Deltas Graphically

Finding Deltas Algebraically

Using the Formal Definition

Theory and Examples

When Is a Number L Not the Limit of f(x) As x → c?

2.4 One-Sided Limits

Approaching a Limit from One Side

Limits at Endpoints of an Interval

Precise Definitions of One-Sided Limits

Limits Involving (sin θ) / θ

Exercises 2.4

Finding Limits Graphically

Finding One-Sided Limits Algebraically

Using limθ→0⁡sin⁡θθ=1

Theory and Examples

Formal Definitions of One-Sided Limits

2.5 Continuity

Continuity at a Point

Continuous Functions

Inverse Functions and Continuity

Continuity of Compositions of Functions

Intermediate Value Theorem for Continuous Functions

A Consequence for Graphing: Connectedness

A Consequence for Root Finding

Continuous Extension to a Point

Exercises 2.5

Continuity from Graphs

Applying the Continuity Test

Limits Involving Trigonometric Functions

Continuous Extensions

Theory and Examples

Solving Equations Graphically

2.6 Limits Involving Infinity; Asymptotes of Graphs

Finite Limits as x → ± ∞

Limits at Infinity of Rational Functions

Horizontal Asymptotes

Oblique Asymptotes

Infinite Limits

Precise Definitions of Infinite Limits

Vertical Asymptotes

Dominant Terms

Exercises 2.6

Finding Limits

Limits of Rational Functions

Limits as x → ∞ or x → − ∞

Infinite Limits

Graphing Simple Rational Functions

Domains and Asymptotes

Inventing Graphs and Functions

Finding Limits of Differences When x → ± ∞

Using the Formal Definitions

Oblique Asymptotes

Additional Graphing Exercises

Chapter 2 Questions to Guide Your Review

Chapter 2 Practice Exercises

Limits and Continuity

Finding Limits

[Technology Exercise] Roots

Continuous Extension

Limits at Infinity

Horizontal and Vertical Asymptotes

Chapter 2 Additional and Advanced Exercises

Precise Definition of Limit

Generalized Limits Involving sin θθ

Oblique Asymptotes

Showing an Equation Is Solvable

More Limits

Limits on Arbitrary Domains

3 Derivatives

3.1 Tangent Lines and the Derivative at a Point

Finding a Tangent Line to the Graph of a Function

Rates of Change: Derivative at a Point

Summary

Exercises 3.1

Slopes and Tangent Lines

Interpreting Derivative Values

Rates of Change

Testing for Tangent Lines

Vertical Tangent Lines

3.2 The Derivative as a Function

Calculating Derivatives from the Definition

Notation

Graphing the Derivative

Differentiability on an Interval; One-Sided Derivatives

When Does a Function Not Have a Derivative at a Point?

