Download 6 Problems with Answers on Calculus and Analytic Geometry - Exam 3 | MATH 221 and more Exams Analytical Geometry and Calculus in PDF only on Docsity! Math 221 – Exam III (50 Minutes) – Friday, April 16 Answers I. (20 points.) Write the formula for a Riemann sum S for a function f(x) on the interval a ≤ x ≤ b and explain in what sense it approximates the definite integral ∫ b a f(x) dx. Answer: A Riemann sum or a function f(x) on the interval a ≤ x ≤ b is an expression of form S = n∑ k=1 f(x̄k)(xk − xk−1) where a = x0 ≤ x̄1 ≤ x1 ≤ x̄2 ≤ x2 ≤ · · · ≤ xn−1 ≤ x̄n ≤ xn = b. It approximates the definite integral in the sense that S ≈ ∫ b a f(x) dx when all the xk − xk−1 are small. II. (30 points.) (a) Find ∫ 2 −1 (3x−2 − 2x10 + 3) dx. Answer: This is Problem 3 on page 256. (b) Find ∫ 3 1 √ 7 + 2t2(8t) dt. Answer: This is Problem 21 on page 256. III. (25 points.) Find y as a function of x if dy dx = −y3(x2 + 2)2 and y = 1 when x = 0. Answer: This is Problem 14 on page 220. 1 IV. (25 points.) (a) Find 5∑ n=1 n cos(nπ). (An exact integer answer is required here.) Answer: This is problem 7 page 226 (b) Write the sum 19∑ i=3 i(i−2) in sigma notation with k = i−2 as the dummy variable. Answer: This is problem 31 p 226. V. (25 points.) Use Riemann sums with four intervals of length one to find positive numbers m and M with 0 < m ≤ ∫ 5 1 ( 3 + 1 x ) dx ≤ M. Answer: This is Problem 35 page 250. VI. (25 points.) Starting from rest, a train increases speed at constant acceleration a1, then travels at a constant speed vm, and finally brakes to a stop at constant de-acceleration −a2. It took 7 minutes to travel 3 miles from Addison to Howard and 3 minutes to travel 1 mile from Howard to Dempster. The train spent one minute in the Howard station. (a) Sketch the graph of the speed v as a function of time t for 0 ≤ t ≤ 11. (The time t = 0 corresponds to the moment when the train leaves Addison.) (b) Find the maximum speed vm and the accelerations a1 and a2 if the train takes 1/2 minute to accelerate to its maximum speed and 1/3 of minute to decelerate from the maximum speed to rest. Show your reasoning.) Answer: this is like Problem Problem 33 page 221. 2
