practice_midterm1_solutions.pdf - Midterm Exam 1 Solutions...

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Midterm Exam 1 — Solutions Math 32A/3, Fall 2014 This is a collection of problems that would not be unreasonable for a real midterm exam. WARNING: the inclusion (or exclusion) of a certain topic or type of problem on this sample exam does not guarantee its inclusion (or exclusion) on the actual exam! 1. (a) Parametrize (in vector form) the line tangent to r ( t ) = h e t/π , sin t, t i at t 0 = π . (b) Find a normal vector for the plane containing the point P = ( - 1 , 0 , 1) and the line parametrized by r ( t ) = h t + 1 , 2 t, - 1 i . Solution: (a) (See, e.g., § 14.2 #29-34) First, the derivative is r 0 ( t ) = h e t/π /π, cos t, 1 i , and so the direction vector of the line is the tangent vector r 0 ( π ) = h e/π, - 1 , 1 i . A point on the line has position vector r ( π ) = h e, 0 , π i . Now, the tangent line has parametrization L ( t ) = r ( π ) + t r 0 ( π ) = h e, 0 , π i + t h e/π, - 1 , 1 i . (b) (See, e.g., § 13.5 #28) We need two non-parallel vectors in the plane, and their cross product will be a normal vector n . One vector is the direction vector of the line, which is u = h 1 , 2 , 0 i . Another vector is the one pointing from the end of r (0) to P , which is v = --→ OP - r (0) = h- 1 , 0 , 1 i - h 1 , 0 , - 1 i = h- 2 , 0 , 2 i Now, a normal vector is n = u × v = ( i + 2 j ) × ( - 2 i
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