Math 41 Final Exam December 9, 2013 Name: SUID#: Circle your section: Valentin Buciumas Jafar Jafarov Jesse Madnick Alexandra Musat Amy Pang 02 (1:15-2:05pm) 08 (10-10:50am) 03 (11-11:50am) 06 (9-9:50am) ACE (1:15-3:05pm) 07 (10-10:50am) 10 (9-9:50am) 04 (1:15-2:05pm) 09 (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 proved in class or the text, but be sure to clearly state the result before using it, and to verify that all hypotheses are satisfied. Please check that your copy of this exam contains 14 numbered pages and is correctly stapled. This is a closed-book, closed-notes exam. No electronic devices, including cellphones, headphones, or calculation aids, will be permitted for any reason. You have 3 hours. Your organizer will signal the times between which you are permitted to be writing, including anything on this cover sheet, and to have the exam booklet open. During these times, the exam and all papers must remain in the testing room. When you are finished, you must hand your exam paper to a member of teaching staff. Paper not provided by teaching staff is prohibited. If you need extra room for your answers, use the back side of the question page or other extra space provided at the front of this packet, and clearly indicate that your answer continues there. Do not unstaple or detach pages from this exam. 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. Formulas for Reference: Signature: V sphere = 4 3 πr3 SA sphere = 4πr 2 A circle = πr 2 V cylinder = πr 2 h V cone = 1 3 πr2 h n i = i=1 n(n + 1) 2 n i 2 = i=1 n(n + 1)(2n + 1) 6 n ( n(n + 1) i 3 = 2 i=1 ) 2
Math 41, Autumn 2013 Final Exam December 9, 2013 Page 1 of 14 1. (10 points) In each part below, use the method of your choice, but show the steps in your computations. (a) Find f (x) if: f(x) = arctan(e x ) csc 5 (3x) + log 2 (2x 2 ) (b) Find dy dx in terms of x and y if: y + xy = 6
Math 41, Autumn 2013 Final Exam December 9, 2013 Page 2 of 14 2. (9 points) A tall, open pot has the shape of a cylinder, with a circular base of radius 4 in. A marble with radius r inches, where 0 < r < 4, is placed in the pot, and the pot is filled with just enough water to cover the marble completely. What radius of marble requires the greatest amount of water to accomplish this? Show all your reasoning.
Math 41, Autumn 2013 Final Exam December 9, 2013 Page 3 of 14 3. (10 points) Find each of the following limits, with justification. If there is an infinite limit, then explain whether it is or. (1 cos x) 2 (a) lim x 0 x sin x (b) lim x) x x(1/
Math 41, Autumn 2013 Final Exam December 9, 2013 Page 4 of 14 4. (18 points) Let f(x) = xe (1/x). (a) Determine, with complete reasoning, whether f has any asymptotes (horizontal or vertical), and give their equations. Compute both one-sided limits for any vertical asymptotes. (b) On what ( interval(s) is f increasing? decreasing? Explain completely. Note: it is a fact that f (x) = 1 + 1 ) e (1/x), which you do not have to prove. x
Math 41, Autumn 2013 Final Exam December 9, 2013 Page 5 of 14 For easy reference, f(x) = xe (1/x) and f (x) = ( 1 + 1 ) e (1/x). x (c) On what interval(s) is f concave upward? downward? Explain completely. (d) Using the information you ve found, sketch the graph y = f(x). Label and provide (x, y) coordinates of any local extrema and inflection points.
Math 41, Autumn 2013 Final Exam December 9, 2013 Page 6 of 14 5. (12 points) For this problem, we try to solve the equation x 3 3x 2 + 1 = 0. (a) Show that the equation has a solution in the interval (0, 1). Explain your reasoning completely; however, you don t have to find the exact value. (b) Use Newton s method with initial guess x 1 = 1 to produce the next approximation x 2 to the solution. Show all of your steps and simplify completely. (c) Now use your result from (b) to produce the next approximation x 3. You do not need to simplify your answer. (d) Explain what happens if Newton s method is instead used with the initial guess x 1 = 0.
