Calculus 3  Hour Exam 2   -   Math 241     Spring, 2010

 

 

 

1.

 

 

Let    w  =  f(x,y),   x  =  uv,    y  =  u2  - v2 .

 

Find  ∂2w/∂v2  in terms of  u, v, and partial derivatives of  f  with respect to  x  and  y.

 

Answer:  ∂2w/∂v2  =  u2 fxx  -  4uv fxy  +  4v2  fyy  -  2 fy

 

 

 

2.

 

Find all critical points of the function,  f(x,y)  =  x2  +  y2  + 4xy  +  2y3   and

characterize them as local minimum, maximum, or saddle points.

 

Answers:  (0, 0) is a saddle point  (-2,1)  is a local minimum

 

 

 

3.

 

 

Use the Lagrange multipliers method to find the absolute minimum and maximum

values of  f(x,y)  =  x2  +  5y2  on the circle  x2  +  4x  + y2  +  3  =  0

 

Answers:   Abs Minimum of 1 at (-1, 0)  Abs Maximum of 10  at  ( -5/2, ± √3/2 )

 

 

 

 

 

4.

 

 

Find equations of the tangent plane and normal line to the surface

 

e x-1  +  2e y-2  +  3e z -3   =  xyz     at the point  ( 1, 2, 3 ).

 

Answers:  -5x  – y  +  z +  4  =  0    and   ( x-1 )/-5   =  ( y-2 )/-1  =  ( z-3 )/1

 

 

 

 

5.

 

 

Using the differential function  f(x,y) =  √(x2  -  y2 ) at  ( 5, 3 ), estimate the value of

√( 4.82  -  3.22 )

 

Answer:  3.6

 

 

 

6.

 

Find the absolute minimum and maximum values of the function, 

f(x,y)  =  x2 +  2x  +  y2  -  y    in the right semi-disk  x2  +  y2  ≤  1,  x  ≥ 0 .

 

Answers:  Abs Min at (0, ½ ) of  - ¼   ,  Abs Max at  ( 2/√5, - 1/√5 ) of  1 + √5