Math 115 - Team Homework Assignment #4, Winter 2017 Due Date: Feb. 23 or 24 (Your instructor will tell you the exact date and time.) This assignment covers material from Chapter 2 and 3 of the course textbook, which is the 6th edition of Calculus: Single Variable by Hughes-Hallett, Gleason, McCallum, et al.. Make sure to understand the material in this chapter before attempting these problems. It is important that you try these problems before your first meeting with your team. Remember, the first meeting with your team is for the members of the team to discuss the arguments used by each member of the team to find their solutions. It is also a time to address any questions that each member of the team may have about the assignment. Remember to follow the guidelines from the \"Doing Team Homework\" and \"Team HW Tutorial\" links in the sidebar of the course website. Do not forget to rotate roles and include a reporter's page each week. This is an assignment that evaluates mathematical skills and your ability to justify carefully your solutions. 1. Biologist introduced a population of brown trout and catfish in a lake. (a) Let A and r be positive constants and B(t) = 4000 10 + Aert be the population (in thousands) of brown trout in the lake t years after they were introduced to the lake. Five years after the fishes were introduced to the late, the biologist estimate that there are hundred thousand brown trout in the lake. If the instantaneous rate of change of the population after 5 years is equal to 1200 brown trouts per year, what are the values of the constants A and r? (b) The population (in thousands) of catfish t years after they were introduced into the lake is given by 1000 S(t) = 5 + 995eg(t) where estimated values of g(t) are shown below t g(t) g 0 (t) g 00 (t) 0 1 0 0 2 1.16 0.21 0.16 4 1.96 0.6 0.22 5 2.67 0.83 0.25 i. What is the population of catfish in the lake five years after it was introduced in the lake? ii. At which of the values of t in the table is the population of catfish increasing the fastest? iii. What is the value of S 00 (4)? What does this tell you about the graph of S(t) near t = 4? 1 2. A company makes 2500 keychains for a thousand dollars. The marketing department estimates that they will sell S(p) of these keychains (in thousands) if the price of each keychain is p dollars. Some of the values of S(p) and its derivative are shown below. p S(p) S 0 (p) 5 2 -0.3 6 1.7 -0.28 7 1.5 -0.25 8 1.15 -0.2 9 0.77 -0.27 10 0.53 -0.19 (a) Let F (p) be the profit of the company when the price of each keychain is p dollars. Find a formula for F (p) in terms of any of the functions introduced above. Remember that the profit of the company is equal to the revenue of sales minus the cost of producing all the keychains. (b) Make a table with values of F (p) for prices between 5 and 10 dollars. (c) Find the exact value of F 0 (8). (d) The company was originally planning to charge $8 for each keychain, but is now considering decreasing that price to $7.50. Write a sentence that interprets your answer to the previous part, explaining what effect the marketing team predicts this change in price will have on the profit. (e) Find the exact value of F 00 (8) if S 00 (8) = 0.023. 3. Let B and C be constants and ( B(x + 3)2 + C f (x) = 4|x 2| x<0 x 0. (a) Describe all possible values of B and C for which the function f (x) is continuous at x = 0 but is not differentiable at x = 0. Use the definition of continuity and differentiability of a function in order to justify your answer carefully. (b) For which values of B and C is the function differentiable at x = 0? Use the definition of the derivative of f (x) at x = 0 in order to justify your answer. (c) With the values of B and C you found in part (b), is the function f (x) continuous for all < x < ? Is it differentiable for all < x < ? Explain. 2