# Single Variable Calculus: Concepts and Contexts (Stewart's by James Stewart

By James Stewart

Stewart's CALCULUS: ideas AND CONTEXTS, FOURTH variation bargains a streamlined method of educating calculus, concentrating on significant thoughts and helping people with certain definitions, sufferer factors, and thoroughly graded difficulties. CALCULUS: techniques AND CONTEXTS is extremely appeared simply because this article deals a stability of thought and conceptual paintings to fulfill extra innovative courses in addition to those people who are more well-off instructing in a extra conventional model. each one name is only one part in a entire calculus path software that rigorously integrates and coordinates print, media, and expertise items for profitable instructing and studying.

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1 2 FIGURE 11 1 4 1 8 1 16 Of course, we know that the man can actually reach the wall, so this suggests that perhaps the total distance can be expressed as the sum of infinitely many smaller distances as follows: 3 1෇ 1 1 1 1 1 ϩ ϩ ϩ ϩ иии ϩ n ϩ иии 2 4 8 16 2 A PREVIEW OF CALCULUS 9 Zeno was arguing that it doesn’t make sense to add infinitely many numbers together. But there are other situations in which we implicitly use infinite sums. 3333 . . d1 d2 d3 d4 . . ෇ d1 d2 d3 dn ϩ 2 ϩ 3 ϩ иии ϩ n ϩ иии 10 10 10 10 Therefore some infinite sums, or infinite series as they are called, have a meaning.

5 ft͞s Similarly, the average velocity in the time interval 2 ഛ t ഛ 3 is average velocity ෇ 24 Ϫ 9 ෇ 15 ft͞s 3Ϫ2 We have the feeling that the velocity at the instant t ෇ 2 can’t be much different from the average velocity during a short time interval starting at t ෇ 2. 2 The average velocities over successively smaller intervals appear to be getting closer to a number near 10, and so we expect that the velocity at exactly t ෇ 2 is about 10 ft͞s. In Chapter 2 we will define the instantaneous velocity of a moving object as the limiting value of the average velocities over smaller and smaller time intervals.

Power Functions A function of the form f ͑x͒ ෇ x a, where a is a constant, is called a power function. We consider several cases. 2 31 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS (i) a ෇ n, where n is a positive integer The graphs of f ͑x͒ ෇ x n for n ෇ 1, 2, 3, 4, and 5 are shown in Figure 11.