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Math 20C Final 12 (No Solutions)

Math 20C Final 12 (No Solutions) - Mathematics 20C Spring...

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Mathematics 20C Spring 2009 Final Exam Name: Section: ID: Be sure to show all work. Answers without full justification are worth no credit. Remember that you can often check your answers. You may use a single page of notes, but no other aids are allowed. This test consists of 8 questions. Each question is worth a total of 5 points. 1 /5 2 /5 3 /5 4 /5 5 /5 6 /5 7 /5 8 /5 /40
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1) Let P and Q be the points P = (1 , 2 , 1) and Q = (0 , 1 , 1). Let u = ( 2 , 1 , 1 ) and v = −→ PQ . (a) Find the cosine of the angle between u and v . (b) Find another vector w which is orthogonal to both u and v .
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2) Let l ( t ) be the line containing the two points (0 , 1 , 0) and (1 , 1 , 1) and let r ( s ) be the parametric curve r ( s ) = ( s 2 ,s,s 3 ) . Notice that r (1) = ( 1 , 1 , 1 ) , and so both paths go through this point. (a) Find the equation of l ( t ). (b) Find the equation of the tangent line (call it v ( s )) of r ( s ) at ( 1 , 1 , 1 ) . (c) Find the equation of the plane containing both l ( t ) and v ( s ).
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3) Let a ( t ) = ( cos t, sin t,e t ) be the acceleration of a particle at time t . (a) Find the velocity of the particle v ( t ) if v (0) = ( 1 , 1 , 1 ) . (b) Find the position of the particle r ( t ) if r (0) = (− 1 , 1 , 0 ) .
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4) Let S be the surface described implicitly by x 2 + z 2 = 3 + y 3 . (a) Find ∂z ∂x and ∂z ∂y
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