NAME: ________________________________ QUIZ SECTION:______ Student ID #: ____________________________ Math 112 -- Winter 2008 Final Exam HONOR STATEMENT: “I affirm that my work upholds the highest standards of honesty and academic integrity at the University of Washington, and that I have neither given nor received any unauthorized assistance on this exam.” SIGNATURE:__________________________________ INSTRUCTIONS: When the exam starts, verify that your exam contains 8 pages (including this cover page). Please turn your cell phone OFF and put it away for the duration of the exam. Unless otherwise indicated, you must show all your work in order to get full credit. The correct answer with incorrect or missing work may result in little or no credit. On problems in which you use a graph, show your work by clearly drawing & labeling any lines and points you use. If you use a graphing calculator or a guess-and-check method when an algebraic method is available, you will not receive full credit. Unless otherwise specified, you may round your final answer to two decimal digits. You are allowed to use a calculator, a ruler, and one sheet of notes. You have 3 hours for this exam. GOOD LUCK! Problem 1 15 Problem 2 10 Problem 3 15 Problem 4 16 Problem 5 16 Problem 6 16 Problem 7 12 Total: 100 1) (15 pts) The following is the graph of a function 𝑓(𝑥). Use this graph and the methods studied in this class to answer the following questions. Draw and label on the graph any lines you use, and be as precise as possible. a) (3 pts) Estimate 𝑓 5.001 −𝑓(5)0.001 . Answer: 𝑓 5.001 −𝑓(5)0.001 ≈______________ b) (3 pts) Find all x where 𝑓′(𝑥) = 5. Answer: 𝑥 = ________________________ (list all) c) (3 pts) Find an interval of length 2 over which the average rate of change of 𝑓(𝑥) is equal to 5. Answer: from 𝑥 =_______ to 𝑥 =________ d) (6 pts) For each part below, circle the correct answer. No justification is needed, but read carefully! i. The value of 𝑓(2) is POSITIVE, NEGATIVE, or ZERO. ii. The value of 𝑓′(2) is POSITIVE, NEGATIVE, or ZERO iii. The value of 𝑓′′ (2) is POSITIVE, NEGATIVE, or ZERO iv. The value of 𝑓(8) is POSITIVE, NEGATIVE, or ZERO v. The value of 𝑓′(8) is POSITIVE, NEGATIVE, or ZERO vi. The value of 𝑓′′ (8) is POSITIVE, NEGATIVE, or ZERO 𝑓(𝑥) 2) (10 pts) A car drives on a straight road. Its distance from a certain point is given by a function 𝑓(𝑡), where the time t is in seconds and 𝑓 𝑡 is in feet. We don’t have a formula for 𝑓(𝑡), but we know that for all a and h, 𝑓(𝑎 + ℎ) − 𝑓(𝑎) = ℎ2 + 2𝑎ℎ + 4ℎ. a) (3 pts) What is the car’s average speed over the interval from t = 1 to t = 5 seconds? Answer:___________ feet per second b) (4 pts) Find the formula in terms of t for 𝑓′(𝑡). Show ALL steps. Answer: 𝑓′ 𝑡 =_______________________________________ c) (3 pts) Suppose 𝑓 1 = 5. Compute 𝑓(4). Answer: 𝑓 4 = _______________________ 3) (15 points) a) Let 𝑓 𝑥 = 𝑥 2−ln 𝑥 𝑥3+𝑒𝑥 . Compute 𝑓 ′ 𝑥 . Do not simplify, but box your final answer. b) Suppose 𝐷 𝑡 = 𝑡 𝑡2 + 5 is the distance function, at time t, for a moving object. Compute the instantaneous speed of this object at 𝑡 = 2. Answer: _______________ c) Let 𝑧 = 3𝑥 2 𝑦 − 𝑦2 𝑥2 +2 ln𝑦. Find 𝜕𝑧 𝜕𝑥. Answer: 𝜕𝑧𝜕𝑥 = ______________________________________ 4) (16 pts) a) An unknown function 𝑦 = 𝑓(𝑟,𝑠) has these two partial derivatives: 𝜕𝑦𝜕𝑟 = 𝑟2𝑠 + 2𝑠, and 𝜕𝑦𝜕𝑠 = 𝑟+𝑠3 𝑠+1. i. (3 pts) Compute the slope of the tangent line to the graph of 𝑦 = 𝑓 1,𝑠 at 𝑠 = 3. Answer:________________ ii. (3 pts) Estimate 𝑓 1.0005,3 −𝑓(1,3)0.0005 Answer: ________________ b) (10 pts) Compute each of the following integrals, and box your final answers. i. 7𝑡 − 3𝑡5 + 2 𝑡 𝑑𝑡 ii. 