Take Home Quiz 14, F09
due Nov. 3rd at the beginning of class
Name ________________________
You can work with others and use references. You should write out your own final solutions. Show your work.
1. a) Use the chain rule to find
z
for z = (e xy ) tan y, x = s+4 t, y = t / s.
t
b) Use partial derivatives to find dy / dx for y6 + x3 y2 = 3 + x
2. (a) For f(x,y) = y e
(b). For f f(x,y) = y e
5x
5x
ey
2
find the gradient of f(x,y) at the point (1, -2).
find the directional derivative of f(x,y) at the point (1, -2) in the direction v = <8, -6>.
3. Provide two geometric interpretations of the gradient – one interpretation should relate to level curves of a function
and another should relate to the rate of increase of the function.
4. Find the following partial derivatives
(a)
  x 


xy  x  y 
(OVER) (OVER) (OVER) (OVER)
(b)

5 xy 2  2 x 3 y 3
xxy


5. Find the critical points and use second derivative tests to identify whether they are maximum, minimums or saddle
points for f(x,y) = x y ( 4 – x – y ).
6. Find the absolute max and min of f(x,y) = 4 + xy – x – 2y over the closed triangle with vertices (1,0), (5,0) and (1,4).
7. Prove or disprove: if
f ( x, y ) 
5 xy
then
4x  2 y 2
2
8. Find the tangent plane to the surface defined by
lim
( x , y ) ( 0 , 0 )
f ( x, y ) = 0
z 2  x 3  y at the point (2,4,2)