course Mth272
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Given Solution: `a The integral as stated here diverges. You need to take the limit as t -> infinity of INT(x^2 e^(-x^3), x from -t to t ). Using the obvious substitution we see that the result is the same as the limiting value as t -> infinity of INT( 1/3 e^(-u), u from -t to t ). Using -1/3 e^(-u) as antiderivative we get -1/3 e^(-t)) - (-1/3 e^(-(-t))); the second term is 1/3 e^t, which approaches infinity as t -> infinity. The first term approaches zero, but that doesn't help. The integral approaches infinity. Note that the integral from 0 to infinity converges: We take the limit as t -> infinity of INT(x^2 e^(-x^3), x from 0 to t ), which using the same steps as before gives us the limit as t -> infinity of -1/3 e^(-t) - (-1/3) e^0. The first term approaches zero, the second is just 1/3. So the limiting value is 1/3. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok just needed to finish ------------------------------------------------ Self-critique rating #$&*2 ********************************************* Question: `qQuery problem 6.6.40 (was 6.6.38) farm profit of $75K per year, 8% continuously compounded, find present value of the farm for 20 years, and forever. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 75000e^-.08t= 75000/-.08 e^-.08t from 0 to 20 I get approx 750,000 confidence rating #$&*2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a The correct antiderivative is [75,000 /-0.08 e^ -0.08t]. For 20 years you evaluate the change in this antiderivative between t = 0 and t = 20, and I believe you obtain $ 748,222.01 To get the present value forever you integrate from 0 to b and let b -> infinity. The integral from 0 to b is 75,000 / (-.08) e^(-(.08 b)) - 75,000 / (-.08) e^(0.08 * 0) = 75,000 / -.08 * (e^(-.08 b) - e^0). e^0 is 1 and as b -> infinity e^(-.08 b) -> 0. So the integral is 75,000 / -.08 ( 0 - 1) = 75,000 / .08 = 937,500. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok ------------------------------------------------ Self-critique rating #$&*2 ********************************************* Question: `qWhat is the present value of the farm for 20 years, and what is its present value forever? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The present value would be the same 750000 and forever would be the 937500 confidence rating #$&*3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a The present value for 20 years is $ 748,222.01 Forever $ 937,500.00 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok ------------------------------------------------ Self-critique rating #$&*3 ********************************************* Question: `qQuery Add comments on any surprises or insights you experienced as a result of this assignment."