Differentiable Functions Are Continuous

Exercises 3.2

Finding Derivative Functions and Values

Slopes and Tangent Lines

Using the Alternative Formula for Derivatives

Graphs

One-Sided Derivatives

Differentiability and Continuity on an Interval

Theory and Examples

3.3 Differentiation Rules

Powers, Multiples, Sums, and Differences

Derivatives of Exponential Functions

Products and Quotients

Second- and Higher-Order Derivatives

Exercises 3.3

Derivative Calculations

Slopes and Tangent Lines

Theory and Examples

3.4 The Derivative as a Rate of Change

Instantaneous Rates of Change

Motion Along a Line: Displacement, Velocity, Speed, Acceleration, and Jerk

Derivatives in Economics and Biology

Exercises 3.4

Motion Along a Coordinate Line

Free-Fall Applications

Understanding Motion from Graphs

Economics

Additional Applications

Analyzing Motion Using Graphs

3.5 Derivatives of Trigonometric Functions

Derivative of the Sine Function

Derivative of the Cosine Function

Simple Harmonic Motion

Derivatives of the Other Basic Trigonometric Functions

Exercises 3.5

Derivatives

Tangent Lines

Trigonometric Limits

Theory and Examples

3.6 The Chain Rule

Derivative of a Composite Function

“Outside-Inside” Rule

Repeated Use of the Chain Rule

The Chain Rule with Powers of a Function

Exercises 3.6

Derivative Calculations

Second Derivatives

Finding Derivative Values

Theory and Examples

Trigonometric Polynomials

3.7 Implicit Differentiation

Implicitly Defined Functions

Derivatives of Higher Order

Lenses, Tangent Lines, and Normal Lines

Exercises 3.7

Differentiating Implicitly

Second Derivatives

Slopes, Tangent Lines, and Normal Lines

Theory and Examples

3.8 Derivatives of Inverse Functions and Logarithms

Derivatives of Inverses of Differentiable Functions

Derivative of the Natural Logarithm Function

Alternative Derivation

The Derivatives of au and loga u

Logarithmic Differentiation

Irrational Exponents and the Power Rule (General Version)

The Number e Expressed as a Limit

Exercises 3.8

Derivatives of Inverse Functions

Derivatives of Logarithms

Logarithmic Differentiation

Finding Derivatives

Powers with Variable Bases and Exponents

Theory and Applications

3.9 Inverse Trigonometric Functions

Inverses of tan x, cot x, sec x, and csc x

The Derivative of y = sin−1 u

The Derivative of y = tan−1 u

The Derivative of y = sec−1 u

Derivatives of the Other Three Inverse Trigonometric Functions

Exercises 3.9

Common Values

Evaluations

Limits

Finding Derivatives

Theory and Examples

3.10 Related Rates

Related Rates Equations

Exercises 3.10

3.11 Linearization and Differentials

Linearization

Differentials

Estimating with Differentials

Error in Differential Approximation

Proof of the Chain Rule

Sensitivity to Change

Exercises 3.11

Finding Linearizations

Linearization for Approximation

Derivatives in Differential Form

Approximation Error

Differential Estimates of Change

Applications

Chapter 3 Questions to Guide Your Review

Chapter 3 Practice Exercises

Derivatives of Functions

Implicit Differentiation

Numerical Values of Derivatives

Applying the Derivative Definition

Slopes, Tangent Lines, and Normal Lines

Analyzing Graphs

Trigonometric Limits

Logarithmic Differentiation

Related Rates

Linearization

Differential Estimates of Change

Chapter 3 Additional and Advanced Exercises

4 Applications of Derivatives

4.1 Extreme Values of Functions on Closed Intervals

Local (Relative) Extreme Values

Finding Extrema

Exercises 4.1

Finding Extrema from Graphs

Absolute Extrema on Finite Closed Intervals

Finding Critical Points

Theory and Examples

4.2 The Mean Value Theorem

Rolle’s Theorem

The Mean Value Theorem

A Physical Interpretation

Mathematical Consequences

Finding Velocity and Position from Acceleration

Proofs of the Laws of Logarithms

Laws of Exponents

Exercises 4.2

Checking the Mean Value Theorem

Roots (Zeros)

