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PHZ 3113 Fall 2011 — Practice Exam 1 DO NOT TURN THE PAGE UNTIL INSTRUCTED TO DO SO Instructions: Attempt all five questions. The maximum possible credit for each question is shown in square brackets. Please try to write your solution neatly and legibly. You will receive credit only for knowledge and understanding that you demonstrate in your written solutions. It is in your best interest to write down something relevant for every question, even if you can’t provide a complete answer. To maximize your score, you should briefly explain your reasoning and show all working. Give all final algebraic answers in terms of variables defined in the problem. During this exam, you may use one formula sheet. You are not permitted (a) to consult any other books, notes, or papers, (b) to use any electronic device, or (c) to communicate with anyone other than the proctor. In accordance with the UF Honor Code, by turning in this exam to be graded, you affirm the following pledge: On my honor, I have neither given nor received unauthorized aid in doing this assignment. Print your name where indicated below, and sign to confirm that you have read and understood these instructions. Please do not write anything else below the line. Name (printed): Signature: Question 1 2 3 4 5 Total Score 1. [20 points] Expand x/(ex − 1) to order x2 for 0 < x 1. 2. [20 points] The equation of state for a van der Waals gas is a p + 2 (v − b) = RT, v where a, b, and R are constants. Consider two experiments on such a gas confined to a cylinder where you may control the pressure p, the molar volume v, and/or the absolute temperature T . (a) Find (∂v/∂p)T . (b) Find (∂v/∂T )p . 3. [20 points] Evaluate the integral Z 0 Z π dy y π dx sin x . x 4. [20 points] Use the method of Lagrange multipliers to find a minimum of the function f = 1/x + 4/y + 9/z among points lying on the plane x + y + z = 12. Choose a solution where x, y, z > 0. 5. [20 points] If ∇ · A = 0 and ∇ · B = 0, show that ∇ × (A × B) = (B · ∇)A − (A · ∇)B. The following are problems of similar difficulty to those above and might have appeared instead. 6. [20 points] Calculate the intervals of convergence of the series ∞ ∞ X X (nx)n xn (a) , (b) . 2+1 n! n n=1 n=1 7. [20 points] Change variables using x = u + v, y = u − v to rewrite ∂2f ∂ 2f − 2 =1 ∂x2 ∂y in terms of u and v. Do not attempt to solve the equation. p 8. [20 points] Suppose that v = 4yx̂ + xŷ + 2zẑ and f (x, y, z) = 1/ x2 + y 2 + z 2 . (a) Find ∇ × v. (b) Evaluate ∇f and ∇2 f . Acknowledgment: This practice exam was adapted from one provided by Prof. Peter Hirschfeld. 2