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charbonneau_1a

charbonneau_1a - ±v ±v =-1(c Three lines that are...

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Duke University Fall 2007 MATH 103-06 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 . (b) What is the area of the triangle PQR . (c) Find the midpoint of the segment PQ . 2. (10 pts) Find the equation of a plane that contains the line x - 2 6 = y - 1 3 = z - 3 2 and is perpendicular to 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) A vector v in three-dimensional space for which
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Unformatted text preview: ±v · ±v =-1. (c) Three lines that are mutually skewed to each other. (d) Three non-coplanar points. 4. (10 pts) Show that lim ( x,y ) → (0 , 0) y 2 x 3 + y 3 + 2 xy doesn’t exist. 5. (10 pts) Find lim ( x,y ) → (0 , 0) x 3 x 2 + y 2 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 (0 , , 0) 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 , , 2). 8. (10 pts) Compute the following partial derivatives. (a) ∂ ∂x ‡ x 2 + xy + e 3 xz · = (b) ∂ ∂z ‡ z sin ( x + y ) + y 2007 x · = 2...
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