MAT285-2001Fall

MAT285-2001Fall - MAT 285 Final Exam Signature Instructions...

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Unformatted text preview: MAT 285 Final Exam . December 18, 2001 Signature: Instructions: Write the answers and show the main steps of your work on this test sheet. There are 11 questions on H pages (includ- ing this cover). Be sure you have all M pages (Hsheets) and that you do all 11 problems! The Final Exam is scored on a basis of 100 points and will count 20% of your final grade. PUT YOUR NAME ON THE TOP OF EACH SHEET - NOW! This Exam has 4 parts corresponding to the 4 major units of the course. You should spend no more than 25 minutes on each part and be sure that you get to the easier problems in each part. Where indicated, you must show your work to get full credit! DO NOT WRITE ON THE REST OF THIS COVER SHEET! Part I: Part II: Part III: Part IV: Total: Have a nice and safe Holiday! 2 MAT 285 Final Exam December 18, 2001 Part I Problem 1 {12 points) Circle the correct answer for each part. a. (3 points) The following table gives the average height h, in feet, of a certain population for various weights w, in pounds. Weight w (lb) — 200 :—:Hez'ght h (ft) I 6.23 Give the slope (with appropriate units) of a linear equation, height as a function of weight, which models these data reason— ably well. 60 ft. per 1b.; 60 lb. per ft.; .017 ft. per 1b.; .017 lb. per ft.; none of these. b. It costs $4,000 per week to run a small widget factory (lights, heat, etc.). The raw materials and labor costs of producing one widget is $7.75. Widgets sell for $16.00 each. (i) (3 points) The weekly cost in dollars for producing q wid- gets is: C(q) = 7.75q + 4, 000; C(q) = 7.75q — 4, 000; C(q) = 8.25q — 4, 000; C(q) = 8.25q + 4, 000; none of these. (ii) (3 points) The weekly profit in dollars from producing and selling q widgets is: P(q) = 7.75q + 4, 000; P(q) = 7.75q — 4, 000, P(q) = 8.25q — 4, 000; P(q) = 8.25q + 4,000; none of these. c. (3 points) The future value after 5 years of an investment ac- count is $5,000. The account earns 3.75% interest per year, compounded continuously. Find the present value of the ac— count to the nearest cent. MAT 285 Final Exam December 18, 2001 3 Problem 2 ( 6 points) The period of a pendulum T seconds) is proportional to the square root of its length a: feet). a. (2 points ) Find the formula for T in terms of :3. Let k denote the proportionality constant. b. (2 points) A pendulum of length x=3 feet has period T=1.44 seconds. Find the constant k to six decimal places. 6. (2 points) Using this model, find to the nearest foot the length of a pendulum that has period T=2.04 seconds. 4 MAT 285 Final Exam December 18, 2001 Problem 3 (7 points) A rectangular pasture with area of 1000 square yards is to be enclosed with fencing that costs $10. 50 per linear yard. The pasture is to be subdivided into two parts of equal area by a fence costing $9.25 per linear yard (see diagram). a. (4 points ) Give the formula for the total cost of the fencing in terms of x and y. b. (3 points) Give the formula for the total cost of the fencing in terms of :3 alone. MAT 285 Final Exam December 18, 2001 5 Part II , (2::2 + 2x) Problem 4 pOZTLtS) Let = [CHECK YOUR INPUT: have you entered the function cor- rectly?] a. (11 points) On the axes below, sketch the graph of this func- tion. The window is —7 g :1: S 7 and —3 S y S 3. Label on the graph all local maxima, local minima, and inflection points. Give their coordinates to two decimal places. Label the horizontal and vertical asymptotes. Give their equations. b. (1 point) At a: = 20 the function is INCREASING DECREASING (circle one) c. (1 point) At 2: = —20 the function is concave UP DOWN (circle one) 6 MAT 285 F inal Exam December 18, 2001 Problem 5 (12 points) a. (4 points) Define f’ as a limit: b. For each of the following questions, circle the correct answer. (i) (4 points) The equation 3:1:2 + 2:1: - y + y2 = 15 defines y implicitly as a function of :r. Differentiating implicitly, we get: @_$+y_ dy_—32:+y' @_—(3x+y)_ dm—3x+y’ Ear—:— x+y’ dm $+y dy_rv—y dx ‘— 2$+y; none of these. (ii) (4 points) q 1' Circle the points at which f(x) is differentiable: p