Spring 2007 Math 151 Common Exam 3 Test Form A PRINT Last Name First Name Signature ID Instructor s Name Section INSTRUCTIONS In Part 1 Problems 1 12 mark the correct choice on your ScanTron form using a 2 pencil For your own records also record your choices on your exam The ScanTrons will be collected after 1 hour they will NOT be returned In Part 2 Problems 13 18 write all solutions in the space provided CLEARLY INDICATE YOUR FINAL ANSWERS No Calculators Permitted 1 Math 151 Multiple Choice 4 points each 1 Which graph illustrates a local maximum which is also an inflection point a b d e c 2 Consider the function f x 3x4 8x3 5 Find the local minima of f x 4 b x 0 and x 3 4 d x 2 e x 3 a x 0 and x 2 c x 1 3 Find the derivative of f x tan 1 ln x 1 1 1 0 1 0 a f x tan b f x x x ln x 2 1 1 x c f 0 x sec 2 ln x d f 0 x 2 x x 1 1 1 e f 0 x tan 1 x x2 1 2 Part I 4 Consider the graph of the function f x 10x6 24x5 15x4 Find the x coordinates of the inflection points a x 0 b x 0 and x 1 c x 1 3 3 d x 1 and x e x 0 x and x 1 5 5 5 50 3i 7 P 5 i 1 a 415 b 40 c 143 5 d 23 e 695 6 Find the region where the function f x 2x5 5x4 10x3 is increasing a x 3 b 1 x 3 d x 1 or 0 x 3 c x 1 or x 3 e 1 x 0 or x 3 3 1 cos3 x x 0 sin2 x sin 2x 7 lim a does not exist b 3 4 c 0 d 3 2 e 1 8 Find the derivative of f x xln x a f 0 x 1 ln x x x c f 0 x x1 x e f 0 x x2 x ln x ln x x x ln x ln x x d f 0 x x b f 0 x 2 4 1 9 cos sin 5 a 3 5 b 3 5 c 3 4 d 3 4 e does not exist 4 10 e3 ln 2 1 ln 5e2 8 e3 ln 5 2 b ln 5 2 e ln 2 1 d 2 8 ln 5 e e3 ln 2 e a c e3 ln 5 1 ln 5 2e e 1 11 Which function is an anti derivative of 1 x2 a 2 1 x2 b tan 1 x c x 1 x2 3 2 d sin 1 x e x2 1 1 12 Find the graph of the consistently increasing function whose derivative is consistently decreasing a b c d e 5 Math 151 Work Out Problems Show your work No credit for unsupported answers will be given Part II 13 The radioactive decay law states that the rate of decrease in the amount of a radioactive isotope is always proportional to the amount remaining The half life of an isotope is the length of time over which the amount of the isotope is reduced by half through radioactive decay Now suppose the initial amount of a given isotope is 7 grams and that 4 grams remain after 15 ln 7 30 ln 2 years a Find the formula for how the amount of isotope depends on time 4 points b Find the half life of the isotope No decimal approximation is required If you obtain an answer like 2e e2 ln then leave it alone 3 points 14 Calculate the following limits 3 points each tan 1 x x 0 sin x a lim ln x b lim x 3 x 6 c lim 1 2x 3 x x 0 15 Consider the function f x x and partition the interval 1 x 4 into 6 equal sub intervals Calculate the Riemann sum for the points x 1 1 x 4 25 9 9 16 x 3 9 4 49 x 5 x 6 4 16 x 2 of evaluation 10 points 16 Fencing is required to enclose a rectangular field with 7500 square feet of area The north side requires higher quality fencing than the other sides If the regular fencing costs 300 per foot and the higher quality fencing costs 500 per foot how long must the north side be in order to minimize the cost 10 points 7 17 A car is accelerating along a straight highway where the acceleration itself is increasing linearly specifically after t seconds the acceleration is given by a t 30t 8 in feet per second per second a Find the velocity v t as a function of t if the initial velocity is v 0 44 ft sec 5 points b Find the position x t as a function of t if the initial position is 100 feet down the road 5 points 2x 1 3 x 2 18 Differentiate the function f x 3 6 points 5x 1 3x 5 2 unwise 8 Hint The direct approach is
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