Josephood7

Calculate all four second-order partial derivatives and check that [math]. Assume the variables are restricted to a domain on which the function is defined. [math]

[math] 
[math] 
[math] 
[math] 

 Consider the partial derivatives [math] [math] Is there a function [math] which has these partial derivatives?
 ? Yes No 

If so, what is it?
[math] 
(Enter none if there is no such function.)

Determine the sign of [math] and [math] at each indicated point using the contour diagram of [math] shown below. (The point [math] is that in the first quadrant, at a positive [math] and [math] value; [math] through [math] are located clockwise from [math], so that [math] is at a positive [math] value and negative [math], etc.)

(a) At point [math],
[math] is  ? positive negative  and
[math] is  ? positive negative .

(b) At point [math],
[math] is  ? positive negative  and
[math] is  ? positive negative .

(c) At point [math],
[math] is  ? positive negative  and
[math] is  ? positive negative .

Solution:

The plane [math] intersects the surface [math] in a certain curve. Find the slope to the tangent line to this curve at the point [math].

[math] 

Find the partial derivatives of the function [math]
[math]
[math]