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Midterm1

# Midterm1 - sphere centered at the orgin 6 Let f x y = x 2 y...

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Math 2411 Feb 15, 2006 Honors Calculus III Spring 2006, Georgia Tech Midterm 1 Time: 50min 1. Show that if the diagonals of a parallelogram have the same length then the parallelogram is a rectangle. 2. Find the distance between the point (3 , 4 , 5) and the plane 2 x + y +3 z = 5. 3. Show that for any collection of n real numbers x 1 , x 2 , . . . , x n , x 1 + x 2 + · · · + x n n q x 2 1 + x 2 2 + · · · + x 2 n . 4. Find the equation of the tangent plane to the surface given by x 3 + y 3 + z 3 = 7 at the point (0 , - 1 , 2). 5. Show that if the velocity of a path is always orthogonal to its position vector, that is x 0 ( t ) is orthogonal to x ( t ) for all t , then the path lies on a
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Unformatted text preview: sphere centered at the orgin. 6. Let f ( x, y ) = ( x 2 + y 2 , x ) and g ( r, θ ) = ( r cos θ, r sin θ ). Compute D ( f ◦ g ). 7. Find the length of the helix x ( t ) = (cos t, sin t, t ) for 0 ≤ t ≤ 2 π . 8. (Extra Credit) Show that the orbit of a planet always lies in a plane which passes through the sun. Each problem is worth 15pts. L A T E X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MG...
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