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# M 408D-Naddaf-Spring 2002-Final - MATH M408D FINAL EXAM...

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MATH M408D -- FINAL EXAM Page 2 of6 1. Please write next to each one of the following 5 statements TRUE or FALSE. Note that they are independent of each other. There is no limit on the number of TRUE statements. No need to justify your answers. (a) Directional derivative of a function is always a multiple of its gradient.Ans:False (b) Gradient of a function is always zero at its maximum or minimum on a bounded domain D. Ans:False (c) Two surfaces are perpendicular ata point iftheir normals at that point are not parallel. Ans: False (d) For a given constant e, and a nice function i . f (x, y, z) = C represents a curve in the three dimentional space. Ans: False (e) Two different level curves of the same function cannot intersect. Ans:True 2. The Celsius temperature T at a point (x, y, z) on the sphere X2 + y2 + Z2 = 1 is given by Find the pointe s) on the sphere at which the temperature is greatest. Ans: From the constraint, solve for y2 in terms of x, z and substitute that into T to create a function of two variables x, z and then use the standard method: -- 'II = 1- x2 - Z2 -tT = 10xz(1- x2 - Z2) After finding the solutions, use the second derivative test to decide which one(s) are the local maxima and among them, find the greatest.

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