By James Stewart
Luck on your calculus path starts off right here! James Stewart's CALCULUS: EARLY TRANSCENDENTALS texts are world-wide best-sellers for a cause: they're transparent, exact, and packed with appropriate, real-world examples. With unmarried VARIABLE CALCULUS: EARLY TRANSCENDENTALS, 7th variation, Stewart conveys not just the application of calculus that can assist you increase technical competence, but additionally offers an appreciation for the intrinsic great thing about the topic. His sufferer examples and integrated studying aids may help you construct your mathematical self assurance and accomplish your targets within the direction!
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Additional info for Single Variable Calculus: Vol. 2, Early Transcendentals
1 that y x 2 dx 13 . So 0 y 1 0 1 1 ͑4 ϩ 3x 2 ͒ dx y 4 dx ϩ 3 y x 2 dx 0 0 4ϩ3и 5 1 3 The next property tells us how to combine integrals of the same function over adjacent intervals: y y=ƒ 5. y c a 0 a FIGURE 15 c b x b b f ͑x͒ dx ϩ y f ͑x͒ dx y f ͑x͒ dx c a This is not easy to prove in general, but for the case where f ͑x͒ ജ 0 and a Ͻ c Ͻ b Property 5 can be seen from the geometric interpretation in Figure 15: The area under y f ͑x͒ from a to c plus the area from c to b is equal to the total area from a to b.
A 8. If m ഛ f ͑x͒ ഛ M for a ഛ x ഛ b, then b m͑b Ϫ a͒ ഛ y f ͑x͒ dx ഛ M͑b Ϫ a͒ a y M y=ƒ m 0 a FIGURE 16 b x If f ͑x͒ ജ 0, then xab f ͑x͒ dx represents the area under the graph of f , so the geometric interpretation of Property 6 is simply that areas are positive. ) Property 7 says that a bigger function has a bigger integral. It follows from Properties 6 and 4 because f Ϫ t ജ 0. Property 8 is illustrated by Figure 16 for the case where f ͑x͒ ജ 0. If f is continuous we could take m and M to be the absolute minimum and maximum values of f on the interval ͓a, b͔.
B) The following formula for the sum of the cubes of the first n integers is proved in Appendix E. Use it to evaluate the limit in part (a). 13 ϩ 2 3 ϩ 3 3 ϩ и и и ϩ n 3 27. 1 that a limit of the form n 1 lim ͚ f ͑x*͒ ⌬x lim ͓ f ͑x *͒ ⌬x ϩ f ͑x *͒ ⌬x ϩ и и и ϩ f ͑x *͒ ⌬x͔ n l ϱ i1 i nlϱ 1 2 n arises when we compute an area. We also saw that it arises when we try to find the distance traveled by an object. It turns out that this same type of limit occurs in a wide variety of situations even when f is not necessarily a positive function.