First Midterm Exam Solution Frenkel Spring 2011

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online