Calculus : An Integrated Approach to Functions and Their Rates of Change, Preliminary Edition

by Gottlieb, Robin J.

• ISBN13: 9780201709292
• ISBN10: 0201709295
• Edition: 1st
• Format: Paperback
• Copyright: 2002-01-01
• Publisher: Addison Wesley

Summary

A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course.

Table of Contents

Preface	vii
PART I Functions: An Introduction	1	(138)
Functions Are Lurking Everywhere	1	(48)
Functions Are Everywhere	1	(4)
Exploratory Problems Calibrating Bottles	4	(1)
What Are Functions? Basic Vocabulary and Notation	5	(10)
Representations of Functions	15	(34)
Characterizing Functions and Introducing Rates of Change	49	(52)
Features of a Function: Positive/Negative, Increasing/Decreasing, Continuous/Discontinuous	49	(12)
A Pocketful of Functions: Some Basic Examples	61	(12)
Average Rates of Change	73	(11)
Exploratory Problems Runners	82	(2)
Reading a Graph to Get Information About a Function	84	(11)
The Real Number System: An Excursion	95	(6)
Functions Working Together	101	(38)
Combining Outputs: Addition, Subtraction, Multiplication, and Division of Functions	101	(7)
Composition of Functions	108	(11)
Decomposition of Functions	119	(7)
Exploratory Problems Flipping, Shifting, Shrinking, and Stretching: Exercising Functions	123	(3)
Altered Functions, Altered Graphs: Stretching, Shrinking, Shifting, and Flipping	126	(13)
PART II Rates of Change: An Introduction to the Derivative	139	(164)
Linearity and Local Linearity	139	(30)
Making Predictions: An Intuitive Approach to Local Linearity	139	(4)
Linear Functions	143	(10)
Modeling and Interpreting the Slope	153	(6)
Exploratory Problem Thomas Wolfe's Royalties for The Story of a Novel	158	(1)
Applications of Linear Models: Variations on a Theme	159	(10)
The Derivative Function	169	(48)
Calculating the Slope of a Curve and Instantaneous Rate of Change	169	(18)
The Derivative Function	187	(7)
Qualitative Interpretation of the Derivative	194	(14)
Exploratory Problems Running Again	206	(2)
Interpreting the Derivative: Meaning and Notation	208	(9)
The Quadratics: A Profile of a Prominent Family of Functions	217	(28)
A Profile of Quadratics from a Calculus Perspective	217	(6)
Quadratics From A Noncalculus Perspective	223	(8)
Exploratory Problems Tossing Around Quadratics	226	(5)
Quadratics and Their Graphs	231	(6)
The Free Fall of an Apple: A Quadratic Model	237	(8)
The Theoretical Backbone: Limits and Continuity	245	(34)
Investigating Limits---Methods of Inquiry and a Definition	245	(13)
Left-and Right-Handed Limits; Sometimes the Approach Is Critical	258	(7)
A Streetwise Approach to Limits	265	(5)
Continuity and the Intermediate and Extreme Value Theorems	270	(9)
Exploratory Problems Pushing the Limit	275	(4)
Fruits of Our Labor: Derivatives and Local Linearity Revisited	279	(24)
Local Linearity and the Derivative	279	(9)
Exploratory Problems Circles and Spheres	286	(2)
The First and Second Derivatives in Context: Modeling Using Derivatives	288	(2)
Derivatives of Sums, Products, Quotients, and Power Functions	290	(13)
PART III Exponential, Polynomial, and Rational Functions--with Applications	303	(118)
Exponential Functions	303	(38)
Exponential Growth: Growth at a Rate Proportional to Amount	303	(6)
Exponential: The Bare Bones	309	(11)
Applications of the Exponential Function	320	(14)
Exploratory Problems The Derivative of the Exponential Function	328	(6)
The Derivative of an Exponential Function	334	(7)
Optimization	341	(32)
Analysis of Extrema	341	(15)
Concavity and the Second Derivative	356	(5)
Principles in Action	361	(12)
Exploratory Problems Optimization	365	(8)
A Portrait of Polynomials and Rational Functions	373	(48)
A Portrait of Cubics from a Calculus Perspective	373	(6)
Polynomial Functions and Their Graphs	379	(12)
Polynomial Functions and Their Graphs	391	(15)
Exploratory Problems Functions and Their Graphs: Tinkering with Polynomials and Rational Functions	404	(2)
Rational Functions and Their Graphs	406	(15)
PART IV Inverse Functions: A Case Study of Exponential and Logarithmic Functions	421	(92)
Inverse Functions: Can What Is Done Be Undone?	421	(18)
What Does It Mean for f and g to Be Inverse Functions?	421	(8)
Finding the Inverse of a Function	429	(5)
Interpreting the Meaning of Inverse Functions	434	(5)
Exploratory Problems Thinking About the Derivatives of Inverse Functions	437	(2)
Logarithmic Functions	439	(28)
The Logarithmic Function Defined	439	(5)
The Properties of Logarithms	444	(5)
Using Logarithms and Exponentiation to Solve Equations	449	(13)
Exploratory Problem Pollution Study	458	(4)
Graphs of Logarithmic Functions: Theme and Variations	462	(5)
Differentiating Logarithmic and Exponential Functions	467	(20)
The Derivative of Logarithmic Functions	467	(6)
Exploratory Problem The Derivative of the Natural Logarithm	468	(5)
The Derivative of bx Revisited	473	(3)
Worked Examples Involving Differentiation	476	(11)
Take It to the Limit	487	(26)
An Interesting Limit	487	(10)