Math 41, Autumn 2013 Final Exam December 9, 2013 Page 7 of 14 6. (10 points) A thin, horizontal copper rod is being subjected to various sources of heating and cooling along its length. At each position x, where x is measured in inches from the left end of the rod, suppose h(x) represents the rate, in Celsius degrees per inch, at which the rod s temperature is changing with respect to changes in position. Approximate values of h(x) were obtained from a heat flux analysis of the rod and listed in the chart below: x 0 1 2 3 4 5 6 7 8 9 10 h(x) 8 4 2 1 3 6 1 3 1 3 5 (a) Without calculating it, what does the quantity 10 0 h(x) dx represent? Express your answer in terms relevant to this situation, and make it understandable to someone who does not know any calculus. Be sure to use any units that are appropriate, and also explain what the sign of this quantity would signify. (b) Use the Right Endpoint Rule with n = 3 to estimate 9 0 h(x) dx; give your answer as an expression in terms of numbers alone, but you do not have to simplify it. (c) Use the Midpoint Rule with n = 2 to estimate 10 2 h(x) dx; again give your final answer in terms of numbers alone.
Math 41, Autumn 2013 Final Exam December 9, 2013 Page 8 of 14 7. (15 points) (a) Suppose g(x) = x(4 x). Let R be the region in the xy-plane bounded by the x-axis and the curve y = g(x), for 0 x 4. Find the area of R by evaluating a limit of a Riemann sum that uses the Right Endpoint Rule; show all reasoning.
Math 41, Autumn 2013 Final Exam December 9, 2013 Page 9 of 14 n ( ) πi πi π (b) Express the limit lim n 2n cos as a definite integral, and then compute its value using 2n 2n i=1 the Evaluation Theorem. Show all the steps of your reasoning.
Math 41, Autumn 2013 Final Exam December 9, 2013 Page 10 of 14 8. (6 points) List the following quantities in increasing order (from smallest to largest). No justification is necessary. (A) (B) (C) (D) (E) (F) 5 1 4e 4 5 e5 e x x dx e + e2 2 + e3 3 + e4 4 e 2 2 + e3 3 + e4 4 + e5 5 e e 3 + e2 4 + e2 e + e3 5 6 + e3 e + e4 7 8 + e4 e + e5 9 10
Math 41, Autumn 2013 Final Exam December 9, 2013 Page 11 of 14 9. (22 points) Show all reasoning when solving each of the problems below; your final answers should not involve integral symbols. t + 1 1 t + 5 if t < 0 x (a) If f(t) = 7e 2t 1, determine a formula for g(x) = f(t) dt. 2 1 + t 2 if t 0 (b) cot θ dθ
Math 41, Autumn 2013 Final Exam December 9, 2013 Page 12 of 14 (c) 1/2 0 arcsin z dz (d) x 3 x 2 1 dx
Math 41, Autumn 2013 Final Exam December 9, 2013 Page 13 of 14 x 2 1 + cos t 10. (12 points) Let g(x) = dt. 1 t (a) Find g (x) and show that g (x) 4 for every x > 0. (b) Give a complete statement of the Mean Value Theorem. 9 1 + cos t (c) Show that dt 8. Justify all steps. (Hint: think about this integral in terms of the 1 t function g, and use the Mean Value Theorem. You can apply the result of part (a) even if you did not prove it.)
Math 41, Autumn 2013 Final Exam December 9, 2013 Page 14 of 14 11. (8 points) Mark each statement below as true or false by circling T or F. No justification is necessary. T F 1 1 dx x 2 = 0. T F a a arctan(x 5 ) dx = 0 for all a > 0. T F 1 3 ( x ) ln dx < 0. 3 T F If f(x) is even and continuous at all x, then h(x) = x 0 f(t)dt is odd. The following four questions refer to the function g(x) = { x 2 sin( 1 x ) if x 0, 0 if x = 0. T F The function g(x) provided above is continuous at x = 0. T F The function g(x) provided above is differentiable at every x 0. T F The function g(x) provided above is differentiable at x = 0. T F For g(x) provided above, the function g (x) is continuous at x = 0.