Single Variable Essential Calculus: Early Transcendentals
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This book is a response to those instructors who feel that calculus textbooks are too big. In writing the book James Stewart asked himself:What is essential for a three-semester calculus course for scientists and engineers? Stewart's SINGLE VARIABLE ESSENTIAL CALCULUS: EARLY TRANSCENDENTALS offers a concise approach to teaching calculus, focusing on major concepts and supporting those with precise definitions, patient explanations, and carefully graded problems. SINGLE VARIABLE ESSENTIAL CALCULUS: EARLY TRANSCENDENTALS is only 850 pages-two-thirds the size of Stewart's other calculus texts (CALCULUS, FIFTH EDITION AND CALCULUS, EARLY TRANSCENDENTALS, Fifth Edition)-yet it contains almost all of the same topics. The author achieved this relative brevity mainly by condensing the exposition and by putting some of the features on the website www.StewartCalculus.com. Despite the reduced size of the book, there is still a modern flavor: Conceptual understanding and technology are not neglected, though they are not as prominent as in Stewart's other books. SINGLE VARIABLE ESSENTIAL CALCULUS: EARLY TRANSCENDENTALS has been written with the same attention to detail, eye for innovation, and meticulous accuracy that have made Stewart's textbooks the best-selling calculus texts in the world.
(d) Shift 3 units to the left. (e) Reﬂect about the x-axis. (f) Reﬂect about the y-axis. (g) Stretch vertically by a factor of 3. (h) Shrink vertically by a factor of 3. x 1 38. f ͑x͒ s3 Ϫ x , t͑x͒ 3x 2 Ϫ 1 t͑x͒ sx 2 Ϫ 1 Unless otherwise noted, all content on this page is © Cengage Learning. SECTION 1.2 ■ Find the functions (a) f ؠt, (b) t ؠf , (c) f ؠf , and (d) t ؠt and their domains. 39–44 39. f ͑x͒ x 2 Ϫ 1, 1 , x t͑x͒ xϩ1 xϩ2 balloon is increasing at a rate
the volume of a sphere varies continuously with its radius because the formula V͑r͒ 43 r 3 shows that V is a polynomial function of r. Likewise, if a ball is thrown vertically into the air with a velocity of 50 ft͞s, then the height of the ball in feet t seconds later is given by the formula h 50t Ϫ 16t 2. Again this is a polynomial function, so the height is a continuous function of the elapsed time. Knowledge of which functions are continuous enables us to evaluate some limits very
Ϫϱ means that for every negative number N there is a positive number ␦ such that if 0 Ͻ x Ϫ a Ͻ ␦, then f ͑x͒ Ͻ N. Deﬁnition 3 can be stated precisely as follows. Խ 7 DEFINITION Խ Let f be a function deﬁned on some interval ͑a, ϱ͒. Then lim f ͑x͒ L xlϱ means that for every Ͼ 0 there is a corresponding number N such that xϾN if TEC Module 1.3/1.6 illustrates Definition 7 graphically and numerically. then Խ f ͑x͒ Ϫ L Խ Ͻ In words, this says that the values of f ͑x͒ can be made
͑21͒2 0.05 Ϸ 277 The maximum error in the calculated volume is about 277 cm3. ■ NOTE Although the possible error in Example 3 may appear to be rather large, a better picture of the error is given by the relative error, which is computed by dividing the error by the total volume: ⌬V dV 4r 2 dr dr Ϸ 4 3 3 V V r r 3 Therefore the relative error in the volume is approximately three times the relative error in the radius. In Example 3 the relative error in the radius is approximately dr͞r
slope of 1, it follows that the reflected curve y ln x crosses the x-axis with a slope of 1. In common with all other logarithmic functions with base greater than 1, the natural logarithm is a continuous, increasing function defined on ͑0, ϱ͒ and the y-axis is a vertical asymptote. If we put a e in 11 , then we have the following limits: 15 FIGURE 15 V EXAMPLE 13 lim ln x ϱ xlϱ lim ln x Ϫϱ x l0ϩ Sketch the graph of the function y ln͑x Ϫ 2͒ Ϫ 1. SOLUTION We start with the graph