MATH 112 Final Exam, Spring Honor Statement

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1 NAME: QUIZ Section: STUDENT ID: MATH 112 Final Exam, Spring 2013 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: Do not open the test until instructed to do so. Please turn your cell phone OFF now. This exam consists of this cover sheet followed by eight problems on eight pages. When the test starts, check that you have a complete exam. This exam is closed book. You may use one double-sided, handwritten page of notes, a ruler, and a non-graphing calculator. Put everything else away. You may not share notes. Unless otherwise indicated, you must show your work and use the methods of this class. An answer with incomplete or missing supporting work may result in no credit, even if correct. Place your final answer in the indicated spaces, or box your final answer. Unless otherwise specified, you may round your final answer to two digits after the decimal. If you need more room, use the backs of the pages and indicate to the grader that you have done so. Problem Total Points Score Total 100

2 1. (13 points) The following is the graph of a function y = f(x). Use this graph to answer the following questions. Draw and label on the graph any lines you use, and be as precise as possible. (a) Estimate f(6.001) f(6) f(6.001) f(6) (b) Find all values of x where f (x) = 5. At x = (c) List the longest interval on which f(x) is concave down. No justification is needed. From x = to x = (d) For each part below, circle the correct answer. No justification is needed, but read each part 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 vii. The value of 3 0 f(x)dx is POSITIVE, NEGATIVE, or ZERO

3 2. (12 points) (a) A car drives on a straight road. Its distance from a certain point is given by a function D(t), where the time t is in seconds and the distance D(t) is in feet. We don t have the formula for D(t), but we know that for all values of a and h, D(a + h) D(a) = ( h 2 + 2ah + 4h ) a + 2h i. Compute the car s average speed over the interval from t = 1 to t = 5 seconds. feet per second ii. Compute the car s instantaneous speed at t = 2 seconds. Show all steps. feet per second (b) Suppose F (t) = t t is the distance another object traveled, in meters, after t minutes. Compute this object s instantaneous speed at t = 2 minutes. meters per minute

4 3. (12 pts) Box your final answers to each of the following questions. (a) Let f(x) = x e2x 3 x 2 + ln(x). Compute f (x). Do not simplify. (b) Compute the following indefinite integral. Simplify your answer. ( 7t 3 t ) t dt (c) Compute the following definite integral. Round your answer to 2 decimal places. 2 1 ( 2 x + e 0.5x ) dx

5 4. (10 points) The demand function for a product is p = 78 x + 1 and the supply function is p = x, where p is the price per unit, in dollars, and x is the number of units. Compute the consumer s surplus and the supplier s surplus under pure competition. Consumer s Surplus = $, and Producer s Surplus =$ (Round your answers to two decimal places.)

6 5. (10 points) (a) (4 pts) Let z = 3x2 y xy2 z + 2 ln(y). Find x + 1 x. z x = (b) (6 pts) An unknown multi-variable function y = f(r, s) has these two partial derivatives: y r = r2 s + 2s, and y s = r + s 3( s + 1) i. Compute the slope of the tangent to the graph of the function y = f(1, s) at s = 3. ii. Estimate f(1.0005, 3) f(1, 3) slope = f(1.0005, 3) f(1, 3)

7 6. (16 pts) You sell Things. You have fixed costs of 2 thousand dollars. Your marginal revenue and marginal cost at q thousand Things are given by: MR(q) = q q + 10 dollars per Thing MC(q) = q + 30 dollars per Thing (a) What are the critical values for the profit function? q = thousand Things. (list all) (b) Use the second derivative test to determine if each of the critical values from part (a) are local maxima or local minima. Show all steps clearly, and box each answer. (c) Compute the maximum profit. Max profit = thousand dollars (d) Use the closed interval method to determine the lowest total cost when you produce between 10 and 40 thousand Things. Show all work/justifications. Lowest TC = thousand dollars

8 7. (12 pts) The following is the instantaneous rate-of-ascent graph, r(t), for a hot air balloon, over a 12 minute period. Let A(t) denote the altitude of the balloon above the ground at t minutes. (a) (2 pts) Find the longest time interval during which the balloon s altitude is increasing. From t = to t = minutes (b) (2 pts) Recall that A(t) denotes the altitude of the balloon. Estimate A (4) and A (4). A (4) =, and A (4) = (c) (3 pts) Estimate the average rate of ascent of the balloon during the time interval from 2 to 7 minutes. meters per minute (d) (2 pts) Find the first time past t = 6 minutes when the balloon returns to the same altitude it had at 6 minutes. At t = minutes. (e) (3 pts) Suppose the balloon starts off at an intial altitude A(0) = 200 meters above ground. Estimate the lowest altitude of the baloon, and at what time it is reached. lowest altitude = meters, at t = minutes

9 8. (15 points) The profit for a crop is related to fertilizer and labor. The profit per acre is: P (x, y) = x 3 2y xy 50 dollars where x is the number of untis of fertilizer, and y is the number of work-hours. (a) (4 pts) Compute the partial derivatives of P. P x (x, y) = P y (x, y) = (b) (5 pts) Find all critical points of the profit. (x, y) = (list both coordinates of all points) (c) (2 pts) Compute the profit at each critical point, and determine the maximum profit. Max profit = $ (d) (4 pts) Suppose you are using 10 units of fertilizer and 20 work-hours per acre. What would increase your profit more: an extra unit of fertilizer, or an extra work-hour? (Circle one and justify your answer.)

Total 100

Total 100 MATH 112 Final Exam Spring 2016 Name Student ID # Section HONOR STATEMENT I affirm that my work upholds the highest standards of honesty and academic integrity at the University of Washington, and that