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Success in your calculus course starts here! James Stewart's CALCULUS texts are world-wide best-sellers for a reason: they are clear, accurate, and filled with relevant, real-world examples. With CALCULUS, Eighth Edition, Stewart conveys not only the utility of calculus to help you develop technical competence, but also gives you an appreciation for the intrinsic beauty of the subject. His patient examples and built-in learning aids will help you build your mathematical confidence and achieve your goals in the course!
functions and have period 2. This means that, for all values of x, sin͑x ϩ 2͒ sin x cos͑x ϩ 2͒ cos x The periodic nature of these functions makes them suitable for modeling repetitive phenomena such as tides, vibrating springs, and sound waves. For instance, in Example 4 in Section 1.3 we will see that a reasonable model for the number of hours of daylight in Philadelphia t days after January 1 is given by the function ͫ L͑t͒ 12 ϩ 2.8 sin The tangent function is related to the sine
can estimate easily that the limit is about 16 . But if we zoom in too much, as in parts (c) and (d), then we get inaccurate graphs, again because of problems with subtraction. 0.2 0.2 0.1 0.1 (a) ͓_5, 5͔ by ͓_0.1, 0.3͔ FIGURE 5 (b) ͓_0.1, 0.1͔ by ͓_0.1, 0.3͔ (c) ͓_10–^, 10–^͔ by ͓_0.1, 0.3͔ (d) ͓_10–&, 10–& ͔ by ͓_0.1, 0.3͔ SECTION 2.2 THE LIMIT OF A FUNCTION V EXAMPLE 3 Guess the value of lim xl0 |||| 69 sin x . x SOLUTION The function f ͑x͒ ͑sin x͒͞x is not defined when x
in 1642, the year of Galileo’s death. When he entered Cambridge University in 1661 Newton didn’t know much mathematics, but he learned quickly by reading Euclid and Descartes and by attending the lectures of Isaac Barrow. Cambridge was closed because of the plague in 1665 and 1666, and Newton returned home to reflect on what he had learned. Those two years were amazingly productive for at that time he made four of his major discoveries: (1) his representation of functions as sums of infinite
b, then lim f ͑t͑x͒͒ f ͑b͒. x la In other words, ( x la ) lim f ͑t͑x͒͒ f lim t͑x͒ xla xla Intuitively, Theorem 8 is reasonable because if x is close to a, then t͑x͒ is close to b, and since f is continuous at b, if t͑x͒ is close to b, then f ͑t͑x͒͒ is close to f ͑b͒. A proof of Theorem 8 is given in Appendix F. n Let’s now apply Theorem 8 in the special case where f ͑x͒ s x , with n being a positive integer. Then n f ͑t͑x͒͒ s t͑x͒ ( ) n f lim t͑x͒ s lim t͑x͒ and xla xla
we have f Ј͑x͒ 12 ͑25 Ϫ x 2 ͒Ϫ1͞2 d ͑25 Ϫ x 2 ͒ dx 12 ͑25 Ϫ x 2 ͒Ϫ1͞2͑Ϫ2x͒ Ϫ Example 1 illustrates that even when it is possible to solve an equation explicitly for y in terms of x, it may be easier to use implicit differentiation. N So f Ј͑3͒ Ϫ x s25 Ϫ x 2 3 3 Ϫ 4 s25 Ϫ 3 2 and, as in Solution 1, an equation of the tangent is 3x ϩ 4y 25. M NOTE 1 The expression dy͞dx Ϫx͞y in Solution 1 gives the derivative in terms of both x and y. It is correct no matter which function y