Calculus: Graphical, Numerical, Algebraic, AP Edition

, by Finney, Ross L.; Demana, Franklin; Waits, Bert K.; Kennedy, Daniel

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Prerequisites for Calculus

(56)

Lines

(9)

Increments

Slope of a Line

Parallel and Perpendicular Lines

Equations of Lines

Applications

Functions and Graphs

(10)

Functions

Domains and Ranges

Viewing and Interpreting Graphs

Even Functions and Odd Functions---Symmetry

Functions Defined in Pieces

Absolute Value Function

Composite Functions

Exponential Functions

(8)

Exponential Growth

Exponential Decay

Applications

The Number e

Parametric Equations

(7)

Relations

Circles

Ellipses

Lines and Other Curves

Functions and Logarithms

(9)

One-to-One Functions

Inverses

Finding Inverses

Logarithmic Functions

Properties of Logarithms

Applications

Trigonometric Functions

(12)

Radian Measure

Graphs of Trigonometric Functions

Periodicity

Even and Odd Trigonometric Functions

Transformations of Trigonometric Graphs

Inverse Trigonometric Functions

Key Terms

(1)

Review Exercises

(2)

Limits and Continuity

(40)

Rates of Change and Limits

(11)

Average and Instantaneous Speed

Definition of Limit

Properties of Limits

One-sided and Two-sided Limits

Sandwich Theorem

Limits Involving Infinity

(8)

Finite Limits as x → ± ∞

Sandwich Theorem Revisited

Infinite Limits as x → α

End Behavior Models

``Seeing'' Limits as x → ± ∞

Continuity

(9)

Continuity at a Point

Continuous Functions

Algebraic Combinations

Composites

Intermediate Value Theorem for Continuous Functions

Rates of Change and Tangent Lines

(11)

Average Rates of Change

Tangent to a Curve

Slope of a Curve

Normal to a Curve

Speed Revisited

Key Terms

(1)

Review Exercises

(3)

Derivatives

(88)

Derivative of a Function

(10)

Definition of a Derivative

Notation

Relationship Between the Graphs of f and f'

Graphing the Derivative from Data

One-sided Derivatives

Differentiability

109

(7)

How f'(a) Might Fail to Exist

Differentiability Implies Local Linearity

Derivatives on a Calculator

Differentiability Implies Continuity

Intermediate Value Theorem for Derivatives

Rules for Differentiation

116

(11)

Positive Integer Powers, Multiples, Sums, and Differences

Products and Quotients

Negative Integer Powers of x

Second and Higher Order Derivatives

Velocity and Other Rates of Change

127

(14)

Instantaneous Rates of Change

Motion along a Line

Sensitivity to Change

Derivatives in Economics

Derivatives of Trigonometric Functions

141

(7)

Derivative of the Sine Function

Derivative of the Cosine Function

Simple Harmonic Motion

Jerk

Derivatives of Other Basic Trigonometric Functions

Chain Rule

148

(9)

Derivative of a Composite Function

``Outside-Inside'' Rule

Repeated Use of the Chain Rule

Slopes of Parametrized Curves

Power Chain Rule

Implicit Differentiation

157

(8)

Implicitly Defined Functions

Lenses, Tangents, and Normal Lines

Derivatives of Higher Order

Rational Powers of Differentiable Functions

Derivatives of Inverse Trigonometric Functions

165

(7)

Derivatives of Inverse Functions

Derivative of the Arcsine

Derivative of the Arctangent

Derivative of the Arcsecant

Derivatives of the Other Three

Derivatives of Exponential and Logarithmic Functions

172

(14)

Derivative of ex

Derivative of αx

Derivative of In x

Derivative of logαx

Power Rule for Arbitrary Real Powers

Calculus at Work

181

(1)

Key Terms

181

(1)

Review Exercises

181

(5)

Applications of Derivatives

186

(76)

Extreme Values of Functions

187

(9)

Absolute (Global) Extreme Values

Local (Relative) Extreme Values

Finding Extreme Values

Mean Value Theorem

196

(9)

Mean Value Theorem

Physical Interpretation

Increasing and Decreasing Functions

Other Consequences

Connecting f' and f'' with the Graph of f

205

(14)

First Derivative Test for Local Extrema

Concavity

Points of Inflection

Second Derivative Test for Local Extrema

Learning about Functions from Derivatives

Modeling and Optimization

219

(14)

Examples from Mathematics

Examples from Business and Industry

Examples from Economics

Modeling Discrete Phenomena with Differentiable Functions

Linearization and Newton's Method

233

(13)

Linear Approximation

Newton's Method

Differentials

Estimating Change with Differentials

Absolute, Relative, and Percentage Change

Sensitivity to Change

Related Rates

246

(16)

Related Rate Equations

Solution Strategy

Simulating Related Motion

Key Terms

255

(1)

Review Exercises

256

(6)

The Definite Integral

262

(58)

Estimating with Finite Sums

263

(11)

Distance Traveled

Rectangular Approximation Method (RAM)

Volume of a Sphere

Cardiac Output

Definite Integrals

274

(11)

Riemann Sums

Terminology and Notation of Integration

Definite Integral and Area

Constant Functions

Integrals on a Calculator

Discontinuous Integrable Functions

Definite Integrals and Antiderivatives

285

(9)

