\fPlease remember to start each question on a new page. 2. (6 points) Below are two improper integrals. If the integral is divergent, Show that it is divergent. If it is not divergent, evaluate it. (a) m 1 [r x 1 + sin(lnx) d3: ('0) 3. Suppose a continuous random variable X has the Probability Density Function 0 $330; as: 02, where a > 0 is a constant. (a) (1 point) Determine P(1 g X 5 2). (b) (1 point) Determine P(1 S X 5 5). (c) (2 points) By considering If; f (:12) d3: or otherwise, determine a. ) (d (4 points) Write down the Cumulative Distribution Function of X. (The answer will be piecewise-dened.) 4. (5 points) You wish to numerically approm'mate 5 :rsinzr: do: 0 using Simpson's Rule. Find a value of n such that your error will be no more than = 00004. As always, fully justify your answer. The theorem we learned bounding the error from Simpson's Rule is given below for reference. [f |f(4)($)| S L for all a S a: s b, then the total error introduced by Simpson's rule when approximating I: f (:r)d:c is bounded by a (b at 180 n4 where n is the number of intervals used (half the number of approximating parabolas). 5. (4 points) The marginal cost of making a quantity q of a product is given by 1 Vq+1 M001) = and the xed costs involved are $1,000. Find the total cost function for the product, TC(q). 6. (4 points) A function y(x) satisfies the differential equation dy y : 2 2 dx and the initial condition y (0) = -4. Give an explicit formula for such a function y(x). (Make sure your final answer is actually a function.)7. (5 points) Each night, your studious roommate studies between 0 and 4 hours. Let T be the continuous random variable representing the length of time your roommate studies each night. You have modelled an approximation F(a') for the Cumulative Dis tribution Function of T, where :1: is time in hours. Use this model to answer the following questions. (a) Describe in words what each of the following quantities represents: 1. F(2) ii. F(3) F(2) iii. 1 F(1) Remark: Remember that the purpose of an exam is to demonstrate understanding. So an answer like \"F(2) is the cumulative distribution function evaluated at 2\" will receive no credit. (b) What is the probability that your roommate studies for about 2.5 hours tomorrow, plus or minus 10 minutes? (Your answer will be in terms of F(:s).) (c) Give exact values for the following quantities: i. F(0) ii. F(4) iii. F(5) 8. In a certain bank account, when the balance is $11, interest accrues continually at a rate of $14 dollars per month. The owner of the account keeps themselves to a budget, withdrawing money continually at a rate of x/Z dollars per month. There is no other activity in the account. (a) (2 points) Set up a differential equation for % summarizing the given information. (b) (3 points) If A(D) = 2500, give an explicit formula for AGE). 9. (4 points) Each option below shows a collection of functions. The a: and y axes have the same scale. Which collection consists of functions satisfying the dierential equation below? (a) Circle the correct collection or give its letter, and explain. (b) Suppose a function y(x) satisfying the differential equation has a local extremum at :L' = 4. What is y(4)? Why