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Exam II S06 - y xy x y 8 Find the orthogonal trajectories...

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Mathematics 3B – Castroconde Name: Exam II (Chapters 9 and 10) I.D.#: Page 1 2 3 4 5 Total Points SHOW YOUR WORK, OR YOU WILL NOT GET ANY CREDIT. 1. Find the length of each curve. a. 3 0 ), ln(cos π = x x y b. 4 2 , 4 ln 2 2 - = x x x y c. 2 0 , 4 2 = y x y
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2. Find the surface area generated when the graph of 1 0 , 1 5 ) ( + = x x x f , is rotated about a. the x-axis. b. the y-axis. 3. Find the center of mass of the lamina of uniform density bounded by 2 4 x y - = and the x-axis.
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4. The demand function of certain commodity is 40 20 ) ( x x p - = . Find the consumer’s surplus when the sales level is 120. 5. Consider the differential equation 0 ) 1 ( = - + y y x . a. Show that every member of the family of functions ) 1 ( x A y + = is a solution of the equation. b. Find a solution of the equation that satisfies the initial condition y(0) = 8. 6. Use Euler’s method with step h = 0.2 to estimate y(1.4) where y(x) is a solution of the initial-value problem y’ = x – xy, y(1) = 0.
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7. Solve each differential equation. a. x dx dy e x 2 = b. 2 ) 1 ( , ln = =
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Unformatted text preview: y xy x y 8. Find the orthogonal trajectories of the family of curves x ke y-= . 9. Assume that the rate of growth of a population of fruit flies is proportional to the size of the population at each instant of time. If 50 fruit flies are present initially and 100 are present after 3 days, how many will be present after 15 days? 10. A tank contains 100 liters of brine with 10 kg of dissolved salt. Pure water enters the tank at a rate of 5 liters per minute. The well-mixed solution drains from the tank at the same rate. How much salt is in the tank after 10 minutes? 11. Suppose that an object has an initial temperature of 250° F and that after one hour its temperature is 200° F. Assuming that the surrounding air is kept at a constant temperature of 72° F, find a model for the temperature T of the object at time t ....
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