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Math1210 Weekly Assignment 10 Spring, 2016 (Definite integrals - FFTC 4.2-4.3) instructor: Uid number: name: Instructions: Please show all of your work as partial credit will be given where appropriate, and there may be no credit given for problems where there is no work shown. All answers should be completely simplified, unless otherwise stated. No calculators or electronics of any kind are allowed. (Total of 60 points possible.) 1. (25 points) Given G(x), find G0 (x) using the First Fundamental Theorem of Calculus and the properties of the definite integral. x Z t2 dt (a) G(x) = 1 sin x Z t2 dt (b) G(x) = 1 2 Z 5 sin t dt (c) G(x) = x x2 Z Z 2t + 5 dt (d) G(x) = (e) G(x) = x Z x t dt 1 Z 0 Z −x2 x2 ] + = [Hint: −x2 −x2 2 x2 0 Page 2 2. (15 points) Suppose that an object is at the origin at time t = 0 and it moves according to the following velocity function √ 2 4−t v(t) = t − 2 3 if 0 6 t 6 2 if 2 6 t 6 5 if 5 6 t 6 7 (a) Draw a precise graph of v(t). (b) Determine the position of the object at time t = 7. [Hint: write the problem in terms of integrals and use the geometric interpretation of the definite integral to find your answer. Do not use the Second Fundamental Theorem.] Page 3 Page 4 3. (20 points) Let f (x) be a continuous (hence integrable) function on the interval [a, b]. Decide whether the following statements are true or false, and justify your answers. Z b f (x) dx ≥ 0. (a) If f (x) ≥ 0 on [a, b], then a Z b f (x) dx ≥ 0, then f (x) ≥ 0 on [a, b]. (b) If a Z b f (x) dx = 0, then f (x) = 0 on [a, b]. Z x f (t) dt is decreasing for a ≤ x ≤ b, then f (x) ≤ 0 on [a, b]. (d) If F (x) = (c) If a a Page 5