UFFLoRIfiA Department of Mathematics MAC2311 Exam 2B Spring 2015 A. Sign your bubble sheet on the back at the bottom in ink. B. In pencil, write and encode in the spaces indicated: 1) Name (last name, first initial, middle initial) 2) UF ID number 3) Section number C. Under "special codes" code in the test ID numbers 2, 2. 1.34567890 1.34567890 D. At the top right of your answer sheet, for "Test Form Code", encode B. A C D E E. 1) This test consists of 15 multiple choice questions, ranging from two to five points in value, plus two white sheets (three pages) of free response questions worth 31 points. The test is counted out of 80 points, and there are 11 bonus points available. 2) The time allowed is 90 minutes. 3) You may write on the test. 4) Raise your hand if you need more scratch paper or if you have a problem with your test. DO NOT LEAVE YOUR SEAT UNLESS YOU ARE FINISHED WITH THE TEST. F. KEEP YOUR BUBBLE SHEET COVERED AT ALL TIMES. G. When you are finished: 1) Before turning in your test check carefully for transcribing errors. Any mistakes you leave in are there to stay. 2) You must turn in your scantron and tearoff sheets to your discussion leader or exam proctor. Be prepared to show your picture I.D. with a legible signature. 3) The answers will be posted in Sakai (e-learning) within one day after the exam. Your discussion leader will return your tearoff sheet with your exam score in discussion. Your score will also be posted in e-learning within one week of the exam.
NOTE: Be sure to bubble the answers to questions 1-15 on your scantron. Questions 1-10 are worth 5 points each. 1. Find the linear approximation of the function f(x) = \1'1 + x at a = 0 and use it to approximate the number ~. a. 1.01 b. 1.02 c. 1.03 d. 2.01 e. 2.02 2. If f(x) = sec(x), find f" c:} a. -3V2 b. -2V2 c. V2 d. 2V2 e. 3V2 3. Find the value(s) of x at which the graph of y horizontal tangent line. 3x ---- has a (4x + 1)2/3 3 1 a. x = -- and j; = -- 10 4 3 1 c. x = -- and x = -- 4 4 e. There is no horizontal tangent line. 3 b. x = -- only 10 3 d. x = -"4 only 4. Use the definition of the derivative to evaluate the limit: 1. 3x - 9 Im- x-t2 X - 2 a. The limit does not exist. b. 0 c. 9ln(3) d. 31n(3) e. In(3) 2B
5. Let f(x) = tan(x) + x + e3x and g(x) be the inverse of f(x). Find g'(l). 1 a. e3 c. 1 d.5 1 e -. 5 6. Let F(x) = f(g(x3)). Find F'(l) if f(-l)=o f'( -1) = 3 f(l) = -2 1'(1) = 4 g(l) = -1 g'(l) = ~ a. 1 b. 2 c. 3 d. ~ 2 3 e. - 2 7. Oil spilled from a ruptured tanker spreads in a circle whose area increases at a constant rate of 1 square miles/hour. How fast is the radius of the spill increasing when the area is 16 square miles? ft. a. - miles/hour 16 1. c. ft miles/hour ft. b. 8 miles/hour d. 1;::;;:miles/hour 8y7f 1. e. 16ft miles/hour 8. Find the slope of the normal line to the curve x3 - y3 + In(2) = In(2xy) at the point (1,1). a. -1 4 b -. 3 3 c -. 4 d ~. 2 e. -2 3B
9. A company has determined that the cost in dollars of producing x items IS C(x) = 0.01x3 + 400x + 300, x where x > O. Estimate the cost of producing the 101st item by using the marginal cost when x = 100. a. $1.96 b. $1.97 c. $2 d. $2.03 e. $2.04 10. The graph of y = f(x) is given below. -i 1 i I! i -1----+ J,....- : i --.. ~ I.. vertical tangent Which of the followingis/are true? P. The graph of f' (x) has a vertical asymptote at x = -2. Q. The graph of l'(x) has x-intercepts at x. -1 and x = 2. R. f is continuous and differentiable at x = -1. a. P only b. Q only c. R only d. None e. P, Q and R 4B
------------------------- Bonus Questions (2 points each): 11. Use the graph of y = J(x) in Question 10, which of the following intervals is l'(x) decreasing? a. (-1,1) b. (-00, -2) c. (1,00) d. (-00,1) 12. Which of the following functions is differentiable at x = O? a. y = Ixl b. y = x2/3 c. y = In(x) d. None of the above ex x < 0 13. If J(x) =, then { cos(x) x > 0 a. J(x) is neither continuous nor differentiable at x = o. b. J(x) is continuous and differentiable at x = o. c. J(x) is continuous but not differentiable at x = o. d. J (x) is differentiable but not continuous at x = o. x-i a. (x - 1)eX-2 b. e c. 3e2 + (x - 1)eX-2 15.![arctan G)] = a. 1 1 b. 2 1+ x2 l+x c. x2 d. 2 l+x 5B
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MAC 2311 Exam 2B, Part II Free Response Name: UFID#: _ Signature: Section #: _ SHOW ALL WORK TO RECEIVE FULL CREDIT. x 1. (8 pomts) Let f(x) =. 3x+ 1 (a) Use the definition of the derivative to find f'(x). (b) Use the Quotient Rule to verify your answer in part (a). f'(x) = _ 7B
2. (6 points) A particle is moving along the graph of y = x2 and let 8 be the distance between the origin and the particle at the point (x, y). (a) Use the distance formula to write an equation relating 8 and x. 82 = -------------------- (b) If the velocity in the x-direction is 3 em/min, find the rate at which the distance 8 is changing as the particle passes through the point (1,1). (include the unit in your answer). 3. (7 points) Let J(x) = (3X)2v'X. (a) Find J'(x). d8 dt (b) Find the equation of the tangent line to y = J (x) at the point (1,9). f'(x) ~ _ 8B y = ----------
Name: UF ID #: _ 4. (10 points) An object has equation of motion given by s(t) = et cos(t), where a ::; t < ]f is in seconds and s is in inches. (a) Find the velocity at time t. (b) At what time(s) is the object at rest? v(t) = _ t = seconds (c) When is the object traveling in a negative direction on the interval [0,]f)? (d) Find the acceleration at time t. --- < t < seconds a(t) = _ (e) When is the object speeding up on the interval [0,]f)? < t < seconds 9B