Pennsylvania State University MATH 141 CALC ANLY GEOM II. Math 141 Written Homework 7 Solutions. Winter 2023

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Math 141 Written Homework 7 Page 1 of 5 Question 1 For each improper integral below, show that it diverges or find its value. (a) Z07 px ln x dx: (b) Z04 lnxx dx: (c) Z41 x21- 9 dx (d) Z31 x21+ 9... dx Question 2 We consider compound interest with a nominal annual rate r compounded n times per year. The value V (t) of an initial investment V0 after t years is given by V = V0 1 + nr nt : (a) For fixed V0; r; and t, the limit lim n!1 V is indeterminate. What type of indeterminate form is it? Math 141 Written Homework 7 Page 3 of 5 (b) Continuously compounded interest is obtained by compounding more and more frequently. Find lim n!1 V . Question 3 The exponential distribution is a probability distribution that describes the time needed for a process to change state. Suppose the number of minutes you wait in line for lunch at the HUB has the probability density function p(x) = Ce-0x=11 x x <≥ 0 0 : (a) In order to be a probability density function, we require Z-1 1 p(x) dx = 1: Use this to solve for the normalization constant C. (b) The probability that the wait time is between a and b minutes is given by Zab p(x) dx. Find the probability that i. you wait between 5 and 8 minutes for your lunch. ii. you wait at least 15 minutes for your lunch. Math 141 Written Homework 7 Page 4 of 5 (c) The mean, or average, wait time is given by x = Z-1 1 xp(x) dx: Calculate the mean wait time for lunch at the HUB. (d) The median m wait time is defined by the equation Z-1 m p(x) dx = 0:5: Calculate the median wait time. [Show More]

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