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Exam_exam_1_(4)

# Exam_exam_1_(4) - (a A function F t = f t g t h t is...

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MATH 241 – Exam 1 – Boyle February 18, 1998 Show your work. Put a box around the result of a computation. 1. (28 points) Let a = (3 , 1 , 0) and b = (2 , 2 , 0). (a) Compute the area of the triangle whose vertices are the points a , b and (0 , 0 , 0). (b) Find a vector equation for the line through the points a and b . (c) Find an equation for the plane which contains the point b and is perpendicular to the vector a . (d) Find a nonzero vector perpendicular to the plane through the points (1 , 1 , 1) , (2 , 3 , 4) and (1 , 1 , 2). 2. (35 points) Over the time interval 0 t π , a particle travels in space with position r ( t ) and velocity v ( t ) = ( - 4sin(2 t ) , 4cos(2 t ) , 2). (a) Assume that the particle has position (2 , 0 , 0) at t = 0. Give a formula for r ( t ). (b) Compute the unit tangent vector T at time t . (c) Compute the the standard normal vector N at time t . (d) The curve C parametrized by r ( t ) lies on a cylinder. Given an equation for that cylinder. (e) What is the length of the curve C ? 3. (20 points) For each part, answer TRUE or FALSE. No explanation required.
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Unformatted text preview: (a) A function F ( t ) = ( f ( t ) , g ( t ) , h ( t )) is continuous if and only if each of the functions f, g and h is continuous. (b) If a particle moves in space at constant speed, then it has zero acceleration. (c) The curve parametrized by r ( t ) = (3sin( e t ) , 3cos( e t )) has curvature 1 / 3 at every point. (d) Some plane contains the curve parametrized by F ( t ) = (4cos(3 t ) , 5sin(3 t ) , 6cos(3 t )). 4. (10 points) Suppose a and b are unit vectors and the angle between them is θ . (a) For which θ in [0 , π ] is | a · b | a minimum? (b) For which θ in [0 , π ] is | a · b | a maximum? 5. (10 points) Give a parametrization for the triangle below with the given orientation. [In the original test there was a picture of a speci±c triangle with directions indicated on the edges.]...
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