Math 242 Additional study problems for the Final Exam – (LL) 1. For the following determine if the integral converges. If the integral converges, compute its value.
a. 2. It is estimated that
b. c. d. years from now, a certain investment will be generating income at the rate of per year, dispensed continuously. If the income is generated in perpetuity and the prevailing annual interest rate remains fixed at 5% compounded continuously, what is the present value of the investment? 3. Use a) The Trapezoidal rule, and b) Simpson’s rule with to approximate the integral . Round each of your answers to 6 decimal places. 4. Determine how many subintervals are required to guarantee accuracy to within 0.0001 of the actual value of the integral using the trapezoidal rule. 5. a. b. 6.
f f 2 f 2 f , , , and . f ( x, y) xy ln(3 y) , Find x dy x 2 yx e2 xy , Find f x , f y , f yy , and f yx . f ( x, y ) x 1 1 2
Suppose the production function of a firm is given by the Cobb-Douglas production function
f ( x, y) 27 x 3 y 3 where x is the number of units of labor and y is the number of units of capital required to produce f ( x, y ) thousand units of the product. Find the marginal productivities of labor and capital when x = 27 and y = 8. 7. Suppose a brewery has a profit function given by P( x, y) 2 x 2 2 xy y 2 2 x 4 y 107 where x is the number (in thousands) of cases of India pale ale, and y is the number (in thousands) of cases of Best bitter, and P is the profit (in thousands of dollars). How many cases of each type of beer should be produced each year to maximize the profit? Show that your answer is a maximum. 8.
1 3 y 3 y . Find all critical points ( x, y) for f ( x, y) . Then use the second 3 derivative test to determine, if possible, if the points yield a maximum or minimum for f ( x, y ) . Let f ( x, y) x 2 2 xy
9. The research department for a computer company has a production function for a particular product...
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