exam1s4 - (b Write down a piecewise parametrization of the...

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Math 241 Exam 1 Sample 4 Directions: Do not simplify unless indicated. No calculators are permitted. Show all work as appro- priate for the methods taught in this course. Partial credit will be given for any work, words or ideas which are relevant to the problem. Please put problem 1 on answer sheet 1 1. Given the following data: P = ( - 1 , 0 , 3) Q = (2 , 5 , 5) ¯ a = 1ˆ ı + 2 ˆ + 0 ˆ k ¯ b = 3ˆ ı + 2 ˆ + 1 ˆ k (a) Find a vector of length 2 pointing from P toward Q . [7 pts] (b) Find the cosine of the angle between ¯ a and ¯ b . [6 pts] (c) Find Pr ¯ a ¯ b . [7 pts] Please put problem 2 on answer sheet 2 2. (a) Find the distance between the point (3 , 2 , 1) and the line with symmetric equations [10 pts] x + 2 3 = z - 3 , y = - 2 (b) Given ¯ r ( t ) = cos( t ı + 3 sin( t ) ˆ ±nd and simplify ¯ T ( π/ 4). [10 pts] Please put problem 3 on answer sheet 3 3. (a) Sketch the VVF ¯ r ( t ) = - ı +2 cos t ˆ +2 sin t ˆ k for 0 t π . Mark the start and end points [10 pts] with their coordinates.
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Unformatted text preview: (b) Write down a piecewise parametrization of the curve shown below with clockwise orientation. [10 pts] The curved part is the parabola y = 4-x 2 . Please put problem 4 on answer sheet 4 4. (a) Give the parametric equations of the line through (1 , 2 , 3) and perpendicular to both vectors [10 pts] 2ˆ ı + 2 ˆ + 0 ˆ k and 3ˆ ı-1 ˆ + 2 ˆ k . (b) Find the point at which the line ¯ r ( t ) = (2 t + 2)ˆ ı + (5-t ) ˆ + ( t + 10) ˆ k passes through the [10 pts] plane 2 x + y-z = 1. Please put problem 5 on answer sheet 5 5. (a) Sketch the plane 2 x + y + 3 z = 12. Mark at least three points with their coordinates. [10 pts] (b) Set up (do not evaluate) the integral for the length of ¯ r ( t ) = t 2 ˆ ı + t 3 ˆ -1 ˆ k for 0 ≤ t ≤ 1. [10 pts]...
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