First Midterm Exam Solution Frenkel Spring 2011

First Midterm Exam Solution Frenkel Spring 2011 - Solutions...

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Solutions to the First Midterm Exam – Multivariable Calculus Math 53, February 25, 2011. Instructor: E. Frenkel 1. Consider the curve in R 2 defined by the equation r = cos(2 θ ) . (a) Sketch this curve. (b) Find the area of the region enclosed by one loop of this curve. 1 2 Z π/ 4 - π/ 4 cos 2 (2 θ ) = 1 4 Z π/ 4 - π/ 4 (1 + cos(4 θ )) = π 8 . 2. (a) Find an equation of the surface consisting of all points in R 3 that are equidistant from the point (0 , 0 , 1) and the plane z = 2. The distance from a point P = ( x,y,z ) to the point (0 , 0 , 1) is p x 2 + y 2 + ( z - 1) 2 , and the distance to the plane z = 2 is z - 2. Hence we obtain the equation p x 2 + y 2 + ( z - 1) 2 = z - 2 , which gives x 2 + y 2 + ( z - 1) 2 = ( z - 2) 2 , and hence z = - x 2 2 - y 2 2 + 3 2 . (b) Sketch this surface. What is it called? This is an elliptic paraboloid which goes downward along the z axis. 3. Show that the function x 50 y 50 x 100 + y 200 does not have a limit at ( x,y ) = (0 , 0). First let’s approach (0
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First Midterm Exam Solution Frenkel Spring 2011 - Solutions...

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