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T161-F-09 - AMS 161 Final Exam A Prof Tucker Fall 2009 1 Do...

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Unformatted text preview: AMS 161 Final Exam A Prof. Tucker Fall 2009 1. Do three of the following problems : a) 0I2[e4x — 3x] dx , b) I x e3x2cos(e3x2)dx, c) I 2x/(x-3)dx ,dx, d) I 2x/sqrt(3x — 2) dx, 2. Do two of the following problems: a) I 4xe3x dx , b) I x7ln(x) dx , c) I xgsin(2xs)dx 3. Evaluate two of the following integrals: a) OI31/(x—2)2 dx, b) 0I °°t2e-t3dz, c) 0I l/(x—l)2 dx. 4. Consider the integral from 0 to 2 of the function sketched at the right. , List the values in INCREASING ORDER of the integral estimates given by the LH = Left-Hand Rule, RH = Right-Hand Rule, MP = Mid-Point Rule, and TP =Trapezoidal Rule in increasing order. Also indicate the position in this ordering of the true integral. 5. Do one of the following problems: a) Set up an integral for the volume of revolution around the y-axis (NOT x-axis) of y = 2/x2 from y=1 to y=4. b) Set up an integral for the volume resulting from revolving the area between the curves y = x+4 and y = x2 between -1 and 1 about the line y = 8. 6. Do two of the following problems (b and c are worth more): JUST SET UP THE INTEGRAL: a) A chain holding a pail of water weighing 25 lbs. hangs from the top of a 40 feet high building down to the street. The chain weighs 2 lbs. per foot. Currently the pail is 10 above the ground. Set up an integral for the work to raise the brick up to the top of the building. b) Water is being pumped into a conically shaped tank (pointed at the top) that has a radius of 20 feet at the bottom and 30 feet high. The tank sits on top of a platform 40 feet high. Set up an integral for the work required to pump water up from ground level to fill the tank Water weights 62.5 pounds per cubic feet. c) A 125-foot high dam is shaped like the symmetric exponential function y = em —1 , x z 0 (and y = e)“3 —1 , x s 0). The water level is at the top of the dam. Set up an integral that gives the total water pressure against the dam. Water weights 62 .5 lbs per cubic foot. 7. a) Determine by direct computation the terms up to x3 in the Taylor series for log(x+l). Show f’ f’f”. 6 b) From part a), determine the terms up to x in the Taylor series for log(3x2 + l). c) Determine the terms up to x7 in the Taylor series for sin(x)log(3x2+1), where sin(x) = x—x3/3! +x5/5! —~x7/7 ! 8. Determine the radius of convergence of the series 4 + 42x/3 + 43x2/32 + 44x3/33 + 45x4/34 + . 9. Solve both DES: a) y' = —3ye'2x, y(0) = 0. b) y" -4y' + 3y = 0, y(0) = 3, y'(0) = 2. 10. Set up a Diff. Eqn. and solve it with given conditions for BOTH of the following two problems. a) Newton’s Law of Heating says that the rate at which the temperature of a hot object cools down to room temperature is proportional to the temperature difference between the object and the room. Hot coffee comes out of coffee maker at 140 “ F. into a cup in a room at 60° F. and in 8 minutes the coffee is 100° F. Determine the temperature of the coffee in the cup as a function of time (in minutes) since coming out of the pot. b.) A reservoir holds 3,000,000 gallons of water. Compound X (which causes skin irritation) has started polluting the water flowing into the reservoir at a concentration of .001 pounds per gallon of water. Each week 200,000 gallons of polluted flow into the reservoir and 200,000 gallons flow out of the reservoir into a nearby town’s drinking water. Initially the reservoir has no compound X. Set up and solve a differential equation for y(t), the amount of Compound X in the reservoir as a function of time (in weeks). ...
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