Finding Functions from Derivatives

Finding Position from Velocity or Acceleration

Applications

Theory and Examples

4.3 Monotonic Functions and the First Derivative Test

Increasing Functions and Decreasing Functions

First Derivative Test for Local Extrema

Exercises 4.3

Analyzing Functions from Derivatives

Identifying Extrema

Theory and Examples

4.4 Concavity and Curve Sketching

Concavity

Points of Inflection

Second Derivative Test for Local Extrema

Graphical Behavior of Functions from Derivatives

Exercises 4.4

Analyzing Functions from Graphs

Graphing Functions

Sketching the General Shape, Knowing y′

Sketching y from Graphs of y′ and y″

Theory and Examples

4.5 Indeterminate Forms and L’Hôpital’s Rule

Indeterminate Form 0/0

Indeterminate Forms ∞/∞ , ∞ ⋅ 0 , ∞ − ∞

Indeterminate Powers

Exercises 4.5

Finding Limits in Two Ways

Applying l’Hôpital’s Rule

Indeterminate Powers and Products

Theory and Applications

4.6 Applied Optimization

Examples from Mathematics and Physics

Examples from Economics

Exercises 4.6

Mathematical Applications

Physical Applications

Business and Economics

Biology

Theory and Examples

4.7 Newton’s Method

Procedure for Newton’s Method

Applying Newton’s Method

Convergence of the Approximations

Exercises 4.7

Root Finding

Dependence on Initial Point

Theory and Examples

4.8 Antiderivatives

Finding Antiderivatives

Initial Value Problems and Differential Equations

Antiderivatives and Motion

Indefinite Integrals

Exercises 4.8

Finding Antiderivatives

Finding Indefinite Integrals

Checking Antiderivative Formulas

Initial Value Problems

Solution (Integral) Curves

Applications

Chapter 4 Questions to Guide Your Review

Chapter 4 Practice Exercises

Finding Extreme Values

Extreme Values

The Mean Value Theorem

Analyzing Graphs

Graphs and Graphing

Using L’Hôpital’s Rule

Optimization

Newton’s Method

Finding Indefinite Integrals

Initial Value Problems

Applications and Examples

Chapter 4 Additional and Advanced Exercises

Functions and Derivatives

Optimization

Limits

Theory and Examples

5 Integrals

5.1 Area and Estimating with Finite Sums

Area

Distance Traveled

Displacement Versus Distance Traveled

Average Value of a Nonnegative Continuous Function

Summary

Exercises 5.1

Area

Distance

Average Value of a Function

Examples of Estimations

5.2 Sigma Notation and Limits of Finite Sums

Finite Sums and Sigma Notation

Limits of Finite Sums

Riemann Sums

Exercises 5.2

Sigma Notation

Values of Finite Sums

Riemann Sums

Limits of Riemann Sums

5.3 The Definite Integral

Definition of the Definite Integral

Integrable and Nonintegrable Functions

Properties of Definite Integrals

Area Under the Graph of a Nonnegative Function

Average Value of a Continuous Function Revisited

Exercises 5.3

Interpreting Limits of Sums as Integrals

Using the Definite Integral Rules

Using Known Areas to Find Integrals

Evaluating Definite Integrals

Finding Area by Definite Integrals

Finding Average Value

Definite Integrals as Limits of Sums

Theory and Examples

5.4 The Fundamental Theorem of Calculus

Mean Value Theorem for Definite Integrals

Fundamental Theorem, Part 1

Fundamental Theorem, Part 2 (The Evaluation Theorem)

The Integral of a Rate

The Relationship Between Integration and Differentiation

Total Area

Exercises 5.4

Evaluating Integrals

Derivatives of Integrals

Area

Initial Value Problems

Theory and Examples

5.5 Indefinite Integrals and the Substitution Method

Substitution: Running the Chain Rule Backwards

Trying Different Substitutions

Exercises 5.5

Evaluating Indefinite Integrals

Initial Value Problems

5.6 Definite Integral Substitutions and the Area Between Curves

The Substitution Formula

Definite Integrals of Symmetric Functions

Areas Between Curves

Integration with Respect to y

Exercises 5.6

Evaluating Definite Integrals

Area

Area Between Curves

Theory and Examples

Chapter 5 Questions to Guide Your Review

Chapter 5 Practice Exercises

Finite Sums and Estimates

Definite Integrals

Area

Initial Value Problems

Evaluating Indefinite Integrals

Evaluating Definite Integrals

Average Values

Differentiating Integrals

Chapter 5 Additional and Advanced Exercises

Theory and Examples

Piecewise Continuous Functions

Limits

Defining Functions Using the Fundamental Theorem

Theory and Examples

6 Applications of Definite Integrals

6.1 Volumes Using Cross-Sections

Slicing by Parallel Planes

Solids of Revolution: The Disk Method

Solids of Revolution: The Washer Method

Exercises 6.1

Volumes by Slicing

Volumes by the Disk Method

Volumes by the Washer Method

Volumes of Solids of Revolution

Theory and Applications

6.2 Volumes Using Cylindrical Shells

Slicing with Cylinders

The Shell Method

Exercises 6.2

Revolution About the Axes

Revolution About the y-Axis

Revolution About the x-Axis

Revolution About Horizontal and Vertical Lines

Choosing the Washer Method or the Shell Method

Theory and Examples

6.3 Arc Length

Length of a Curve y = f(x)

