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Calculus I – Review for Test 4 Show all work in the blue book. No more than one problem per page. 1. Definitions and important theorems... a) definition of an antiderivative b) definition of a definite integral c) Fundamental Theorem of Calculus - both parts d) Definition of a Riemann Sum of f(x) on [a,b] for a partition P. 2. For the function f x =x 3−8 on the interval [0, 6] a. Compute the Riemann Sum using the following partition: subintervals are [0, 3] [3, 5], and [5, 6], and w1=1, w2=3, and w3=5. b. Compute the definite integral of f(x) from 0 to 6. c. Compute the exact area enclosed between the graph of f(x) and the x-axis on [0, 6]. Examine the graph of f(x) for a hint on how to proceed. 3 The acceleration of a particle at time = t is given by the formula a=6 t−4 . If it is known that v = 11 and s = 10 when t = 1: a. find formulas for velocity (v) and position (s) of the particle. b. find the speed, direction of motion, and postion of the particle at t = 4. 4. Compute the following integrals: 2 a. 1 b. ∫ 3 x x 3 − cos2 x dx ∫ x sin x 27 dx d. ∫ 4 x 2 x 3−7 dx f. ∫ sin x cos x dx 4 c. 5 −∫ x2 dx 2 e. 12 x 2 dx ∫ x 3−52 1 5. For f(x) = x2 - 12, use two iterations of Newton´s Method to estimate a zero of this function. Use an initial value x0 = 6, and determine x1 and x2. Show all work! 6. Use a change of variables: ∫ x−1 x33 dx