Math 132 Fall 2003 Final Exam Name ID No calculators with a CAS are allowed Be sure your calculator is set for radians not degrees if you do any calculus computations with trig functions For Parts I II and III please mark your answer on the answer card Part I Multiple Choice 5 points problem 1 Solve the initial value problem dy 4x3 y dx A y 6ex F y x4 4 B y e7x G y x7 4 y 0 6 C y ex 4 H y 4x7 1 D y 4ex 7 I y 4xe7x E y 7x4 J No Solution 2 Find the radius of convergence and the interval of convergence of the series X n x 1 n bn n 1 A R 0 I 1 B R 1 I 1 1 C R 1 I 1 1 D R 1 I 1 1 E R 1 I 1 1 F R b I 1 b 1 b G R b I 1 b 1 b H R b I 1 b 1 b I R b J R I 1 b 1 b I 2 b 0 3 Find the volume of the following solid The base is the region in the xy plane bounded by the curves x 3y 3 x 3y 3 and x 0 Cross sections perpendicular to the x axis are isosceles right triangles with one side on the base Answers are in terms of units3 A 1 B 1 3 C 4 D 3 2 E 4 3 F 3 G 2 3 H 6 I 2 J 3 4 3 4 A bacteria population grows with constant relative growth rate i e its growth rate is proportional to its size At noon there are 1000 bacteria By 3 00 PM there are 10 000 bacteria At approximately what time will the population be 100 000 A 5 00 PM B 5 30 PM C 6 00 PM D 6 30 PM E 7 00 PM F 7 30 PM G 8 00 PM H 8 30 PM I 9 00 PM J 9 30 PM 5 Determine the values of p for which the integral Z 2 ln x p dx converges x A p 1 B p 1 C p 0 D p 0 E p 1 F p 1 G p 0 H p 0 I 1 p 0 J p 1 4 6 Which of the following describes the area bounded by the curves y A 7 x2 2 5 10 x2 2 B 1 5 10 2 x 2 C 7x x2 2 D 10 x2 10 ln x 7 F 7 x 2 10 2 x 10 x2 10 x 5 10 2 x 2 7 x 2 H 10 ln x 7x I 2 5 x2 2 2 E 13 7 x 3 G 5 5 2 5 2 5 x2 2 2 5 1 J 7x 2 5 x2 2 2 5 10 x and y 7 x 7 Using an infinite series we can approximate Z 1 x cos 0 x dx How many non zero terms must we add so the approximation has error less than 0 00001 A 1 term F 6 terms B 2 terms G 7 terms C 3 terms H 8 terms D 4 terms I 9 terms E 5 terms J 10 terms 8 A rocket full of fuel weighs 10 000 lbs at launch After launch the rocket gains altitude and loses weight as the fuel burns Assume that the rocket loses 1 lb of fuel for every 15 feet of altitude gained What is the work done in raising the rocket 30 000 feet Answers are given in millions of foot pounds Note You need not account for the change in gravity that occurs with the change in altitude A 10 B 30 C 95 D 105 E 145 F 255 G 270 H 300 I 330 J 450 6 9 Find the power series representation for the function f x A X 1 n xn 1 x x2 x3 x4 n 1 4n 2 42 2 43 3 44 4 45 1 n xn 1 x x2 x3 5 4n 2 42 43 44 4 n 0 B X n 0 C X nxn 4n n 1 D X x 2x2 3x3 4x4 2 3 4 4 4 4 4 xn x x2 x3 x4 4 5 4n 1 42 43 4 4 1 n 1 n 1 E X 1 n n 0 F 4n X xn 1 n 1 I n 4n 2 X nxn n 1 H X 4n 1 X n 0 1 n 1 2 3 2 4 3 x x x 42 43 2 44 3 45 x 2x2 3x3 4x4 2 3 4 4 4 4 4 1 n 1 n 1 J xn 1 x2 x3 x4 x 2 3 4n 4 4 4 X n 1 xn n 0 G 1 4 x 2 1 x x2 x3 5 42 43 44 4 1 4n 1 nxn 1 1 2 3 4 3 x 4 x2 5 x3 2 4 4 4 4 x x2 x3 xn 1 2 3 4n 4 4 4 7 10 Determine which of the following series are absolutely convergent AC conditionally convergent CC or divergent D X X X sin n n n2 3 i ii 1 n n iii 1 n 3 e n 1 n n n 1 n 1 A i CC ii AC iii D B i CC ii CC iii AC C i AC ii D iii D D i D ii CC iii AC E i CC ii AC iii AC F i AC ii D iii CC G i AC ii AC iii CC H i D ii D iii D I i AC ii AC iii D J i CC ii D n 2 iii CC 8 11 Determine the following i Find the sum of the series 1 e ii e2 e3 e4 2 3 4 The function f x ln 5 2x has a Taylor series expansion centered at the point x 1 so ln 5 2x X n 0 Find c2 A i 1 ee ii c2 B i ee ii c2 C i cos ee ii c2 2 25 D i sin ee ii c2 2 25 E i F i G 1 1 e 2 49 2 49 ii c2 20 343 1 ii c2 ln 7 1 e 1 2 i sin e ii c2 e 5 1 ee H i cos I i ee ii c2 2 7 J i 1 ee ii c2 2 7 ii c2 2 5 9 cn x 1 n 12 Find the arc length of the curve defined by the parametric equations Z t Z t cos w sin w x t dw y t dw 1 t w w 2 1 1 A F B 1 1 2 13 If g x G ln Z 3 2 C 1 2 D cos 1 sin 1 H 1 16 3 I 1 E 4 2 4 …
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