Unformatted text preview: AMS 161 _ Final Exam A _ Prof. Tucker Spring 2010 ' 1. Do three of the following problems :
I a) 0J3[e3x— 3x]dx ,1»)! xze3x3sin(e3x3)dx, c) I 4x/(x+1)dx ,dx. d)! 5x/sqrt(2.x+2)dx, 2. Do two of the following problems: a) I 3xe4x dx , b) I x7ln(x) dx , c) I x3sin(2x2)dx
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3. Evaluate two of the following integrals: a) OJ 1/(.1c—2)3 dx, b) (J tﬁe‘ﬂdt, c) 0] l/(Jcl)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 — sqrt(2/x) from y=1 to
y=3. b) Set up an integral for the volume resulting from revolving the area between the curves y — x+3 and
y = x3 1 between— 1 and 1 about the line 32— 5. ' 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 50 lbs. hangs from the top of a 80 feet high building down to the street. The chain weighs 5 lbs. per foot. Currently the pail is 20 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 10 feet at the
bottom and 12 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 250foot high dam is shaped like the symmetric eXponential function y— = em2 —1,x 1?» 0 (and y— = 6"?“2 —l, x s 0). The water level 18 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 3) Determine by direct computation the terms up to x3 b) From part a), determine the terms up to x6 1n the Taylor series for log(2 + 2x2 ). 0) Determine the terms Up to x7 In the Taylor series for sin(x)log(2+ 2x2 ), where sin(x)= x—ex3/3! Hrs/5! —x7/7! 1n the Taylor series for 120g(2 +x) Show f’ ,f’ ,f’”. 8. Determine the radius of convergence of the series 1 + 5x/I! + 52x2/2!2+ 53x3/3! + 54x4/4! +
9. Solve both DEs: a) y' = 2326”, y(0) = 5. b) y" 5y' + 4y = 0, y(0) = 5, y'(0) = 2. 10. Set up a Diff. Eqn. and solve it with given conditionsfor BOTH of the following two problems. a) N ewton’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 c'omes
out of coffee maker at 130 “ F. into a cup in a room at 70” 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 2,000,000 gallons of water. Compound X (which causes skin irritation) has started
polluting the water ﬂowing into the reservoir at a concentration of .002 pounds per gallon of water. Each week
100,000 gallons of polluted flow into the reservoir and 100,000 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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 Spring '08
 Tucker

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