Introducing Differential Equations	497	(16)
Exploratory Problems Population Studies	507	(6)
PART V Adding Sophistication to Your Differentiation	513	(46)
Taking the Derivative of Composite Functions	513	(22)
The Chain Rule	513	(8)
The Derivative of xn where n is any Real Number	521	(2)
Using the Chain Rule	523	(12)
Exploratory Problems Finding the Best Path	528	(7)
Implicit Differentiation and its Applications	535	(24)
Introductory Example	535	(3)
Logarithmic Differentiation	538	(3)
Implicit Differentiation	541	(9)
Implicit Differentiation in Context: Related Rates of Change	550	(9)
PART VI An Excursion into Geometric Series	559	(34)
Geometric Sums, Geometric Series	559	(34)
Geometric Sums	559	(7)
Infinite Geometric Series	566	(6)
A More General Discussion of Infinite Series	572	(3)
Summation Notation	575	(4)
Applications of Geometric Sums and Series	579	(14)
PART VII Trigonometric Functions	593	(118)
Trigonometry: Introducing Periodic Functions	593	(34)
The Sine and Cosine Functions: Definitions and Basic Properties	594	(9)
Modifying the Graphs of Sine and Cosine	603	(12)
The Function f(x) = tan x	615	(4)
Angles and Are Lengths	619	(8)
Trigonometry--Circles and Triangles	627	(56)
Right-Triangle Trigonometry: The Definitions	627	(8)
Triangles We Know and Love, and the Information They Give Us	635	(10)
Inverse Trigonometric Functions	645	(6)
Solving Trigonometric Equations	651	(6)
Applying Trigonometry to a General Triangle: The Law of Cosines and the Law of Sines	657	(10)
Trigonometric Identities	667	(4)
A Brief Introduction to Vectors	671	(12)
Differentiation of Trigonometric Functions	683	(28)
Investigating the Derivative of sin x Graphically, Numerically, and Using Physical Intuition	683	(5)
Differentiating sin x and cos x	688	(7)
Applications	695	(8)
Derivatives of Inverse Trigonometric Functions	703	(4)
Brief Trigonometry Summary	707	(4)
PART VIII Integration: An Introduction	711	(72)
Net Change in Amount and Area: Introducing the Definite Integral	711	(32)
Finding Net Change in Amount: Physical and Graphical Interplay	711	(14)
The Definite Integral	725	(6)
The Definite Integral: Qualitative Analysis and Signed Area	731	(7)
Properties of the Definite Integral	738	(5)
The Area Function and Its Characteristics	743	(18)
An Introduction to the Area Function fxa f(t) dt	743	(4)
Characteristics of the Area Function	747	(10)
The Fundamental Theorem of Calculus	757	(4)
The Fundamental Theorem of Calculus	761	(22)
Definite Integrals and the Fundamental Theorem	761	(14)
The Average Value of a Function: An Application of the Definite Integral	775	(8)
PART IX Applications and Computation of the Integral	783	(136)
Finding Antiderivatives--An Introduction to Indefinite Integration	783	(22)
A List of Basic Antiderivatives	783	(4)
Substitution: The Chain Rule in Reverse	787	(11)
Substitution to Alter the Form of an Integral	798	(7)
Numerical Methods of Approximating Definite Integrals	805	(22)
Approximating Sums: Ln, Rn, Tn, and Mn	805	(15)
Simpson's Rule and Error Estimates	820	(7)
Applying the Definite Integral: Slice and Conquer	827	(26)
Finding ``Mass'' When Density Varies	827	(16)
Slicing to Find the Area Between Two Curves	843	(10)
More Applications of Integration	853	(24)
Computing Volumes	853	(12)
Are Length, Work, and Fluid Pressure: Additional Applications of the Definite Integral	865	(12)
Computing Integrals	877	(42)
Integration by Parts---The Product Rule in Reverse	877	(9)
Trigonometric Integrals and Trigonometric Substitution	886	(12)
Integration Using Partial Fractions	898	(5)
Improper Integrals	903	(16)
PART X Series	919	(132)
Series	919	(64)
Approximating a Function by a Polynomial	919	(15)
Error Analysis and Taylor's Theorem	934	(7)
Taylor Series	941	(11)
Working with Series and Power Series	952	(12)
Convergence Tests	964	(19)
Differential Equations	983	(68)
Introduction to Modeling with Differential Equations	983	(8)
Solutions to Differential Equations: An Introduction	991	(11)
Qualitative Analysis of Solutions to Autonomous Differential Equations	1002	(16)
Solving Separable First Order Differential Equations	1018	(6)
Systems of Differential Equations	1024	(21)
Second Order Homogeneous Differential Equations with Constant Coefficients	1045	(6)
Appendices	1051	(82)
Appendix A Algebra	1051	(34)
Introduction to Algebra: Expressions and Equations	1051	(5)
Working with Expressions	1056	(14)
Solving Equations	1070	(15)
Appendix B Geometric Formulas	1085	(2)
Appendix C The Theoretical Basis of Applications of the Derivative	1087	(8)
Appendix D Proof by Induction	1095	(4)
Appendix E Conic Sections	1099	(12)
Characterizing Conics from a Geometric Viewpoint	1100	(1)
Defining Conics Algebraically	1101	(5)
The Practical Importance of Conic Sections	1106	(5)
Appendix F L'Hopital's Rule: Using Relative Rates of Change to Evaluate Limits	1111	(10)
Indeterminate Forms	1111	(10)
Appendix G Newton's Method: Using Derivatives to Approximate Roots	1121	(6)
Appendix H Proofs to Accompany Chapter 30, Series	1127	(6)
Index	1133