Properties of Definite Integrals

Average Value of a Function

Mean Value Theorem for Definite Integrals

Connecting Differential and Integral Calculus

Fundamental Theorem of Calculus

294

(12)

Fundamental Theorem, Part 1

Graphing the Function ∫ x/α f(t)dt

Fundamental Theorem, Part 2

Area Connection

Analyzing Antiderivatives Graphically

Trapezoidal Rule

306

(14)

Trapezoidal Approximations

Other Algorithms

Error Analysis

Key Terms

315

(1)

Review Exercises

315

(4)

Calculus at Work

319

(1)

Differential Equations and Mathematical Modeling

320

(58)

Slope Fields and Euler's Method

321

(10)

Differential Equations

Slope Fields

Euler's Method

Antidifferentiation by Substitution

331

(10)

Indefinite Integrals

Leibniz Notation and Antiderivatives

Substitution in Indefinite Integrals

Substitution in Definite Integrals

Antidifferentiation by Parts

341

(9)

Product Rule in Integral Form

Solving for the Unknown Integral

Tabular Integration

Inverse Trigonometric and Logarithmic Functions

Exponential Growth and Decay

350

(12)

Separable Differential Equations

Law of Exponential Change

Continuously Compounded Interest

Radioactivity

Modeling Growth with Other Bases

Newton's Law of Cooling

Logistic Growth

362

(16)

How Populations Grow

Partial Fractions

The Logistic Differential Equation

Logistic Growth Models

Key Terms

372

(1)

Review Exercises

372

(4)

Calculus at Work

376

(2)

Applications of Definite Integrals

378

(41)

Integral As Net Change

379

(11)

Linear Motion Revisited

General Strategy

Consumption Over Time

Net Change from Data

Work

Areas in the Plane

390

(9)

Area Between Curves

Area Enclosed by Intersecting Curves

Boundaries with Changing Functions

Integrating with Respect to y

Saving Time with Geometry Formulas

Volumes

399

(13)

Volume As an Integral

Square Cross Sections

Circular Cross Sections

Cylindrical Shells

Other Cross Sections

Lengths of Curves

412

(7)

A Sine Wave

Length of Smooth Curve

Vertical Tangents, Corners, and Cusps

Applications from Science and Statistics

419

(15)

Work Revisited

Fluid Force and Fluid Pressure

Normal Probabilities

Calculus at Work

430

(1)

Key Terms

430

(1)

Review Exercises

430

(4)

Sequences, L'Hopital's Rule, and Improper Integrals

434

(38)

Sequences

435

(9)

Defining a Sequence

Arithmetic and Geometric Sequences

Graphing a Sequence

Limit of a Sequence

L'Hopital's Rule

444

(9)

Indeterminate Form 0/0

Indeterminate Forms ∞/∞, ∞ 0, and ∞ - ∞

Indeterminate Forms 1∞, 00, ∞0

Relative Rates of Growth

453

(6)

Comparing Rates of Growth

Using L'Hopital's Rule to Compare Growth Rates

Sequential versus Binary Search

Improper Integrals

459

(13)

Infinite Limits of Integration

Integrands with Infinite Discontinuities

Test for Convergence and Divergence

Applications

Key Terms

470

(1)

Review Exercises

470

(2)

Infinite Series

472

(58)

Power Series

473

(11)

Geometric Series

Representing Functions by Series

Differentiation and Integration

Identifying a Series

Taylor Series

484

(11)

Constructing a Series

Series for sin x and cos x

Beauty Bare

Maclaurin and Taylor Series

Combining Taylor Series

Table of Maclaurin Series

Taylor's Theorem

495

(8)

Taylor Polynomials

The Remainder

Remainder Estimation Theorem

Euler's Formula

Radius of Convergence

503

(10)

Convergence

nth-Term Test

Comparing Nonnegative Series

Ratio Test

Endpoint Convergence

Testing Convergence at Endpoints

513

(17)

Integral Test

Harmonic Series and p-series

Comparison Tests

Alternating Series

Absolute and Conditional Convergence

Intervals of Convergence

A Word of Caution

Key Terms

526

(1)

Review Exercises

526

(3)

Calculus at Work

529

(1)

Parametric, Vector, and Polar Functions

530

(88)

Parametric Functions

531

(7)

Parametric Curves in the Plane

Slope and Concavity

Arc Length

Cycloids

Vectors in the Plane

538

(10)

Two-Dimensional Vectors

Vector Operations

Modeling Planar Motion

Velocity, Acceleration, and Speed

Displacement and Distance Traveled

Polar Functions

548

(14)

Polar Coordinates

Polar Curves

Slopes of Polar Curves

Areas Enclosed by Polar Curves

A Small Polar Gallery

Key Terms

559

(1)

Review Exercises

560

(2)

APPENDIX

Formulas from Precalculus Mathematics

562

(4)

Mathematical Induction

566

(3)

Using the Limit Definition

569

(8)

Proof of the Chain Rule

577

(1)

Conic Sections

578

(25)

Hyperbolic Functions

603

(9)

A Brief Table of Integrals

612

(6)

Glossary

618

(11)

Selected Answers

629

(51)

Applications Index

680

(4)

Index

684

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Calculus: Graphical, Numerical, Algebraic, AP Edition

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