Calculus : Concepts and Applications

by Foerster, Paul

• ISBN13: 9781559536547
• ISBN10: 1559536543
• Edition: 2nd
• Format: Hardcover
• Copyright: 2004-05-01
• Publisher: Kendall Hunt

Summary

The acclaimed Calculus: Concepts and Applications is now available in a new edition, revised to reflect important changes in the Advanced Placement curriculum, and updated to incorporate feedback from instructors throughout the U.S.With over 40 years of experience teaching AP Calculus, Paul Foerster developed Calculus: Concepts and Applications with the high school student in mind, but with all the content of a college-level course. Like the previous edition, the second edition follows the AP Calculus curriculum for both AB and BC levels.In Calculus: Concepts and Applications, students start off with calculus! Review of precalculus occurs at various points when it's needed. The text combines graphing-calculator technology with a unique, real-world application approach, and presents calculus as a study of just four fundamental concepts: limits, derivatives, definite integrals, and indefinite integrals. Students learn these concepts using algebraic, numerical, graphical, and verbal approaches. As a result, students with a wider range of abilities can be successful in calculus, not just those who are strong in algebra. The accompanying set of Explorations in the Instructor's Resource Book, designed for cooperative group work, gives students hands-on experience with new topics before they are formally introduced.In this new edition, derivatives of transcendental functions, related rates, as well as area and volume applications of the definite integral are introduced earlier. Additionally, the Instructor's Resource Book includes projects utilizing the CBL'„¢, The Geometer's Sketchpad ®, and Fathom Dynamic Statistics'„¢ software, giving students extended opportunities to explore and understand calculus in depth.

Table of Contents

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)
Calculus Journal	24	(1)
Chapter Review and Test	25	(6)
Properties of Limits	31	(40)
Numerical Approach to the Definition of Limit	33	(1)
Graphical and Algebraic Approaches to the Definition of Limit	34	(6)
The Limit Theorems	40	(5)
Continuity and Discontinuity	45	(7)
Limits Involving Infinity	52	(8)
The Intermediate Value Theorem and Its Consequences	60	(4)
Chapter Review and Test	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 Functions---The Chain Rule	102	(5)
Proof and Application of Sine and Cosine Derivatives	107	(8)
Exponential and Logarithmic Functions	115	(7)
Chapter Review and Test	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)
Related Rates	174	(6)
Chapter Review and Test	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)
Chapter Review and Test	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)
Chapter Review and Test	306	(5)
Cumulative Review: Chapters 1--6	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 Real-World 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 Predator-Prey Population Problems	348	(11)
Chapter Review and Test	359	(6)
Cumulative Review: Chapters 1-7	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 Curve---Arc Length	401	(6)
Area of a Surface of Revolution	407	(7)
Lengths and Areas for Polar Coordinates	414	(9)
Chapter Review and Test	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 Parts---A 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)
Improper Integrals	481	(7)
Miscellaneous Integrals and Derivatives	488	(5)
Integrals in Journal	493	(1)
Chapter Review and Test	494	(5)
The Calculus of Motion---Averages, 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)
Minimal Path Problems	514	(6)
Maximum and Minimum Problems in Motion and Elsewhere	520	(2)
Vector Functions for Motion in a Plane	522	(16)
Chapter Review and Test	538	(7)
The Calculus of Variable-Factor Products	545	(42)
Review of Work---Force Times Displacement	547	(1)
Work Done by a Variable Force	548	(5)
Mass of a Variable-Density Object	553	(5)
Moments, Centroids, Center of Mass, and the Theorem of Pappus	558	(9)
Force Exerted by a Variable Pressure---Center of Pressure	567	(6)
Other Variable-Factor Products	573	(7)
Chapter Review and Test	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 Series---The Ratio Technique	613	(8)
Convergence of Series at the Ends of the Convergence Interval	621	(14)
Error Analysis for Series---The Lagrange Error Bound	635	(8)
Chapter Review and Test	643	(5)
Cumulative Reviews	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