Calculus

by Varberg, Dale; Purcell, Edwin J.; Rigdon, Steven E.

• ISBN13: 9780130811370
• ISBN10: 0130811378
• Edition: 8th
• Format: Hardcover
• Copyright: 2000-01-01
• Publisher: PEARSON

Summary

This the shortest mainstream calculus book available. The authors make effective use of computing technology, graphics, and applications, and provide at least two technology projects per chapter. This popular book is correct without being excessively rigorous, up-to-date without being faddish.Maintains a strong geometric and conceptual focus. Emphasizes explanation rather than detailed proofs. Presents definitions consistently throughout to maintain a clear conceptual framework. Provides hundreds of new problems, including problems on approximations, functions defined by tables, and conceptual questions.Ideal for readers preparing for the AP Calculus exam or who want to brush up on their calculus with a no-nonsense, concisely written book.

Table of Contents

Preface	xiii
Preliminaries	1	(36)
The Real Number System	1	(5)
Decimals, Calculators, Estimation	6	(4)
Inequalities	10	(4)
Absolute Values, Square Roots, Squares	14	(5)
The Rectangular Coordinate System	19	(4)
The Straight Line	23	(6)
Graphs of Equations	29	(4)
Chapter Review	33	(4)
Graphing	35	(1)
Solving Equations by Zooming	36	(1)
Functions and Limits	37	(62)
Functions and Their Graphs	37	(6)
Operations on Functions	43	(6)
The Trigonometric Functions	49	(11)
Introduction to Limits	60	(5)
Rigorous Study of Limits	65	(7)
Limit Theorems	72	(5)
Limits Involving Trigonometric Functions	77	(4)
Limits at Infinity, Infinite Limits	81	(5)
Continuity of Functions	86	(7)
Chapter Review	93	(2)
Additional Problems	95	(4)
Shifting and Scaling the Graph of a Function	97	(1)
Limits	98	(1)
The Derivative	99	(62)
Two Problems with One Theme	99	(8)
The Derivative	107	(6)
Rules for Finding Derivatives	113	(7)
Derivatives of Trigonometric Functions	120	(3)
The Chain Rule	123	(5)
Leibniz Notation	128	(5)
Higher-Order Derivatives	133	(6)
Implicit Differentiation	139	(5)
Related Rates	144	(7)
Differentials and Approximations	151	(5)
Chapter Review	156	(2)
Additional Problems	158	(3)
Secant and Tangent Lines	160	(1)
Linear Approximation to a Function	160	(1)
Applications of the Derivative	161	(48)
Maxima and Minima	161	(7)
Monotonicity and Concavity	168	(6)
Local Maxima and Minima	174	(5)
More Max-Min Problems	179	(9)
Economic Applications	188	(4)
Sophisticated Graphing	192	(6)
The Mean Value Theorem	198	(4)
Chapter Review	202	(2)
Additional Problems	204	(5)
Reflection and Refraction of Light	206	(1)
An Optimization Problem	207	(2)
The Integral	209	(64)
Antiderivatives (Indefinite Integrals)	209	(6)
Introduction to Differential Equations	215	(6)
Sums and Sigma Notation	221	(6)
Introduction to Area	227	(7)
The Definite Integral	234	(8)
The First Fundamental Theorem of Calculus	242	(9)
The Second Fundamental Theorem of Calculus and the Mean Value Theorem for Integrals	251	(7)
Evaluating Definite Integrals	258	(8)
Chapter Review	266	(2)
Additional Problems	268	(5)
Riemann Sums	270	(1)
Accumulation Functions	271	(2)
Applications of the Integral	273	(46)
The Area of a Plane Region	273	(7)
Volumes of Solids: Slabs, Disks, Washers	280	(7)
Volumes of Solids of Revolution: Shells	287	(6)
Length of a Plane Curve	293	(7)
Work	300	(5)
Moments, Center of Mass	305	(7)
Chapter Review	312	(2)
Additional Problems	314	(5)
Volume in an Elliptical Cylinder	316	(1)
Arc Length	317	(2)
Transcendental Functions	319	(52)
The Natural Logarithm Function	319	(6)
Inverse Functions and Their Derivatives	325	(6)
The Natural Exponential Function	331	(5)
General Exponential and Logarithmic Functions	336	(5)
Exponential Growth and Decay	341	(6)
First-Order Linear Differential Equations	347	(4)
The Inverse Trigonometric Functions and Their Derivatives	351	(8)
The Hyperbolic Functions and Their Inverses	359	(6)
Chapter Review	365	(1)
Additional Problems	366	(5)
Special Functions	368	(1)
Population Growth and Least Squares	369	(2)
Techniques of Integration	371	(32)
Integration by Substitution	371	(6)
Some Trigonometric Integrals	377	(4)
Rationalizing Substitutions	381	(5)