q r Circle the points at which f(x) is continuous: p q r MAT 285 Final Exam December 18, 2001 7 (iii) (4 points) The graph of f is pictured above. Which of the pictures below is the graph of the derivative of f A B C None of the below. A7? Fifi 6% 8 MAT 285 Final Exam December 18, 2001 Part III Problem 6 (9 points) For each of the following questions, circle the correct answer. a. The global minimum of f = 13—3332—24r+5 on the interval -3 S 1: g 7.1 occurs at: a: = —2; (I: = 7; :1: = —; a: = 7.1; none of these. b. The cost (in dollars) of producing 11) Widgets is given by the equation C(w) = 0.011112 —— 0.611) + 240. At the production level of 60 Widgets, the marginal cost is: $4 per widget 60 cents per widget 0.4 Widgets 60 cents equal to the average cost none of these c. A spherical balloon is leaking helium. Recall that the volume 4 V of a shpere of radius r is given by V = -7rr3. If the volume is decreasing at a rate of 4 cubic inches per hour when the radius of the balloon is 5 inches, then: 1 the rate of change of the radius is E square inches per hour; 7T . . 1 . the rate of change of the radius is —§5—— square inches per hour; 77 . . 1 . the rate of change of the radius is 55— inches per hour; 7r . . 1 . the rate of change of the radius 1s —§5—— square 1nches per hour; 7r none of these. MAT 285 Final Exam December 18, 2001 9 Problem 7 (8 points) The population of a country is modelled by 100 th l 't' t. Pt = -—-—— e 0923 ic func ion ( ) 1 +996_0.3t, years since 1920 and P(t) is in millions. where t is the number of a. {2 points) Find the intial population. b. (5’ points) When is the population growing fastest and at what rate? c. (3 points) The farmers of the country can provide sufiicent food for a population of 75,000,000 people. In what year will the population reach 75,000,000 people? 10 MAT 285 Final Exam December 18, 2001 Problem 8 {8 points) You are to design a set of four identical 3- sided study camels, each with an area of 12 sq. ft. mm Find the dimensions of the camels for the configuration of minimum wall length. MAT 285 Final Exam December 18, 2001 11 Part IV Problem 9 (9 points) For each of the following questions, circle the correct answer. a. Let 2 = g(m,y) be given. Which of the following statements 3 expresses the inequality: a—ilmyo < 0? At the point (930, yo), z is increasing in the positive a: direction; At the point (2:0, yo), 2 is decreasing in the positive x direction; At the point (230,310), 2 is increasing in the positive y direction; At the point ($0,310), 2 is decreasing in the positive y direction; none of these. b. The Cobb-Douglas Production function for a small printing firm is P = 0.8N0‘6V0‘4 where N is the number of workers, V is the value of the equipment in thousands of dollars and P is the production level in thousands of pages per day. Suppose that this firm has a labor force of 150 and the value of its equipment is $1,000,000. If the work force is increased slightly production will: increase at the rate of 1,025 pages per day; decrease at the rate of 1,025 pages per worker; increase at the rate of $1,025 per day; increase at the rate of 1,025 pages per day per worker; remain the same; none of these. 12 MAT 285 Final Exam December 18, 2001 c. Consider the function 21 2:, y = M, pictured below in the 25 3D window —10 g a: £10; —10 S y _<_ 10; -5 S 2 S 10. Circle the letter which corresponds to the slice of this surface for y = 4. (The cross section perpendicular to the y-axis at y = 4.) function: A B C D E F MAT 285 Final Exam December 18, 2001 13 Problem 10 (8 points) Minimize f(x, y) = 2:132 + By2 + 4x - y + 10 subject to the constraint :r + y = 28 using the method of LaGmnge multipliers. Give the minumum value of f (53,31) subject to the con- straint x + y = 28 SHOW THE MAIN STEPS OF YOUR WORK 14 MAT 285 Final Exam December 18, 2001 Problem 11 (8 points) Let f(z,y) = x3 + 51:2 + 3/2 — z — 2y + 2. Find all local maxima and minima and verify that they are indeed local maxima or minima. SHOW THE MAIN STEPS OF YOUR WORK ...
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MAT285-2001Fall - MAT 285 Final Exam Signature Instructions...

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