Calculus Preliminary Edition Robert Decker UNIVERSITY OF HARTFORD Dale Varberg HAML1NE UNIVERSITY Prentice Hall, Upper Saddle River, New Jersey 07458
Annotated Table of Contents.For your convenience, we have annotated this table of contents in order to highlight some specific areas of the text where traditional presentation has been rejuvenated with what we believe to be new and innovative approaches. Preface xi 0 Preliminaries The properties and graphs of linear, polynomial and trigonometric functions are presented, and some simple mathematical models involving these functions are introduced. Examples are given of numerical and graphical and algebraic approaches to problem solving in settings that do not require the use of sophisticated technology. 0.1 Graphs and Equations 1 0.2 Functions 9 In Example 2, a quadratic mathematical model of a water balloon is introduced and approached numerically, graphically and algebraically. 0.3 The Straight Line and Linear Functions 19 - Example 4 involves developing d linear model for an automobile trip; numerical, graphical and algebraic methods are used to answer some questions. 0.4 The Trigonometric Functions 27 Example 5 demonstrates how graphical/numerical solutions to equations involving trigonometric functions are related to algebraic solutions involving inverse trigonometric functions. 0.5 Chapter Review 38 ~ "^. Calculus: A First Look 41 Limits, derivatives and the exponential and logarithmic functions are introduced and explored. Numerical estimation of limits and derivatives is emphasized. 1.1 Introduction to Limits: Part I 41 This section introduces a method of estimating limits numerically and of determining the accuracy of the estimate. Lab 1: Exploring Limits 50 1.2 Introduction to Limits: Part II 52 Limits involving infinity are approached numerically. Continuity is introduced and, in Example 4, discussed in the context of a practical mathematical model of the voltage supplied by a battery. 1.3 The Derivative: Two Problems with One Theme 60 Derivatives and instantaneous velocity are estimated numerically, and some simple derivative formulas are found. 1.4 Exponential Functions 70 The natural exponential function is developed as the exponential function whose slope and y-coordinate are the same at each point.
Vlii Annotated Table of Contents Lab 9: Area and Distance 308 5.4 The Definite Integral and Numerical Integration 310 The definite integral is defined and explored numerically using both midpoint Riemann sums and the built-in numerical methods of calculators and computers. Lab 10: Area Functions and the Fundamental Theorem of Calculus 319 5.5 The Fundamental Theorem of Calculus 321 Exact definite integrals are found by hand and with the use of computer algebra. 5.6 More Properties of the Definite Integral 329 5.7 Substitution 339 5.8 Chapter Review 345 Applications of the Integral - - 349 The definite integral is applied to numerous situations. 6.1 The Area of a Plane Region 349 6.2 Volumes of Solids: Slabs, Disks, Washers 358 6.3 Length of a Plane Curve 366 Parametric representation of curves is introduced, including calculator and computer graphing of such curves. In the lab project at,the end of the section, a problem involving a guitar string is used to show when it is helpful to be able to evaluate a definite integral exactly, and when a numerical approximation is necessary. Lab 11: Arc Length 376 6.4 Work 378 6.5 Moments; Center of Mass 384 6.6 Chapter Review 392 Transcendental Functions and Differential Equations 395 The exponential, logarithmic and trigonometric functions are explored in greater detail, as well as the relationship between exponential functions and differential equations. 1.1 Inverse Functions and Their Derivatives 395 7.2 A Different Approach to Logarithmic and Exponential Functions 402 This section shows that by defining the natural log function using a definite integral, and the natural exponential function as its inverse, it now becomes relatively easy to demonstrate several properties of the log and exponential functions that were previously left unproven. In the lab at the end, a model of falling objects that encounter air resistance is developed andfitto real data taken of a falling balloon. Lab 12: Falling Objects 411 7.3 General Exponential and Logarithmic Functions 413 7.4 Exponential Growth and Decay 419 This section explores the relationship of the exponential function to certainfirst-orderdifferential equations. 7.5 Numerical and Graphical Approaches to Differential Equations 426 Euler's method of numerically solving differential equations is presented, and the numerical methods built in to some calculators and computer algebra systems are investigated. In the lab at the end of the section, advantages and disadvantages of exact versus numerical solutions are discovered in the context of retirement planning. Lab 13: Planning Your Retirement Numerically 433 7.6 The Trigonometric Functions and Their Inverses 435 7.7 The Hyperbolic Functions and Their Inverses 445 7.8 Chapter Review 451
Annotated Table of Contents IX 8 Techniques of Integration 453 Several methods offindingindefinite and definite integrals are presented and applied to variety of situations. 8.1 Substitution and Tables < of Integrals 453 8.2 Integration by Parts 462 Lab 14: Integration by Parts 472 8.3 Some Trigonometric Integrals 473 8.4 Integration of Rational Functions 481 Simple partial fraction decomposition (both by hand and with computer algebra) is presented. In Example 4, the logistic population model of population growth is developed by solving the appropriate differential equation and is further explored in the lab at the end of the section. Lab 15: Population Models 488 8.5 Indeterminate Forms 490 L'Hopital 's Rule is presented. 8.6 Improper Integrals 496 In the lab at the end of the section, a possible numerical approach to improper integrals is discovered and applied to the problem of estimating the heights of the tallest man and woman in the United States. Lab 16: Probability and Improper Integrals 507 8.7 Chapter Review " 509 9 Infinite Series 513 Infinite sequences and infinite series of constants and functions are presented and applied to a variety of situations. Both exact and numerical methods are presented and compared. 9.1 Infinite Sequences and Dynamical Systems 513 Explicitly and recursively defined sequences are given equal emphasis. The lab at the end introduces a discrete model of population growth, which parallels the continuous model from Lab i5 and leads to the study of chaotic systems. Lab 17: Discrete Population Models 525 9.2 Infinite Series 527 Lab 18: Bouncing Balls and Infinite Series 536 9.3 Alternating Series and Absolute Convergence 538 9.4 Taylor's Approximation to Functions 544 Lab 19: Taylor Series and Fourier Series 551 9.5 Applications of Taylor's Formula 553 This section uses order-of-convergence arguments to introduce Newton's method offindingroots and Simpson's method of numerical integration. Lab 20: Newton's Method 562 9.6 General Power Series 563 Both by-hand and computer algebra methods of finding power series are discussed. Lab 21: The Gamma Function and Taylor Series Approximations 573 9.7 Representing Functions with Power Series 575 9.8 Chapter Review 583 10 Conies, Polar Coordinates and Parametric Curves 587 Parabolas, ellipses and hyperbolas are defined and studied. Polar coordinate and parametric representations of these and other curves are explored.
X Annotated Table of Contents 10.1 Conic Sections 587 10.2 Translation of Axes 597 10.3 The Polar Coordinate - System and Graphs 602 Lab 22: Graphs in Polar Coordinates 609 10.4 Technology and Graphs of Polar Equations 611 This section explores advantages and pitfalls of using calculators and computers to generate polar graphs. Calculator/computer graphs are used to study more complicated polar graphs. 10.5 Calculus in Polar Coordinates 617 This section develops area and arclength formulas for curves given in polar coordinates and then applied to the motion of Earth in the lab at the end of the section. Lab 23: Orbits of the Planets 624 10.6 Plane Curves: Parametric Representation 627 By-hand as well as calculator/computer-based methods of graphs of parametric equations are studied. 10.7 Chapter Review 636