Fall 07 - MAC 2311 ~— EXAM 2 FALL 2007 A Sign your...

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Unformatted text preview: MAC 2311 ~— EXAM 2 FALL 2007 A. Sign your scantron sheet in ink in the white area on the back. B. Write and encode in the spaces indicated: 1) Your name (last name, first initial, middle initial) 2) Your UP ID number 3) Your discussion section number C. Under ”Special Codes”, encode the test number 2,1 . D. At the top right of your scantron sheet, for ”Test Form Code”, encode A. a B C I) E E. This test consists of 8 four—point multiple choice questions, 4 two-point mul— . tiple choice questions, 2 two-point bonus multiple choice questions, and four pages (two pages front and back) of free response questions worth forty points. The time allowed is 90 minutes. F. WHEN YOU ARE FINISHED: 1) Before you turn in your test, check carefully for transcribing errors. Your responses cannot be changed after you turn in your exam. 2) You must turn in your scantron and free-response portion of the exam to the proctor. Note: Problems 1—8 are worth four points each. 1. Find the equation of the tangent line to the graph of the functiorr f (w) = :c + (at — 2 same) at the point with :c—coordinate 0. Amex—l B.y“—:1+:E C.y=«-~1n—a: D.y=1-«~3m I E.y=4sc+1 2. Suppose the equation e” = 2: — y - 1 defines y implicitly as a function of 9:. What is the slope of the tangent line to the curve at the point ( 2, 0) ? D.-~ E 1 1 . . C." .—— A1 B2 . 2 3 4 3. In a certain country, the number N of people that contract a certain iflness within radius r mfles from a polluted lake is given by N(r) = 2000x/100 —— 2r . Calculate the rate at which the number of 111 persons is changing with respect to the radius when that radius is 18 miles. A. —125 persons / mi 3. M225 persons / mi C. —250 persons / mi D. m200 persons/mi E. ~160 persons/mi 4. Evaluate the limit: lim M “H1 51: + 111(cc) m 1 5. The depth (in feet) of water in a reservoir at time t days from now is given by the function D(t) z: 20t — t3 — 18 Where D n 0 represents its normal depth. Choose the time 1: below at which the depth is greater than normal, but is decreasing. A.1 13.2 (3.3 1:14 '11:) 6. Let f, g, and h be functions such that 11(3) 2 “236) Find h’(2) given that: 9($2)' 9(4) e 2 WM) = W2 f(4) : m1 9’01): "é" 3 3 3 3 A. ””21 B. 7 c. 3 E). E 11.1 '7. Given the function f(3:) 22“ tan(29:), evaluate the second derivative f ”(1) . A. 8 sec2(2) tan(2) B. 8 sec(2) tan(2) C. 8393029) D. 2 'sec(2) tan2 (2) E. 4 sec2 (2) tan2(2) 3:2 8. At which of these w~values does the function g(r:;) x W have a horizontal tangent line. Note: Problems 9—12 are worth two points each. 9. Which of the following derivatives are correct? d 2 m 2 d x __ w d e W 3—1 I. 33:— (in(:c)) .... E I}, dx 7? w 7r [1n(7r)] III. a; cc m- 82: A. I only B. III only C. I and II only D. I and III Only E. II and 111 only 263 10. The graph of the function f (2:) == has which horizontal asymptotes ? 2wex Ayaionl Bywayml Gym—20111 Y Y Dym—Zyxo ‘Eyzwzy=1 11. Suppose the depth 5' of sand in centimeters on a certain area of Venice Beach is a function of the height :0 of the tide in feet given by 8(m) 2 25+ 1. Suppose m the height of the tide after 15 days is given by 33(t) m 4.2? _ t2. Find the rate of change of the depth of the sand with respect to time after t m 1 day. 2 1 1 A. —§ cm/day B. m5 cam/day C. W6» cm/day 1 3 D. m1 cm/day E. —E cm/day ' __1__ 12. Evaluate the h'mit: lim (1 + 3x?) 111(3) . $"PDO A. e B. 62 C. e3 D. 3 E. 1 Bonus!!(2 points each) 13. In your homework and projects, you enccmntered the Logistic function. What did we call its positive horizontal asymptote? A. Carrying Capacity 13. Point of No Return C. Doomsday D. Dynamic Equilibrium , 14. In our lecture, we discussed a function for which implicit differentiation failed at the point (0, 0 ). It was called. . . A. Curve of Triton B. A Cycloid C. A Swaflowtail D. Devii’s Curve MAC 2311 ~— EXAM 2 Free Response NAME . SECTION UF ID ' TEST FORM CODE A YOU MUST SHOWALL OF YOUR WORK TO RECEIVE CREDITII 1. A particle moves in a straight line so that its position in inches after time t 1 seconds is given by the function 505) m 31761 a) Find a formula for the velocity of the particle at every time t. b) At what time(s) does the particle have zero velocity? if none, write none. c) Find the acceleration of the particle after 1 second. Include units. 2. For the following functions, calculate and factor the derivative, and then list just the $~va1ues at which it has horizontal and vertical tangent lines. If there are none, write ”none”. Be mindful of the domain. 1 a} 9(93) = :c m 122:3 HTLS: VTLS: b} h($) ____ a”flog—334% HTLS: VTLS: NAME SECTION 3. Water fills and then drains from a tank so that the amount remaining (in 20 l't ftt"th d" thw————————-~——.A h 1 ers) a er mmu es ave passe 1s given y ( ) 10 * 6 x/t-H roug sketch of the graph is provided below. We) a 4; a) What is the initial amount of water? What does the volume tend toward as time passes (to infinity)? ‘ b) Calculate the average rate of change in volume from the end of the first minute to the end of the sixteenth minute. Include units. c) Calculate the rate of change in volume after exactly four minutes have passed, and also after sixteen minutes have passed. Include units. (1) After how many minutes does the tank switch from being filled with I water to draining? What is the largest amount of water in the tank? 4. Write a formula for the derivatives of the fofldwing functions: a} g(x) = 10g5(ac + 3) + tan—103:) 2 b) Mm) = :L‘ 22: ' (Use logarithmic differentiation.) ‘1 + costs) a: < 0 c) f (m) 2 _ $+2 m>0 Bonus) Use limits to determine whefller we can choose some number a so that the f (as) in part (c) is continuous and/0r differentiable at x m 0 by defin~ ' ins f (0) m a- ...
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