Dealing with Discontinuities in dy/dx

The Differential Formula for Arc Length

Exercises 6.3

Finding Lengths of Curves

[Technology Exercise] Finding Integrals for Lengths of Curves

Theory and Examples

6.4 Areas of Surfaces of Revolution

Defining Surface Area

Revolution About the y-Axis

Exercises 6.4

Finding Integrals for Surface Area

Finding Surface Area

6.5 Work

Work Done by a Constant Force

Work Done by a Variable Force Along a Line

Hooke’s Law for Springs: F = kx

Lifting Objects and Pumping Liquids from Containers

Exercises 6.5

Springs

Work Done by a Variable Force

Pumping Liquids from Containers

Work and Kinetic Energy

6.6 Moments and Centers of Mass

Masses Along a Line

Thin Wires

Masses Distributed over a Plane Region

Thin, Flat Plates

Plates Bounded by Two Curves

Centroids

Exercises 6.6

Mass of a wire

Thin Plates with Constant Density

Thin Plates with Varying Density

Centroids of Triangles

Thin Wires

Plates Bounded by Two Curves

Chapter 6 Questions to Guide Your Review

Chapter 6 Practice Exercises

Volumes

Lengths of Curves

Areas of Surfaces of Revolution

Work

Centers of Mass and Centroids

Chapter 6 Additional and Advanced Exercises

Volume and Length

Surface Area

Work

Centers of Mass

7 Integrals and Transcendental Functions

7.1 The Logarithm Defined as an Integral

Definition of the Natural Logarithm Function

The Derivative of y = ln x

The Graph and Range of ln x

The Integral ∫ 1/u du

The Inverse of ln x and the Number e

The Derivative and Integral of ex

Laws of Exponents

The General Exponential Function ax

Logarithms with Base a

Derivatives and Integrals Involving loga x

Summary

Exercises 7.1

Integration

Initial Value Problems

Theory and Applications

Calculations with Other Bases

7.2 Exponential Change and Separable Differential Equations

Exponential Change

Separable Differential Equations

Unlimited Population Growth

Radioactivity

Heat Transfer: Newton’s Law of Cooling

Exercises 7.2

Verifying Solutions

Initial Value Problems

Separable Differential Equations

Applications and Examples

7.3 Hyperbolic Functions

Definitions and Identities

Derivatives and Integrals of Hyperbolic Functions

Inverse Hyperbolic Functions

Useful Identities

Derivatives of Inverse Hyperbolic Functions

Exercises 7.3

Values and Identities

Finding Derivatives

Integration Formulas

Evaluating Integrals

Inverse Hyperbolic Functions and Integrals

Applications and Examples

Chapter 7 Questions to Guide Your Review

Chapter 7 Practice Exercises

Integration

Solving Equations with Logarithmic or Exponential Terms

Theory and Applications

Chapter 7 Additional and Advanced Exercises

8 Techniques of Integration

8.1 Integration by Parts

Product Rule in Integral Form

Evaluating Definite Integrals by Parts

Exercises 8.1

Integration by Parts

Using Substitution

Evaluating Integrals

Theory and Examples

Reduction Formulas

Integrating Inverses of Functions

8.2 Trigonometric Integrals

Products of Powers of Sines and Cosines

Eliminating Square Roots

Integrals of Powers of tan x and sec x

Products of Sines and Cosines

Exercises 8.2

Powers of Sines and Cosines

Integrating Square Roots

Powers of Tangents and Secants

Products of Sines and Cosines

Assorted Integrations

Applications

8.3 Trigonometric Substitutions

Exercises 8.3

Using Trigonometric Substitutions

Assorted Integrations

Complete the Square Before Using Trigonometric Substitutions

Initial Value Problems

Applications and Examples

8.4 Integration of Rational Functions by Partial Fractions

