M23Fall02 Sample Exam 1

M23Fall02 Sample Exam 1 - , 4 ,-8). 5. Find the length of...

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Mathematics 23 Midterm Exam, September 26, 2002 Answer all questions and be sure to show all work . No notes, books, or calculators are allowed. 1. a. Describe in words the three dimensional region represented by the inequality x 2 + y 2 1. b. Write inequalities to describe the half-space consisting of all points to the right of the xz plane. 2. Find a unit vector that is orthogonal to both i + j and i + k . 3. Find an equation of the plane that passes through the point (1 , 4 , 2) and contains the line x = 3 t,y = 2 + t,z = 2 - t . 4. Consider the curve given by the parametric equations x = t 2 ,y = t 2 ,z = t 3 . a. Find the unit tangent vector at the point (4 , 4 , - 8). b. Find parametric equations for the tangent line at (4
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Unformatted text preview: , 4 ,-8). 5. Find the length of the curve r ( t ) = 2ti + e t j + e-t k , 0 t 1. 6. Find the velocity vector v ( t ) of a particle that has acceleration a ( t ) = 4 ti + 6 tj + k,v (0) = 0. 7. Consider the curve r ( t ) = t 2 i + tk. a. Find the curvature of the curve at the point (4 , , 2). b. Find the point (in xyz coordinates) where the curvature is largest. 8. Sketch the region bounded by the surfaces z = x 2 + y 2 and x 2 + y 2 = 1. Describe both surfaces using cylindrical coordinates. 1...
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