Calculus 3  Hour Exam 2   -   Math 241     Fall, 2010

 

 

1a.

 

 

State, with hypotheses, the second derivative test.

 

 

 

 

 

1b.

 

For   f(x,y)  =  ey ( y2 – x2 ),  find   fx,  fy,  fxx,  fxy,  fyy  and all critical points.

 

Answers:  find   fx,  = ey(-2x),  fy  =  2yey + y2 ey,   fxx  =  -2ey,    fxy  =  -2xey

  fyy  =  (2 + 4y + y2 )ey

 

Critical points:   (0,0)  and  (0, -2).

 

 

 

1c.

 

 

Classify the critical points as max, min, saddle, or indeterminate.

 

Answer:  (0,0) saddle     (0 -2) max

 

 

 

2a.

 

 

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

labeling each branch and intercept.

 

 

 

2b.

 

 

Use Lagrange multipliers to find the maximal value of  f(x,y) subject to the

constraint   y -  x2  =  -2.   Show work.

 

Answer:  Max  =  96.

 

 

 

3a.

 

For the region   22  ≤  x2 + y2 + z2  ≤ 42,    2  ≤  y  ≤  3,  sketch a slice for some y

between  2  and  3.

 

 

 

 

3b.

 

 

Set up an integral to find the volume of the region.       

Answer:

 

      y = 3     θ = 2π          r = √(16 – y2 )

          ∫            ∫                  ∫     r dr  dθ dy

      y = 2     θ = 0           r = 0

 

 

 

3c.

 

 

 

Compute the volume.         Answer:  V  =  29π/3

 

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