General Description of the Method

Determining Coefficients by Differentiating

Exercises 8.4

Expanding Quotients into Partial Fractions

Nonrepeated Linear Factors

Repeated Linear Factors

Irreducible Quadratic Factors

Improper Fractions

Evaluating Integrals

Initial Value Problems

Applications and Examples

8.5 Integral Tables and Computer Algebra Systems

Integral Tables

Reduction Formulas

Integration with a CAS

Nonelementary Integrals

Exercises 8.5

Using Integral Tables

Substitution and Integral Tables

Using Reduction Formulas

Applications

8.6 Numerical Integration

Trapezoidal Approximations

Simpson’s Rule: Approximations Using Parabolas

Error Analysis

Exercises 8.6

Estimating Definite Integrals

Estimating the Number of Subintervals

Estimates with Numerical Data

Theory and Examples

Applications

8.7 Improper Integrals

Infinite Limits of Integration

The Integral ∫1∞d⁢xxp

Integrands with Vertical Asymptotes

Improper Integrals with a CAS

Tests for Convergence and Divergence

Exercises 8.7

Evaluating Improper Integrals

Testing for Convergence

Theory and Examples

Chapter 8 Questions to Guide Your Review

Chapter 8 Practice Exercises

Integration by Parts

Partial Fractions

Trigonometric Substitutions

Trigonometric Integrals

Numerical Integration

Improper Integrals

Assorted Integrations

Chapter 8 Additional and Advanced Exercises

Evaluating Integrals

Applications

The Gamma Function and Stirling’s Formula

9 Infinite Sequences and Series

9.1 Sequences

Representing Sequences

Convergence and Divergence

Calculating Limits of Sequences

Using L’Hôpital’s Rule

Commonly Occurring Limits

Recursive Definitions

Bounded Monotonic Sequences

Exercises 9.1

Finding Terms of a Sequence

Finding a Sequence’s Formula

Convergence and Divergence

Recursively Defined Sequences

Theory and Examples

9.2 Infinite Series

Geometric Series

The nth-Term Test for a Divergent Series

Combining Series

Adding or Deleting Terms

Reindexing

Exercises 9.2

Finding nth Partial Sums

Series with Geometric Terms

Repeating Decimals

Using the nth-Term Test

Telescoping Series

Convergence or Divergence

Geometric Series with a Variable x

Theory and Examples

9.3 The Integral Test

Nondecreasing Partial Sums

The Integral Test

Error Estimation

Exercises 9.3

Applying the Integral Test

Determining Convergence or Divergence

Theory and Examples

9.4 Comparison Tests

The Limit Comparison Test

Exercises 9.4

Direct Comparison Test

Limit Comparison Test

Determining Convergence or Divergence

Theory and Examples

9.5 Absolute Convergence; The Ratio and Root Tests

The Ratio Test

The Root Test

Exercises 9.5

Using the Ratio Test

Using the Root Test

Determining Convergence or Divergence

Convergence or Divergence

Theory and Examples

9.6 Alternating Series and Conditional Convergence

Conditional Convergence

Rearranging Series

Summary of Tests to Determine Convergence or Divergence

Exercises 9.6

Convergence of Alternating Series

Absolute and Conditional Convergence

Error Estimation

Determining Convergence or Divergence

Theory and Examples

9.7 Power Series

Power Series and Convergence

The Radius of Convergence of a Power Series

Operations on Power Series

Exercises 9.7

Intervals of Convergence

Using the Geometric Series

Theory and Examples

9.8 Taylor and Maclaurin Series

Series Representations

Taylor and Maclaurin Series

Taylor Polynomials

Exercises 9.8

Finding Taylor Polynomials

Finding Taylor Series at x = 0 (Maclaurin Series)

