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APPM 2350 FINAL EXAM FALL 2007 INSTRUCTIONS: Electronic devices, books, and crib sheets are not permitted. Write your (1) name, (2) instructor’s name, and (3) recitation number on the front of your bluebook. Work all problems. Show your work clearly. Note that a correct answer with incorrect or no supporting work may receive no credit, while an incorrect answer with relevant work may receive partial credit. 1. (30 Points) Find the absolute maximum and minimum values of the function f ( x,y ) = 20 - 16 x - 4 y + 4 x 2 + y 2 on the closed region bounded by the lines x = 4, y = x , and y = 0. Be sure to clearly give both the locations and values of the absolute extremum. 2. (30 points) Consider the curve y = x for 0 x 3. Find the point on the curve closest to, and furthest from, the point (2 , 0). Although this problem can be worked using simple Calculus I principles, you need to work this problem using Calculus III principles! Clearly explain your approach as you work through the problem! (Hint: you may find it useful to graph the constraint curve in the x - y plane as well as the level curves of the objective function.) 3. (30 Points) Consider a particle moving along a path in space through a temperature field T ( x,y,z ). At a particular instant in time (and only at that instant in time) the position of the particle is r = 2 i + j + k , its velocity is v = 2 i + 2 j + k , and its acceleration is a = 2 j . At the location (2 , 1 , 1) the value of the temperature is T (2 , 1 , 1) = 5 and the gradient of the temperature field at the same location is T (2 , 1 , 1) = 3 i + 2
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