Thomas' Calculus: Early Transcendentals (13th Edition)
George B. Thomas Jr., Maurice D. Weir
Format: PDF / Kindle (mobi) / ePub
This text is designed for a three-semester or four-quarter calculus course (math, engineering, and science majors).
Thomas’ Calculus: Early Transcendentals, Thirteenth Edition, introduces readers to the intrinsic beauty of calculus and the power of its applications. For more than half a century, this text has been revered for its clear and precise explanations, thoughtfully chosen examples, superior figures, and time-tested exercise sets. With this new edition, the exercises were refined, updated, and expanded—always with the goal of developing technical competence while furthering readers’ appreciation of the subject. Co-authors Hass and Weir have made it their passion to improve the text in keeping with the shifts in both the preparation and ambitions of today's learners.
a function at x0 leads to division by zero, which is undefined. We encountered this last circumstance when seeking the instantaneous rate of change in y by considering the quotient function ¢y>h for h closer and closer to zero. Here’s a specific example where we explore numerically how a function behaves near a particular point at which we cannot directly evaluate the function. y 2 2 y ϭ f (x) ϭ x Ϫ 1 xϪ 1 1 –1 0 1 x EXAMPLE 1 How does the function y ƒsxd = x2 - 1 x - 1 behave near x
conclusion in part (a) by graphing g near u0 = 0 . 74. Let Gstd = s1 - cos td>t 2 . a. Make tables of values of G at values of t that approach t0 = 0 from above and below. Then estimate limt:0 Gstd . b. Support your conclusion in part (a) by graphing G near t0 = 0 . 75. Let ƒsxd = x 1>s1 - xd . a. Make tables of values of ƒ at values of x that approach x0 = 1 from above and below. Does ƒ appear to have a limit as x : 1 ? If so, what is it? If not, why not? b. Support your conclusions in part (a)
to x = 2, and x2 + x - 6 5 = lim ƒsxd = . 2 4 x:2 x:2 x - 4 The graph of ƒ is shown in Figure 2.45. The continuous extension F has the same graph except with no hole at (2, 5>4). Effectively, F is the function ƒ with its point of discontinuity at x = 2 removed. lim FIGURE 2.45 (a) The graph of ƒ(x) and (b) the graph of its continuous extension F(x) (Example 10). Intermediate Value Theorem for Continuous Functions Functions that are continuous on intervals have properties that make them
increasingly steep (Figure 3.2). We see this situation again as a : 0 - . As a moves away from the origin in either direction, the slope approaches 0 and the tangent levels off to become horizontal. Rates of Change: Derivative at a Point The expression ƒsx0 + hd - ƒsx0 d , h h Z 0 is called the difference quotient of ƒ at x0 with increment h. If the difference quotient has a limit as h approaches zero, that limit is given a special name and notation. DEFINITION The derivative of a function ƒ
University of Montevallo Bobby Winters, Pittsburg State University Dennis Wortman, University of Massachusetts—Boston 7001_AWLThomas_ch01p001-057.qxd 10/1/09 2:23 PM Page 1 FPO 1 FUNCTIONS OVERVIEW Functions are fundamental to the study of calculus. In this chapter we review what functions are and how they are pictured as graphs, how they are combined and transformed, and ways they can be classified. We review the trigonometric functions, and we discuss misrepresentations that can occur when