𝑥 + 1 2𝑑𝑥20 5) (16 pts) You sell Things. The marginal revenue and marginal cost at q thousand Things are given by: 𝑴𝑹 𝒒 = −𝒒𝟐 + 𝟏𝟏𝒒 + 𝟏𝟎 dollars 𝑴𝑪 𝒒 = −𝒒 + 𝟑𝟎 dollars You have fixed costs of 2 thousand dollars. a) (3 pts) Find the formula for the variable cost of producing q thousand Things. Answer: 𝑉𝐶(𝑞) = _____________________________ b) (5 pts) What is your profit (or loss) if you sell 1 thousand Things? Answer: Profit OR Loss (circle one) of ________________ _________________________ c) (4 pts) What quantity results in the largest profit? Answer: q = _____________ thousand Things d) (4 pts) What is the largest profit? Answer: ______________________ Units: _______________________ 6) (16 pts) The following is the rate-of-ascent graph, 𝑎(𝑡), for a hot air balloon, over a 12 minute period. Let 𝑨 𝒕 represent the altitude of the balloon above the ground at t minutes. a) Estimate the average rate-of-ascent of the balloon during the last minute (𝑡 = 11 to 𝑡 = 12). Answer: ___________________ meters per minute. b) Find the longest time interval during which the balloon’s altitude is increasing. Answer: from 𝑡 =_____ to 𝑡 =_____ c) Estimate 𝐴′(4). Answer: 𝐴′ 4 =_________________ d) Estimate 𝑎 𝑡 𝑑𝑡64 Answer: _________________ e) Find the first time past 𝑡 = 6 minutes when the balloon returns to the same altitude it had at 𝑡 = 6 minutes. Answer: at 𝑡 =_________ f) The critical points of the altitude 𝐴(𝑡) are at 𝑡 = ____________________________ (list all, no need to justify) g) The local minima of the altitude 𝐴(𝑡) are at 𝑡 =_____________________________(list all, no need to justify) h) Suppose 𝐴(0) = 150 meters. What is the lowest altitude of the balloon and at what time is it achieved? Answer: _________ meters, at 𝑡 =________minutes. Rat e-of -Asc en t (m ete rs p er min ute ) time (minutes) 𝑎(𝑡) 7) (12 pts) You produce and sell two types of birthday cards: Cute and Funny. Every Cute card you sell results in a profit of $0.50, while each Funny card you sell generates $0.30 in profit. Each Cute card takes 48 in2 of paper and 5 mg of ink to produce, and each Funny card takes 64 in2 of paper and 2 mg of ink to produce. Your supply of paper is limited to 19,200 in2 per day, and your supply of ink is limited to 1500 mg. Let x be the number of Cute cards, and let y be the number of Funny cards you produce in a day. a) (2 pts) Give the formula for your daily profit, as a function of x and y. Profit 𝑃 𝑥,𝑦 =_________________________________. b) (2 pts) List your constraints: c) (3 pts) Sketch the feasible region. d) (3 pts) Compute & list all vertices (round coordinates to the nearest whole number). (𝑥,𝑦) = ____________________________________________ e) (2pts) Find the maximal profit. Answer: Max profit is $____________. Alexandra Nichifor NAME: ________________________________, Student ID #: _________________