prac1asol - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 200 7 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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18.02 Practice Exam 1 A Solutions Problem 1. a) −−� = ı +ˆ k; −−� = 1 ı 1 k. OQ ˆ � + ˆ OR ˆ � + ˆ 2 2 1 1 OQ OR , 1 , 2 2 b) cos = −−� · −−� = 1 , 1 , 1 ⊥ · 2 2 = . OQ OR 3 3 2 | −−� | | −−� | 3 Problem 2. Velocity: V = d dt R = →− 3 sin t, 3 cos t, 1 . Speed: | V | = 9 sin 2 t + 9 cos 2 t + 1 = 10. Problem 3. 1 1 2 1 1 2 1 2 3 1 a) Minors: 2 2 2 . Cofactors: 2 2 2 . Inverse: 1 2 5 . 2 3 5 6 3 5 6 2 2 6 3 b) X = A 1 B = 4 . 4 Problem 4. Q = OQ = 0 , L sin ; R = OR top of the ladder: −−� bottom of the ladder: −−� = →− L cos �, 0 . 1 Midpoint: −−� = ( −−� OR ) = →− OP 2 OQ + −−� L L cos �, sin . 2 2 L L Parametric equations: x = cos , y = sin . 2 2 Problem 5. ˆ ˆ ı ˆ k 1 1 a) −−− P 0 P 1 × −−− P 0 P 2 = 1 1 1 = ı +ˆ k. Area =
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