Math 41 Second Exam November 4, 2010 Name: SUID#: Circle your section: Olena Bormashenko Ulrik Buchholtz John Jiang Michael Lipnowski Jonathan Lee 03 (11-11:50am) 07 (10-10:50am) 02 (1:15-2:05pm) 04 (1:15-2:05pm) ACE 06 (1:15-2:05pm) 09 (2:15-3:05pm) 08 (10-10:50am) 10 (11-11:50am) Complete the following problems. In order to receive full credit, please show all of your work and justify your answers. You do not need to simplify your answers unless specifically instructed to do so. You may use any result from class that you like, but if you cite a theorem be sure to verify the hypotheses are satisfied. You have 2 hours. This is a closed-book, closed-notes exam. No calculators or other electronic aids will be permitted. If you finish early, you must hand your exam paper to a member of teaching staff. Please check that your copy of this exam contains 12 numbered pages and is correctly stapled. If you need extra room, use the back sides of each page. If you must use extra paper, make sure to write your name on it and attach it to this exam. Do not unstaple or detach pages from this exam. It is your responsibility to arrange to pick up your graded exam paper from your section leader in a timely manner. You have only until Thursday, November 18, to resubmit your exam for any regrade considerations; consult your section leader about the exact details of the submission process. Please sign the following: On my honor, I have neither given nor received any aid on this examination. I have furthermore abided by all other aspects of the honor code with respect to this examination. Signature: The following boxes are strictly for grading purposes. Please do not mark. Question: 1 2 3 4 5 6 7 8 Total Points: 13 12 9 10 13 10 17 10 94 Score:
Math 41, Autumn 2010 Second Exam November 4, 2010 Page 1 of 12 1. (13 points) Differentiate, using the method of your choice. (a) p(t) = ln(sec t + tan t) + log 2 (2 + t) (b) f(x) = 10 sin πx + x + x
Math 41, Autumn 2010 Second Exam November 4, 2010 Page 2 of 12 (c) g(x) = arctan(x cos x )
Math 41, Autumn 2010 Second Exam November 4, 2010 Page 3 of 12 2. (12 points) Consider the curve with equation x 2 y 2 + xy = 2. (a) Find an expression for dy. (Your answer can be in terms of both x and y.) dx (b) There is a unique point on the curve with x = 1 and positive y; find this point and write the equation of the tangent line to the curve at this point.
Math 41, Autumn 2010 Second Exam November 4, 2010 Page 4 of 12 For easy reference, the curve has equation x 2 y 2 + xy = 2. (c) Find all points on the curve where the slope of the tangent line is 1.
Math 41, Autumn 2010 Second Exam November 4, 2010 Page 5 of 12 3. (9 points) (a) Use linear approximation to estimate e1.1 1.1, showing your reasoning. Express your answer as a simplified rational multiple of e; that is, give your answer in the form a e, where a and b are whole b numbers. (b) Is your estimate in part (a) larger or smaller than the actual value? Justify your answer.
Math 41, Autumn 2010 Second Exam November 4, 2010 Page 6 of 12 4. (10 points) A caffeinated beverage is brewed by placing water in an inverted cone-shaped filter of radius 3 cm and height 5 cm, and allowing it to leak through the cone s vertex into a cylindrical cup of the same radius and height. (You may assume the cone is suspended well above the top of the cup.) The leaking is such that the water level in the cone filter decreases at a rate of 0.5 cm/sec. How fast is the water level in the cup rising when the water level in the cone filter is 2 cm above the vertex? Formulas for reference: the volumes of a circular cone and a circular cylinder, each of height h and base radius r, are given, respectively, by V cone = 1 3 πr2 h, V cyl = πr 2 h.
Math 41, Autumn 2010 Second Exam November 4, 2010 Page 7 of 12 5. (13 points) Find each of the following limits, with justification. If there is an infinite limit, then explain whether it is or. f(x) 1 (a) lim, where f is such that the tangent line to f at x = 4 has equation 2x 4y = 4. x 4 x 4 (You can assume that f and all of its derivatives are continuous near x = 4.) (b) lim x ln x x
Math 41, Autumn 2010 Second Exam November 4, 2010 Page 8 of 12 (c) lim x 0 cos(x)1/ sin2 (x) +
Math 41, Autumn 2010 Second Exam November 4, 2010 Page 9 of 12 6. (10 points) Let f(x) = 9x 2/3 x 5/3. (a) Find the critical numbers of f, and characterize each as a point of local minimum, local maximum, or neither. Show all steps of your reasoning. (b) Find the absolute maximum and minimum values of f on the interval [1, 8]. Justify completely. (Hint: it might be useful to note that f(x) = x 2/3 (9 x).)
Math 41, Autumn 2010 Second Exam November 4, 2010 Page 10 of 12 7. (17 points) Consider the function g(x) = xe x. (a) Determine if g has any asymptotes (horizontal and vertical), with complete reasoning. If there is any vertical asymptote, compute both corresponding one-sided limits. (b) On what interval(s) is g increasing? decreasing? Explain completely.
Math 41, Autumn 2010 Second Exam November 4, 2010 Page 11 of 12 (c) On what interval(s) is g concave up? concave down? Explain completely. (d) Using the information you ve found, sketch the graph y = g(x). Label and provide the (x, y) coordinates of any local extrema and inflection points.
Math 41, Autumn 2010 Second Exam November 4, 2010 Page 12 of 12 8. (10 points) At any point near a fixed electric charge, the electrostatic potential due to the charge is equal to the amount of charge (in coulombs) divided by the distance to the charge (in centimeters). Suppose two charges, one measuring 80 coulombs and the other 20 coulombs, are placed 10 cm apart. At what point on the line between the charges is the total electrostatic potential minimized? Justify completely.
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