Finding Taylor and Maclaurin Series

Theory and Examples

9.9 Convergence of Taylor Series

Estimating the Remainder

Using Taylor Series

Exercises 9.9

Finding Taylor Series

Error Estimates

Theory and Examples

9.10 Applications of Taylor Series

The Binomial Series for Powers and Roots

Evaluating Nonelementary Integrals

Arctangents

Evaluating Indeterminate Forms

Euler’s Identity

Exercises 9.10

Taylor Series

Approximations and Nonelementary Integrals

Indeterminate Forms

Using Table 9.1

Theory and Examples

Euler’s Identity

Chapter 9 Questions to Guide Your Review

Chapter 9 Practice Exercises

Determining Convergence of Sequences

Convergent Series

Determining Convergence of Series

Power Series

Maclaurin Series

Taylor Series

Nonelementary Integrals

Using Series to Find Limits

Theory and Examples

Chapter 9 Additional and Advanced Exercises

Determining Convergence of Series

Choosing Centers for Taylor Series

Theory and Examples

10 Parametric Equations and Polar Coordinates

10.1 Parametrizations of Plane Curves

Parametric Equations

Cycloids

Brachistochrones and Tautochrones

Exercises 10.1

Finding Cartesian from Parametric Equations

Finding Parametric Equations

Distance Using Parametric Equations

10.2 Calculus with Parametric Curves

Tangent Lines and Areas

Length of a Parametrically Defined Curve

Length of a Curve y = f(x)

