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Final Fall 2005 - Math 122 Final Name 1 2 3 4 5 6 7 8 9 10...

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Math 122 Final December 13, 2005 Name 1 2 3 4 5 6 7 8 9 10 Total / 200 Directions: 1. No books, notes or passing go and collecting $200. You may use a cal- culator to do routine arithmetic computations. You may not use your calculator to store notes or formulas. You may not share a calculator with anyone. 2. You should show your work, and explain how you arrived at your an- swers. A correct answer with no work shown (except on problems which are completely trivial) will receive no credit. If you are not sure whether you have written enough, please ask. 3. You may not make more than one attempt at a problem. If you make several attempts, you must indicate which one you want counted, or you will be penalized. 4. You may leave as soon as you are finished, but once you leave the exam, you may not make any changes to your exam.
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1. (20 points) (a) Z dx e x e 2 x + 4 Hint: Z du u 2 u 2 ± a 2 = u 2 ± a 2 a 2 u + C (b) Z x + 1 3 x 2 + 2 x - 16 dx (c) Z x sec 2 x dx (d) Z sin 3 x cos 3 x dx .
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2. (20 points) (a) Z 1 + sin x 1 - sin 2 x dx . (b) Z (sec 3 3 x )(tan 3 3 x ) dx (c) Z 1 - x 2 x 2 + x dx (d) Z 1 (4 + x 2 ) 3 / 2 dx
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3. (20 points) (a) For the differential equation dy dx = F ( x, y ), where F ( x, y ) is given below: x 0 1 2 3 4 0 1 0 -1 -1 0 1 0 1 0 1 0 y 2 -1 0 -1 1 1 3 -1 1 0 1 0 4 1 1 -1 -1 1 Use Euler’s method with h = 1 and y (0) = 0 to approximate y (4). (b) Find the solution for the following differential equations. i. dy dx = - 4 xy 2 y (0) = 1 ii. dy dx - y = e 2 x y (0) = 3
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4. (20 points) A tank contains 30 gallons of pure water. Water containing 2 pounds of Grape Kool-aid mix dissolved per gallon enters the tank at 5 gallons per minute. The well-stirred mixture drains out at 5 gallons per minute. (a) Write a differential equation (with initial condition) for how much Grape Kool-aid mix is in the tank at any time.
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