Calculus 3  Hour Exam 1   -   Math 241     Fall, 2010

 

 

1a.

 

 

Find the angle between  (1, -2, -5)  and  (3,2,1)

 

                                                 Answer:   θ = cos-1 (-6/(√30 √14)

 

 

 

1b.

 

Find the angle between  (1,1,1) x (1,2,3)  and  (0,1,1)

 

                                                      Answer:  θ = cos-1 (-1/√12 )

 

 

 

1c.

 

 

Find   the volume of the parallelpiped determined by  <1,1,1>,  <1,2,3>, <0,1,1>

 

                                                       Answer:  V  =  1

 

 

 

2a.

 

 

Draw the level curves for   f(x,y)  =  16x2  -  y2   for  k = - 1,  k = 0, and  k  = 1

 

                                                                     Answer:  16x2  -  y2   = k

 

 

2b.

 

 

Let   g(x,y,z)  =  z  -  f(x,y)  where  f(x,y)  =  16x2  -  y2     Find  grad(g).

 

                                                               Answer:  grad(g)  =  < -32x, 2y, 1 >

 

 

 

2c.

 

Find the tangent plane to the surface  g(x,y,z) = 0  at the point  (1,0,16).

 

                    Answer:    <  x – 1>, y – 0, z – 16  >  ∙  < -32, 0, 1 >  =  0

 

 

 

 

2d.

 

At the point  (1,0)  in what direction is  f(x,y) changing most rapidly?

 

                                                                        Answer:  In direction < 1,0 >.

 

 

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