charbonneau_1b

# charbonneau_1b - dicular(c Three lines that are mutually...

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Duke University Fall 2007 MATH 103-11 Intermediate Calculus Midterm #1: September 20, 2007. No calculator allowed. No books or notes allowed. You have 75 minutes to complete this exam. I have adhered to the Duke Community Standards in completing this exam. Signature: Name: 1. (10 pts) Given the points P = (1 , 2 , 3), Q = (5 , 6 , 7), R = (1 , 1 , 1). (a) Find the vectors ± PQ , ± PR and ± PQ × ± PR . Find also ± PQ · ± PR . (b) What is the area of the parallelogram with sides ± PQ and ± PR . (c) Find points A and B cutting the segment PQ in three equal parts. 2. (10 pts) Find the equation of a line that is perpendicular to the line x - 2 6 = y - 1 3 = z - 3 2 and is contained in the plane x + y + z = 3. 3. (10 pts) Give examples of the following or explain why they don’t exist? 1

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(a) A line intersecting the x -axis and parallel to the line given by the parametric equation ( x, y, z ) = (2 , 3 , 4) + t (1 , 1 , 1). (b) Four vectors in three-dimensional space that are mutually perpen-
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Unformatted text preview: dicular. (c) Three lines that are mutually skewed to each other. (d) Three non-intersecting planes. 4. (10 pts)Find lim ( x,y ) → (0 , 0) x 5 + y 5 x 4 + 2 x 2 y 2 + y 4 if it exists and justify your answer. 5. (10 pts) Find lim ( x,y ) → (0 , 0) x 2 x 2 + y 2 if it exists and justify your answer. 6. (10 pts) For the curve ± r ( t ) = (3 t-t 3 , 3 t 2 , 3 t + t 3 ), compute at the point (2 , 3 , 4) the curvature κ , the unit tangent vector ± T and the unit normal vector ± N . 7. (10 pts) Find the distance between the plane of equation x + y + z = 1 and the point P = (0 , , 3). 8. (10 pts) Compute the following partial derivatives. (a) ∂ ∂x ‡ x 2 + xy + ln (3 xz ) · = (b) ∂ 2 ∂x∂y ‡ x 2 + cos( x ) sin(8 y ) + x z y · = 2...
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charbonneau_1b - dicular(c Three lines that are mutually...

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