Name: Math 112 - Winter 2014 Exam 1 January 30, 2014 Section: Student ID Number: PAGE 1 13 PAGE 2 11 PAGE 3 12 PAGE 4 14 Total 50 After this cover page, there are 5 problems spanning 4 pages. Please make sure your exam contains all of this material. You are allowed to use a scientific calculator (no graphing calculators and no calculators with calculus abilities), a ruler, and one hand-written 8.5 by 11 inch page of notes. Put your name on your sheet of notes and turn it in with the exam. You must show your work on all problems. The correct answer with no supporting work may result in no credit. If you use a guess-and-check, or calculator, method when an algebraic method is available, you may not receive full credit. If you need more room, use the backs of the pages and indicate to the grader that you have done so. Raise your hand if you have a question. There are multiple versions of the exam so if you copy off a neighbor and put down the answers from another version we will know you cheated. Any student found engaging in academic misconduct will receive a score of 0 on this exam. All suspicious behavior will be reported to the academic misconduct board. In such an instance, you will be required to meet in front of a board of professors to explain your actions. DO NOT CHEAT OR DO ANYTHING THAT LOOKS SUSPICIOUS! WE WILL REPORT YOU AND YOU MAY BE EXPELLED! You have 50 minutes to complete the exam. Budget your time wisely. SPEND NO MORE THAN 10 MINUTES PER PAGE! GOOD LUCK!
1. (13 points) Show your intermediate work wherever possible. You do not have to simplify your final answers for your derivatives. Put a box around your final answer. (a) (4 points) Find y for the function y = (3x 2 1) 4 ( 5 2x 3 + 4 ) (b) (4 points) Find f (x) for the function f(x) = 4x 3 + 7 5 x 3 2(3x x 5 ) (c) (5 pts) Find the equation for the tangent line to y = x 2 + 3 + 5 at x = 1. x3
2. The graph of the function y = f(x) is given below: Approximate (as accurately as possible) the answers to the following questions from the graph. (a) (4 pts) Circle ALL of the following that are TRUE: i. f(x) is positive at x = 1.25 iii. f (x) is positive at x = 1.25 ii. f(x) is increasing at x = 1.25 iv. f (x) is increasing at x = 1.25 (b) (3 pts) Find a value of h such that f(1.75 + h) f(1.75) h = 0. ANSWER: h = (c) (4 pts) Assume that f(x) is Profit. Also assume that x is in thousands of items and f(x) is in thousands of dollars. Estimate the value of Marginal Profit at x = 2.5. And fill in the blanks of the sentence below interpreting your value for MP (2.5). ANSWER: MP (2.5) = If the quantity changes from items to items, then profit will approximately change by dollars.
3. (6 pts) The slope of the secant line from x to x + h for some function f(x) is always given by f(x + h) f(x) the formula = 3h 2 + 9x 2 + 9hx + 8. Answer the following questions. h (a) If f(1) = 10, then what is the value of f(3)? (b) Find the slope of the tangent line at x = 4. ANSWER: f(3) = ANSWER: f (4) = 4. (6 pts) You are starting a business selling a new product. From studying similar products, you expect the demand function to be given by p = 200 2x dollars and you expect the overall average cost per item to be given by AC(x) = 8 + 15 + x dollars, where x is in items. Find x the quantity and price that correspond to maximum profit. (Hint: First give all the business functions T R, T C, etc.) ANSWERS: quantity = price = items dollars
5. (14 pts) Two balloons are let go right next to each other at the same time. The height of the balloons are given by the functions: A(t) = 40t 3t 2 and B(t) = 15t 2 t 3, where height is in feet and time, t, is in minutes. For all parts, assume t 0. Give units for all answers. (a) (3 pts) At time t = 2 minutes, is the distance between the two balloons increasing or decreasing? (As part of your answer, you must compute the values of the functions and their derivatives and give your justification) The distance between the balloons is (CIRCLE ONE): Increasing or Decreasing (b) (3 pts) Find the largest interval of times t for which the rate of ascent (i.e. speed) graph for Balloon B is increasing. ANSWER: From t = to t= ( units) (c) (4 pts) Find the maximum height reached by Balloon B. ANSWER: ( units) (d) (4 pts) Find all times at which Balloon A is traveling 4 ft/min faster than Balloon B. (Round your final answer(s) to two digits after the decimal) ANSWER: ( units)