practice_final.pdf - Sample Final Exam Math 32A/3 Fall 2014...

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Sample Final Exam Math 32A/3, Fall 2014 This is a collection of problems that would not be unreasonable for a real midterm exam. WARNING: the inclusion (or exclusion) of a certain topic or type of problem on this sample exam does not guarantee its inclusion (or exclusion) on the actual exam! 1. Recall that the cycloid generated by a circle of radius R is the curve with parametrization r ( t ) = h R ( t - sin t ) , R (1 - cos t ) i . Find the length of this curve when 0 t 2 π . [The identity 1 - cos t = 2 sin 2 ( t/ 2) may be useful.]
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2. Find and classify all critical points of the function f ( x, y ) = 1 4 x 4 + y 4 + 1 2 x 2 - 2 y 2 . [Hint: there are three critical points.]
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3. (a) Describe the set of all vectors perpendicular to both u = h 1 , 0 , 1 i and v = h 0 , 2 , 3 i . (b) Let f ( x, y ) = 1 2 log( x 2 + y 2 ). At the point P = ( x, y ) 6 = (0 , 0), find a unit vector u such that the directional derivative D u f ( P ) is maximal. What is the maximal value of D u f ( P )?
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4. Decompose the acceleration vector a ( t ) of r ( t ) = h 2 3 t 3 , 2 t i into tangential and normal compo- nents. That is, compute each quantity in the expression a = a T T + a N N as a function of t .
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5.
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