Integration by Parts	386	(6)
Integration of Rational Functions	392	(6)
Chapter Review	398	(5)
Integration Using a Computer Algebra System	400	(1)
The Logistic Differential Equation	401	(2)
Indeterminate Forms and Improper Integrals	403	(26)
Indeterminate Forms of Type 0/0	403	(6)
Other Indeterminate Forms	409	(5)
Improper Integrals: Infinite Limits of Integration	414	(6)
Improper Integrals: Infinite Integrands	420	(5)
Chapter Review	425	(1)
Additional Problems	426	(3)
Probability Density Functions	427	(1)
The Normal Distribution	428	(1)
Infinite Series	429	(50)
Infinite Sequences	429	(6)
Infinite Series	435	(7)
Positive Series: The Integral Test	442	(5)
Positive Series: Other Tests	447	(6)
Alternating Series, Absolute Convergence, and Conditional Convergence	453	(5)
Power Series	458	(4)
Operations on Power Series	462	(5)
Taylor and Maclaurin Series	467	(8)
Chapter Review	475	(4)
Using Infinite Series to Approximate π	477	(1)
Euler's Derivation	478	(1)
Numerical Methods, Approximations	479	(38)
The Taylor Approximation to a Function	479	(8)
Numerical Integration	487	(7)
Solving Equations Numerically	494	(5)
The Fixed-Point Algorithm	499	(5)
Approximations for Differential Equations	504	(7)
Chapter Review	511	(6)
Maclaurin Polynomials	513	(1)
Numerical Integration	514	(1)
Bisection, Newton's, and Fixed-Point Methods	515	(2)
Conics and Polar Coordinates	517	(42)
The Parabola	517	(5)
Ellipses and Hyperbolas	522	(5)
More on Ellipses and Hyperbolas	527	(4)
Translation of Axes	531	(5)
Rotation of Axes	536	(3)
The Polar Coordinate System	539	(6)
Graphs of Polar Equations	545	(5)
Calculus in Polar Coordinates	550	(5)
Chapter Review	555	(4)
Rotations in the Plane	558	(1)
Another Kind of Rose	558	(1)
Geometry in the Plane, Vectors	559	(36)
Plane Curves: Parametric Representation	559	(8)
Vectors in the Plane: Geometric Approach	567	(4)
Vectors in the Plane: Algebraic Approach	571	(6)
Vector-Valued Functions and Curvilinear Motion	577	(5)
Curvature and Acceleration	582	(8)
Chapter Review	590	(5)
Hypocycloids	592	(2)
Measuring Home Run Distance	594	(1)
Geometry in Space, Vectors	595	(38)
Cartesian Coordinates in Three-Space	595	(4)
Vectors in Three-Space	599	(5)
The Cross Product	604	(5)
Lines and Curves in Three-Space	609	(4)
Velocity, Acceleration, and Curvature	613	(6)
Surfaces in Three-Space	619	(4)
Cylindrical and Spherical Coordinates	623	(5)
Chapter Review	628	(5)
Curves in Three-Space	630	(1)
The Ferris Wheel and the Corkscrew Roller Coaster	631	(2)
The Derivative in n-Space	633	(52)
Functions of Two or More Variables	633	(7)
Partial Derivatives	640	(5)
Limits and Continuity	645	(5)
Differentiability	650	(6)
Directional Derivatives and Gradients	656	(5)
The Chain Rule	661	(5)
Tangent Planes, Approximations	666	(4)
Maxima and Minima	670	(6)
Lagrange's Method	676	(5)
Chapter Review	681	(4)
Newton's Method for Two Equations in Two Unknowns	683	(1)
Visualizing the Directional Derivative	684	(1)
The Integral in n-Space	685	(46)
Double Integrals over Rectangles	685	(6)
Iterated Integrals	691	(4)
Double Integrals over Nonrectangular Regions	695	(6)
Double Integrals in Polar Coordinates	701	(5)
Applications of Double Integrals	706	(5)
Surface Area	711	(4)
Triple Integrals (Cartesian Coordinates)	715	(7)
Triple Integrals (Cylindrical and Spherical Coordinates)	722	(5)
Chapter Review	727	(4)
Newton's Law of Gravitation	728	(1)
Monte Carlo Integration	729	(2)
Vector Calculus	731	(42)
Vector Fields	731	(4)
Line Integrals	735	(6)
Independence of Path	741	(7)
Green's Theorem in the Plane	748	(6)
Surface Integrals	754	(5)
Gauss's Divergence Theorem	759	(6)
Stokes's Theorem	765	(4)
Chapter Review	769	(4)
Line Integrals and Work	770	(1)
Parametrized Surfaces	771	(2)
Differential Equations	773	(16)
Linear Homogeneous Equations	773	(5)
Nonhomogeneous Equations	778	(4)
Applications of Second-Order Equations	782	(4)
Chapter Review	786	(3)
Vibrating Spring	787	(1)
Phase Portraits	787	(2)
Appendix	789	(1)
A.1 Mathematical Induction	789	(3)
A.2 Proofs of Several Theorems	792	(3)
A.3 A Backward Look	795
Answers to Odd-Numbered Problems	A-1	(1)
Index	I-1	(1)
Photo Credits	P-1