Harvard University - MATH 2413Midterm1_v2_solution

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1. True/False questions: circle T or F. Each question is worth 4 points if you answer correctly, 0 points if you don’t. You receive 2 points if you don’t circle neither T or F. a) F The tangen... t line to r1(t) = ht; 2t2; −t3i at the point (1; 2; −1) is perpendicular to the tangent line to r2(t) = ht; t + 1; ln ti at the point (1; 2; 0). b) F If r(t) = hsin t; cos t; t3i, then the unit tangent vector is T(t) = hcos t; − sin t; 3t2i. c) T The intersection of the plane defined by the equation 2x − 2y + z − 3 = 0 and of the line defined by the parametric equation 8<: x = t + 2 y = −t + 3 t 2 R z = −3t − 2 is a point. d) F Consider the curve in the plane given by the equation y = x3 + x. Its curvature at the point (1; 2) is equal to 1=3. e) T If the curve r(s) is parametrized with respect to arc length then necessarily: jr0(s)j = 1: Page 2 of 72. a) Show that the planes defined by the following equations are parallel: 2x − y + z + 2 = 0; −4x + 2y − 2z − 1 = 0: Answer: The normal vectors are h2; −1; 1i, h−4; 2; −2i. Notice h−4; 2; −2i = −2h2; −1; 1i. This implies the normal vectors of the planes parallel, meaning the planes are parallel. b) Find the distance between these two planes. Since the planes are parallel, we can choose any point on the first plane and compute the distance of that point to the second plane. M(0; 2; 0) is on the first plane. Applying the distance formula to M and the second plane one gets the distance d is d = j − 4 ·p0 + 2 42 + 2 · 22−+ 2 2 ·20 − 1j = p324 = p46 [Show More]

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