The Arc Length Differential

Areas of Surfaces of Revolution

Exercises 10.2

Tangent Lines to Parametrized Curves

Implicitly Defined Parametrizations

Area

Lengths of Curves

Surface Area

Centroids

Theory and Examples

10.3 Polar Coordinates

Definition of Polar Coordinates

Polar Equations and Graphs

Relating Polar and Cartesian Coordinates

Exercises 10.3

Polar Coordinates

Polar to Cartesian Coordinates

Cartesian to Polar Coordinates

Graphing Sets of Polar Coordinate Points

Polar to Cartesian Equations

Cartesian to Polar Equations

10.4 Graphing Polar Coordinate Equations

Symmetry

Slope

Converting a Graph from the rθ-Plane to the xy-Plane

Exercises 10.4

Symmetries and Polar Graphs

Slopes of Polar Curves in the xy-Plane

Concavity of Polar Curves in the xy-Plane

Graphing Limaçons

Graphing Polar Regions and Curves in the xy-Plane

10.5 Areas and Lengths in Polar Coordinates

Area in the Plane

Length of a Polar Curve

Exercises 10.5

Finding Polar Areas

Finding Lengths of Polar Curves

Theory and Examples

Chapter 10 Questions to Guide Your Review

Chapter 10 Practice Exercises

Identifying Parametric Equations in the Plane

Finding Parametric Equations and Tangent Lines

Lengths of Curves

Surface Areas

Polar to Cartesian Equations

Cartesian to Polar Equations

Graphs in Polar Coordinates

Area in Polar Coordinates

Length in Polar Coordinates

Chapter 10 Additional and Advanced Exercises

Polar Coordinates

Theory and Examples

11 Vectors and the Geometry of Space

11.1 Three-Dimensional Coordinate Systems

Distance and Spheres in Space

Exercises 11.1

Geometric Interpretations of Equations

Geometric Interpretations of Inequalities and Equations

Distance

Inequalities to Describe Sets of Points

Spheres

Theory and Examples

11.2 Vectors

Component Form

Vector Algebra Operations

Unit Vectors

Midpoint of a Line Segment

Applications

Exercises 11.2

Vectors in the Plane

Vectors in Space

Geometric Representations

Length and Direction

Direction and Midpoints

Theory and Applications

11.3 The Dot Product

Angle Between Vectors

Orthogonal Vectors

Dot Product Properties and Vector Projections

Work

Exercises 11.3

Dot Product and Projections

Angle Between Vectors

Theory and Examples

Equations for Lines in the Plane

Work

Angles Between Lines in the Plane

11.4 The Cross Product

The Cross Product of Two Vectors in Space

|u × v| Is the Area of a Parallelogram

Determinant Formula for u × v

Torque

Triple Scalar or Box Product

Exercises 11.4

Cross Product Calculations

Triangles in Space

Triple Scalar Products

Theory and Examples

Area of a Parallelogram

Area of a Triangle

Volume of a Tetrahedron

11.5 Lines and Planes in Space

Lines and Line Segments in Space

The Distance from a Point to a Line in Space

An Equation for a Plane in Space

Lines of Intersection

The Distance from a Point to a Plane

Angles Between Planes

Exercises 11.5

Lines and Line Segments

Planes

Distances

Angles

Intersecting Lines and Planes

Theory and Examples

11.6 Cylinders and Quadric Surfaces

Cylinders

Quadric Surfaces

General Quadric Surfaces

Exercises 11.6

Matching Equations with Surfaces

Drawing

Cylinders

Ellipsoids

Paraboloids and Cones

Hyperboloids

Hyperbolic Paraboloids

Assorted

Theory and Examples

Viewing Surfaces

Chapter 11 Questions to Guide Your Review

Chapter 11 Practice Exercises

Vector Calculations in Two Dimensions

Vector Calculations in Three Dimensions

Lines, Planes, and Distances

Quadric Surfaces

Chapter 11 Additional and Advanced Exercises

12 Vector-Valued Functions and Motion in Space

12.1 Curves in Space and Their Tangents

Limits and Continuity

Derivatives and Motion

Differentiation Rules

Vector Functions of Constant Length

Exercises 12.1

Motion in the Plane

Motion in Space

Tangents to Curves

Theory and Examples

12.2 Integrals of Vector Functions; Projectile Motion

Integrals of Vector Functions

The Vector and Parametric Equations for Ideal Projectile Motion

Exercises 12.2

Integrating Vector-Valued Functions

Initial Value Problems

Motion Along a Straight Line

Projectile Motion

Theory and Examples

12.3 Arc Length in Space

Arc Length Along a Space Curve

Speed on a Smooth Curve

Unit Tangent Vector

Exercises 12.3

Finding Tangent Vectors and Lengths

Arc Length Parameter

Theory and Examples

12.4 Curvature and Normal Vectors of a Curve

Curvature of a Plane Curve

Circle of Curvature for Plane Curves

Curvature and Normal Vectors for Space Curves

Exercises 12.4

Plane Curves

Space Curves

More on Curvature

12.5 Tangential and Normal Components of Acceleration

The TNB Frame

Tangential and Normal Components of Acceleration

Exercises 12.5

Finding Tangential and Normal Components

Finding the TNB Frame

Physical Applications

Theory and Examples

12.6 Velocity and Acceleration in Polar Coordinates

Motion in Polar and Cylindrical Coordinates

Planets Move in Planes

Kepler’s First Law (Ellipse Law)

Kepler’s Second Law (Equal Area Law)

Kepler’s Third Law (Time Distance Law)

Exercises 12.6

Chapter 12 Questions to Guide Your Review

Chapter 12 Practice Exercises

Motion in the Plane

Projectile Motion

Motion in Space

Theory and Examples

Chapter 12 Additional and Advanced Exercises

Applications

Motion in Polar and Cylindrical Coordinates

13 Partial Derivatives

13.1 Functions of Several Variables

Domains and Ranges

Functions of Two Variables

Graphs, Level Curves, and Contours of Functions of Two Variables

Functions of Three Variables

Computer Graphing

Exercises 13.1

Domain, Range, and Level Curves

Matching Surfaces with Level Curves

Functions of Two Variables

Finding Level Curves

Sketching Level Surfaces

Finding Level Surfaces

13.2 Limits and Continuity in Higher Dimensions

Limits for Functions of Two Variables

Continuity

Functions of More Than Two Variables

Extreme Values of Continuous Functions on Closed, Bounded Sets

Exer