prac1asol

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

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.02 Practice Exam 1 A – Solutions Problem 1. 1ˆ −� − −� 1 − a) OQ = ˆ + ˆ + k; OR = ˆ + ˆ + k. ı�ˆ ı� 2 2 � −� −� − − →1, 1, 1⊥ · 1 , 1, 1 ⊥ OQ · OR 22 2 2 b) cos � = −� −� = = . � ⎦3 − − 3 |OQ | |OR | 32 Problem 2. � dR � Velocity: V = = →−3 sin t, 3 cos t, 1⊥. dt Problem 3. ⎤ � � Speed: |V | = ⎤ ⎦ 9 sin2 t + 9 cos2 t + 1 = � 10. 1 1 2 1 −1 2 1 2 1⎥ ⎥ � ⎥ � 2 ⎣. Inverse: � −1 −2 a) Minors: � −2 −2 −2 ⎣. Cofactors: � 2 −2 2 −3 −5 −6 −3 5 −6 2 2 −3 ⎥ � −1 b) X = A B = � 4 ⎣. −4 Problem 4. ⎤ � � ⎤ −3 � 5 ⎣. −6 � −� − −� − Q = top of the ladder: OQ = →0, L sin �⊥; R = bottom of the ladder: OR = →−L cos �, 0⊥. −� −� − − −� − Midpoint: OP = 1 (OQ + OR ) = →− L cos �, L sin �⊥. 2 2 2 Parametric equations: x = − L cos �, y = L sin �. 2 2 ⎡ Problem 5. ⎡ ⎡ˆ ı ˆ � ⎡ −− − −− − ⎡ a) P0 P1 × P0 P2 = ⎡ −1 −1 ⎡ ⎡ 0 −2 c) Parametric equations for the line: x = −1 + t, y = t, z = t. Substituting: −1 + 4t = 3, t = 1, intersection point (0, 1, 1). ˆ⎡ k⎡ ⎡ ˆ ı� 1 ⎡ = ˆ + ˆ + 2k. ⎡ 1⎡ −− −� −− −� ˆ b) Normal vector: P0 P1 × P0 P2 = ˆ + ˆ + 2k. ı� � −− −� −− −� Area = 1 |P0 P1 × P0 P2 | = 1 6. 2 2 Equation: x + y + 2z = 3. Problem 6. d�� �� �� �� a) (R · R) = V · R + R · V = 2R · V . dt d�� � �� �� b) Assume |R| is constant: then (R · R) = 2R · V = 0, i.e. R�V . dt d�� �� �� �� �� � c) R · V = 0, so (R · V ) = V · V + R · A = 0. Therefore R · A = −|V |2 . dt 1 ...
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