A Note to the Student from the Author 

xiii  

Limits, Derivatives, Integrals, and Integrals 


1  (30) 

The Concept of Instantaneous Rate 


3  (3) 

Rate of Change by Equation, Graph, or Table 


6  (8) 

One Type of Integral of a Function 


14  (4) 

Definite Integrals by Trapezoids, from Equations and Data 


18  (6) 


24  (1) 


25  (6) 


31  (40) 

Numerical Approach to the Definition of Limit 


33  (1) 

Graphical and Algebraic Approaches to the Definition of Limit 


34  (6) 


40  (5) 

Continuity and Discontinuity 


45  (7) 

Limits Involving Infinity 


52  (8) 

The Intermediate Value Theorem and Its Consequences 


60  (4) 


64  (7) 

Derivatives, Antiderivatives, and Indefinite Integrals 


71  (58) 

Graphical Interpretation of Derivative 


73  (1) 

Difference Quotients and One Definition of Derivative 


74  (4) 

Derivative Functions, Numerically and Graphically 


78  (7) 

Derivative of the Power Function and Another Definition of Derivative 


85  (7) 

Displacement, Velocity, and Acceleration 


92  (8) 

Introduction to Sine, Cosine, and Composite Functions 


100  (2) 

Derivatives of Composite FunctionsThe Chain Rule 


102  (5) 

Proof and Application of Sine and Cosine Derivatives 


107  (8) 

Exponential and Logarithmic Functions 


115  (7) 


122  (7) 

Products, Quotients, and Parametric Functions 


129  (58) 

Combinations of Two Functions 


131  (1) 

Derivative of a Product of Two Functions 


132  (5) 

Derivative of a Quotient of Two Functions 


137  (5) 

Derivatives of the Other Trigonometric Functions 


142  (4) 

Derivatives of Inverse Trigonometric Functions 


146  (7) 

Differentiability and Continuity 


153  (7) 

Derivatives of a Parametric Function 


160  (9) 

Graphs and Derivatives of Implicit Relations 


169  (5) 


174  (6) 


180  (7) 

Definite and Indefinite Integrals 


187  (80) 

A Definite Integral Problem 


189  (1) 

Linear Approximations and Differentials 


190  (7) 

Formal Definition of Antiderivative and Indefinite Integral 


197  (7) 

Riemann Sums and the Definition of Definite Integral 


204  (7) 

The Mean Value Theorem and Rolle's Theorem 


211  (10) 

The Fundamental Theorem of Calculus 


221  (6) 

Definite Integral Properties and Practice 


227  (6) 

Definite Integrals Applied to Area and Other Problems 


233  (9) 

Volume of a Solid by Plane Slicing 


242  (10) 

Definite Integrals Numerically by Grapher and by Simpson's Rule 


252  (7) 


259  (8) 

The Calculus of Exponential and Logarithmic Functions 


267  (48) 

Integral of the Reciprocal Function: A Population Growth Problem 


269  (1) 

Antiderivative of the Reciprocal Function and Another Form of the Fundamental Theorem 


270  (10) 

The Uniqueness Theorem and Properties of Logarithmic Functions 


280  (8) 

The Number e, Exponential Functions, and Logarithmic Differentiation 


288  (7) 

Limits of Indeterminate Forms: l'Hospital's Rule 


295  (6) 

Derivative and Integral Practice for Transcendental Functions 


301  (5) 


306  (5) 

Cumulative Review: Chapters 16 


311  (4) 

The Calculus of Growth and Decay 


315  (54) 

Direct Proportion Property of Exponential Functions 


317  (1) 

Exponential Growth and Decay 


318  (6) 

Other Differential Equations for RealWorld Applications 


324  (9) 

Graphical Solution of Differential Equations by Using Slope Fields 


333  (8) 

Numerical Solution of Differential Equations by Using Euler's Method 


341  (7) 

The Logistic Function, and PredatorPrey Population Problems 


348  (11) 


359  (6) 

Cumulative Review: Chapters 17 


365  (4) 

The Calculus of Plane and Solid Figures 


369  (62) 

Cubic Functions and Their Derivatives 


371  (1) 

Critical Points and Points of Inflection 


372  (13) 

Maxima and Minima in Plane and Solid Figures 


385  (10) 

Volume of a Solid of Revolution by Cylindrical Shells 


395  (6) 

Length of a Plane CurveArc Length 


401  (6) 

Area of a Surface of Revolution 


407  (7) 

Lengths and Areas for Polar Coordinates 


414  (9) 


423  (8) 

Algebraic Calculus Techniques for the Elementary Functions 


431  (68) 

Introduction to the Integral of a Product of Two Functions 


433  (1) 

Integration by PartsA Way to Integrate Products 


434  (4) 

Rapid Repeated Integration by Parts 


438  (6) 

Reduction Formulas and Computer Algebra Systems 


444  (5) 

Integrating Special Powers of Trigonometric Functions 


449  (5) 

Integration by Trigonometric Substitution 


454  (6) 

Integration of Rational Functions by Partial Fractions 


460  (6) 

Integrals of the Inverse Trigonometric Functions 


466  (3) 

Calculus of the Hyperbolic and Inverse Hyperbolic Functions 


469  (12) 


481  (7) 

Miscellaneous Integrals and Derivatives 


488  (5) 


493  (1) 


494  (5) 

The Calculus of MotionAverages, Extremes, and Vectors 


499  (46) 

Introduction to Distance and Displacement for Motion Along a Line 


501  (1) 

Distance, Displacement, and Acceleration for Linear Motion 


502  (6) 

Average Value Problems in Motion and Elsewhere 


508  (6) 


514  (6) 

Maximum and Minimum Problems in Motion and Elsewhere 


520  (2) 

Vector Functions for Motion in a Plane 


522  (16) 


538  (7) 

The Calculus of VariableFactor Products 


545  (42) 

Review of WorkForce Times Displacement 


547  (1) 

Work Done by a Variable Force 


548  (5) 

Mass of a VariableDensity Object 


553  (5) 

Moments, Centroids, Center of Mass, and the Theorem of Pappus 


558  (9) 

Force Exerted by a Variable PressureCenter of Pressure 


567  (6) 

Other VariableFactor Products 


573  (7) 


580  (7) 

The Calculus of Functions Defined by Power Series 


587  (68) 

Introduction to Power Series 


589  (1) 

Geometric Sequences and Series as Mathematical Models 


590  (7) 

Power Series for an Exponential Function 


597  (1) 

Power Series for Other Elementary Functions 


598  (7) 

Taylor and Maclaurin Series, and Operations on These Series 


605  (8) 

Interval of Convergence for a SeriesThe Ratio Technique 


613  (8) 

Convergence of Series at the Ends of the Convergence Interval 


621  (14) 

Error Analysis for SeriesThe Lagrange Error Bound 


635  (8) 


643  (5) 


648  (7) 
Final Examination: A Guided Tour Through Calculus 

655  (4) 
Appendix: Summary of Properties of Trigonometric Functions 

659  (2) 
Answers to Selected Problems 

661  (94) 
Glossary 

755  (6) 
Index of Problem Titles 

761  (6) 
General Index 

767  (10) 
Photograph Credits 

777  