Final Exam Math 220 May 12 2007 Instructions This exam is worth 200 points Put one problem on each answer sheet use the back if necessary and put your name your TA s name your section number and the problem number on each page Only sign the honor pledge on the rst sheet Show all of your work and justify your answers Scienti c but not graphing calculators are allowed 2 For the function f x y z exy z2 x y 2 compute the following 1 Compute the following cid 19 cid 18 ex 2 x2 cid 2 ln t t 1 e 2 cid 3 a b d dx d dt a b f x f z 3 Compute the following integrals Z cid 18 1 x2 2 x 1 cid 19 dx cid 0 3x2 e2x cid 1 dx a b Z 2 0 4 a A certain radioactive isotope is undergoing exponential decay with decay constant 0 05 Find the half life of the isotope b Elizabeth invests 5 000 in an account gaining 6 annual interest compounded contin uously i How many years will it take until the balance in the account is 7 500 ii At what rate will the balance be growing at that time Turn over the page for the remainder of the questions 1 5 a Let h x x3 12x 4 h x is decreasing maximum or neither i Use calculus to nd all intervals where h x is increasing and all intervals where ii Give any critical points of h x and classify each as a relative minimum a relative b Find the equation of the line tangent to the graph of g x 3x2 2x ln x at x 1 6 Samuel wants to build a rectangular garden in his back yard The out side fence of the garden will be wood that costs 2 per foot In addition Samuel wants a brick wall down the center of his garden which will cost 6 per foot see the gure to the right Samuel has 800 to spend on the fence and wall nd the maximum total area of Samuel s garden Tell how you know that your answer is the maxi mum area 7 a Approximate the area under the curve k x ln x from x 1 to x 3 using a left hand b Sketch the bounded region enclosed by the graphs of y x2 20 and y x2 12 and Riemann sum with n 4 compute its area 8 A ball is thrown straight up into the air with velocity v t 32t 128 feet per second where t is measured in seconds a Find the value of t for which the height is maximized b Give a physical interpretation for v t dt and compute its value Z 4 1 9 Let f x y 2x4 8xy 2y4 a Use the rst derivative test to locate all of the points where f x y could have a relative minimum or relative maximum b Use the second derivative test to classify each point from part a as a relative minimum a relative maximum or a saddle point The following formula may help you apply the second derivative test D x y 2f x2 2f y2 cid 18 2f x y cid 19 2 10 Use the method of Lagrange multipliers to nd the minimum value of the function f x y y2 x2 subject to the constraint y 3 2x 2
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