examfinalsols03

# examfinalsols03 - Math 212 Multivariable Calculus - Final...

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Math 212 Multivariable Calculus - Final Exam Instructions: You have 3 hours to complete the exam ( 12 problems ). This is a closed book, closed notes exam. Use of calculators is not permitted. Show all your work for full credit. Please do not forget to write your name and your instructor’s name on the blue book cover, too. Print your instructor’s name : Print your name : Upon ﬁnishing please sign the pledge below: On my honor I have neither given nor received any aid on this exam. Signature : Problem Max Points Your Score Problem Max Points Your Score 1 8 7 8 2 8 8 8 3 8 9 9 4 8 10 9 5 8 11 9 6 8 12 9 Total 100 1

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(8 points) Find the equation of the tangent line to the path c ( t ) = (cos t, sin t, t 2 ) at t = π . Solution. The derivative of this path is given by c ± ( t ) = ( - sin ( t ) , cos ( t ) , 2 t ) The tangent line to c ( t ) at the point t = π is given by L ( t ) = c ( π ) + t · c ± ( π ) = ( cos ( π ) , sin ( π ) , π 2 ) + t · ( - sin ( π ) , cos ( π ) , 2 π ) = ( - 1 , 0 , π 2 ) + t · (0 , - 1 , 2 π ) = ( - 1 , - t, π 2 + 2 πt ) (Or you may use L ( t ) = c ( π ) + ( t - π ) · c ± ( π ).) [2] (8 points) Let f : R 3 R be deﬁned by: f ( x, y, z ) = 3 z + e x 2 - y 2 Let C be the set of the heads of unit vectors v in R 3 such that f increases at 1 / 3 of its maximum rate of change in the direction v starting from (0 , 0 , 1) . Find the equation(s) which determine(s) the set C . (Hint : C is a circle in R 3 .) Solution. The gradient of f is given by f = ( 2 x exp ( x 2 - y 2 ) , - 2 y exp ( x 2 - y 2 ) , 3 ) f (0 , 0 , 1) is a vector in the direction of greatest increase for the function f at the point (0 , 0 , 1). So f (0 , 0 , 1) = (0 , 0 , 3) = 3 · (0 , 0 , 1) Calculate the maximimum rate of change of f by evaluating the directional derivative of f . Df (0 , 0 , 1) = f · (0 , 0 , 1) = (0 , 0 , 3) · (0 , 0 , 1) = 3 1 / 3 of this rate is just 1, so the goal of this problem is to ﬁnd and equation for the unit vectors v = ( x, y, z ) such that Df v = 1 at the point (0 , 0 , 1).
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## This note was uploaded on 10/04/2010 for the course MATH 166 taught by Professor Hotlz during the Spring '08 term at HCCS.

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examfinalsols03 - Math 212 Multivariable Calculus - Final...

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