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C4_C5_Review - Sample Problems for Chapters 4 and...

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Sample Problems for Chapters 4 and 5 (Final-Exam Review) MAC2311, Section 17, Instructor: Ms. Hoa Nguyen ([email protected]) Problem 1 ( similar to 8 of Section 4.1 ) Find the critical numbers of the function f ( x ) = x 2 - 9. Then find the domain of this function. Problem 2 ( similar to 13 of Section 4.4 ) Evaluate the limit: lim x →-∞ xe x . Problem 3 ( similar to 16 of Section 4.4 ) Evaluate the limit: lim x →∞ ( x 2 e 5 x 2 - x 2 ). Problem 4 ( 7 of Make-up Test 3 ) Evaluate the limit: lim x 0 + x x . Problem 5 ( similar to 11 of Section 4.9 ) Find f ( t ) given that f ( t ) = 4 cos t + sin t and f (0) = 0. Problem 6 ( similar to 7 of Section 5.1 ) The area A of the region S that lies under the graph of the continuous function f ( x ) from a to b is the limit of the sum of the areas of approximating rectangles: A = lim n →∞ n i =1 f ( x i ) x a) Use this definition to find an expression for the area under the curve y = x 2 from 0 to 1 as a limit. b) Evaluate the limit in part (a) by using the following formula for the sum of the squares of the first n integers: 1 2 + 2 2 + · · · + n 2 = n ( n + 1)(2 n + 1) 6 c) Evaluate the limit in part (a) by computing the definite integral: A = lim n →∞ n i =1 f
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