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21c_02f_fe

# 21c_02f_fe - Bender Math 21C Final Exam Fall 2002 Put your...

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Bender Math 21C Final Exam Fall 2002 Put your name, ID number, and section number (or time) on your blue book. You may have TWO PAGES of notes. NO CALCULATORS are allowed. You must show your work to receive credit. Please start each problem on a new page. 1. (15 pts) The equation z = 4 - x 2 - y 2 describes a surface. Write down an iterated integral for the area of that part of the surface that lies above the xy -plane. You need not evaluate the integral. 2. (20 pts) Change (1 , 3 , 2) from rectangular coordinates to (a) cylindrical coordinates and (b) spherical coordinates. Your answers should give all angles exactly in radians and should not contain any inverse trig functions (that is, functions such as cos - 1 ). 3. (20 pts) Here are two skew lines in parametric form: h x ( t ) , y ( t ) , z ( t ) i = h 1 , 1 , 0 i t, h x ( t ) , y ( t ) , z ( t ) i = h 0 , 1 , 2 i t + h 1 , 1 , 1 i . (a) Find a vector v that is perpendicular to both lines. (b) Compute the minimum distance between the lines.

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