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midterm1_solns

# midterm1_solns - Weds April 23 2008 Page 1/7 MATH 32A FIRST...

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Weds April. 23, 2008 Page 1/7 MATH 32A: FIRST MIDTERM EXAMINATION SOLUTIONS Spring 2008 – Copyright Dr. Frederick Park 1

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Weds April. 23, 2008 Page 2/7 1. (20 points) Find a vector function that represents the curve C of intersection of the surface z 2 = x 2 + y 2 and the plane 2 z = 1+ y for z 0. Graph the curve and indicate the direction in which the curve C is traced as the parameter t increases from your parametrization of C. Solution: There are many different ways to do this. The clearest way is to note that if we set z = c = constant, then x 2 + y 2 = z 2 = c 2 defines a circle of radius z = c . Thus, cross sections of x 2 + y 2 = z 2 wrt the x-y plane are circles. If we set y = 0, then x 2 + y 2 = z 2 becomes z 2 = x 2 or | z | = | x | , since z 0, this implies z = | x | . Similarly z = | y | when x = 0. Thus, the surface defined by x 2 + y 2 = z 2 is a cone w/ vertex emanating from the origin. Now, from conic sections, we know that a plane intersected in a non orthogo- nal manner is an ellipse. Thus C is an ellipse. Now, if we set x = z cos t and y = z sin t . Then for z = c = constant, this defines a circle of radius z 2 = c 2 oriented in the clockwise direction. If we plug these into the given equation for the plane we obtain 2 z = 1 + y = 1 + z sin t .
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midterm1_solns - Weds April 23 2008 Page 1/7 MATH 32A FIRST...

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