Fall 2006 Math 151 Common Exam 1B Thu, 28/Sep/2006Name (print):Signature:Instructor:Section #Seat #For official use only!QN PTS1–1314151617181920TotalInstructions1. In Part 1 (Problems 1–13), mark the correct choice on your ScanTron form using a No. 2 pencil. For your ownrecord, also mark your choices on your exam! ScanTrons will be collected from all examinees after 90 minutesand will not be returned.2. Be sure to write your name, section number, and version of the exam (1A or 1B) on your ScanTron.3. In Part 2 (Problems 14–20), present your solutions in the space provided. Show all your work neatly and concisely,and indicate your final answer clearly. You will be graded not merely on the final answer, but also on the qualityand correctness of the work leading up to it.4. Neither calculators nor computers are permitted on this exam.5. Please turn off all cell phones so as not to interrupt other students.1Part 1: Multiple Choice (52 points)Read each question carefully. Each problem in Part 1 is worth 4 points.1. The curve given parametrically by the equations x = cos t, y = sin2t, 0 ≤ t ≤π/2, forms(a) part of a hyperbola.(b) part of a circle.(c) a line segment.(d) part of a parabola.(e) none of the above.2. Which of the following statements is true about the graph of f(x)=x2+ 1(x + 3)4 − x2(x + 3)?(a) The graph has one vertical asymptote x = 2 and no horizontal asymptotes.(b) limx→2−f(x)= limx→2+f(x).(c) The graph has vertical asymptotes x = 2, x = −2, x = −3, and horizontal asymptote y = −1.(d) The graph has vertical asymptotes x = 2, x = −2, and horizontal asymptote y = −1.(e) limx→−3f(x)does not exist.3. Evaluate limx→−∞√x2+ 4x4x + 1.(a) −14(b) −∞(c)14(d) 0(e) −124. One of the following intervals contains a number c such that c3+ 2c − 12 = π . Which one?(a)(4, ∞)(b)(3, 4)(c)(0, 1)(d)(1, 2)(e)(2, 3)5. Refer to the graph of y = f (x) given below, then decide which of the following statements is false.–10123y–2 –1 1 2 3x(a) limx→2−f (x) = 2 (b) limx→1f (x) = 2 (c) limx→0f (x) = 0 (d) limx→1−f (x) = 1 (e) limx→1+f (x) = 26. The following statements pertain to the figure below. Which statements are true?(i) a + b = c (ii) a − b = c (iii) b − a = c (iv)|b|+|c|=|a|(a) Only statement (ii) is true.(b) Only statement (iii) is true.(c) Statements (ii) and (iii) are both true.(d) Only statement (i) is true.(e) Statements (iii) and (iv) are both true.bca7. If a =h1, −4iand b =h2, 3i, find 3a − 2b.(a)h−4, 17i(b)h3, −1i(c)h5, −6i(d)h−1, −18i(e)h7, −6i38. Evaluate limh→0√9 + h − 3h.(a) does not exist(b) ∞(c) 0(d) 1(e)169. Suppose v and w are vectors with|v|= 2 and|w|= 3. If the angle between v and w is 60 degrees,compute v ·(2w − 3v).(a) 12 − 3√2(b) 6√3 − 12(c) 0(d) −6(e) 2√1310. Suppose f is a function defined on an open interval containing the point 3 and that f(3)= 2.Which of the following statements is always true?(a) If limx→3f(x)= 2, then f is differentiable at x = 3.(b) If limx→3f(x)= 2, then f is continuous at x = 3.(c) If limx→3f(x)exists, then f is continuous at x = 3.(d) Either limx→3−f(x)or limx→3+f(x)must exist.(e) limx→3f(x)= 2.411. Let a = 2i − 3j and b = i + 2j. Find compab, the scalar projection of b onto a.(a) −4√1313(b)4√55(c) −4(d)4√1313(e) −4√5512. Find limx→2+|3x − 6|6 − 3x.(a) does not exist(b) ±1(c) 1(d) 0(e) −113. Let f be the function defined by f(x)=x2+ 3, if x > 0;2x + 3, if x < 0;1x + 1, if x = 0.Which of the following statements is true?(a) f is continuous everywhere.(b) f is not continuous at x = −1.(c) f is not continuous at x = 0 because limx→0f(x)does not exist.(d) f is not continuous at x = 0 because limx→0f(x)does not equal f(0).(e) All of the above statements are false.5Part 2: Work-Out Problems (48 points)Partial credit is possible. SHOW ALL STEPS!14. Differentiate; you need not simplify.(a) [3 points] y = 4 −2x+3x2(b) [3 points] f(x)= 3x +4√2x5(c) [3 points] g(s)=7s − 38s + 2(d) [3 points] h(t)=t3− 2t5− 7t + 6t7+ t5+ t3+ t15. A particle is moving in the x y-plane. Its position at time t is given by the position vectorr(t)=4t2i +t3− 9t − 2j. Distances are in feet; times are in seconds.(a) [2 points] At what point is the particle at time t = 2? Write your answer in the form(a, b).(b) [3 points] What is the particle’s velocity at time t = 2?(c) [2 points] What is the particle’s speed at time t = 2? You need not simplify.(d) [2 points] At what time(s) will the particle’s velocity be vertical (parallel to j)?616. (a) [3 points] Find parametric equations for the line that contains the points(4, 9)and(−6, 2).(b) [2 points] The graph of 3x + 2y = 5 is a line L. Which of the following statements is true?Circle the true statement.i. The vector 2i − 3j is parallel to the line L.ii. The vector 2i − 3j is perpendicular to the line L.iii. The vector 2i − 3j has no special relationship with L.(c) [2 points] Find a unit vector that is orthogonal to the vector −2i + 5j.17. Find the following limits.(a) [3 points] limx→−2x2− 4x2− x − 6(b) [3 points] limx→∞1 + 2x + 3x21 − x − x2718. [5 points] Find an equation of the tangent line to the graph of f(x)=2x − 7x2+ 6x − 4at the point where x = 1.Write your answer in the form Ax + By = C, where A, B, C are integers.19. [4 points] Suppose f is a function such that f(3)= f0(3)= 5. Find limx→3f(x)2− 25x − 3.Your work needs to justify your answer. Do not skip steps.820. [5 points] Let f be the function defined by f(x)=√x + 8. Use the definition of the derivative to find f0(1),the derivative of f at x = 1. Your work must clearly show that you know the definition of
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