Advance Calculus

1. Findthepartialderivativesof

 

 

f(x,y)=

x2 −1

 

xy

 

with respect to x and y, where f is defined on the positive quadrant

 

R

2

++ = {(x, y) : x >0, y >0}.

 

 

2. Let

 

 

wherex>0.

f(x, y)= x(y2 −xy + 1)

 

 

1. Show that the set of points (x, y) suchthat

 

f(x, y)= 2

 

DOES NOT defines a function y = g(x) locally around x = 2.

1. Considerthesetofpoints(x,y)suchthat

 

f(x, y)= 2

 

and that y >1. Show that the set defines a function y = g(x) locally around x = 2 by plotting the set. (You can use https://www.desmos.com/calculator or other softwares.)

 

1. Find gl(x) evaluated at x = 2 in part(b).

 

3. Letf(x,y)=min{x,y},definedonthex≥0,y≥0,wheremin{x,y}isthefunction thatspitsthesmallerofthetwonumbersxand(e.g.min{1,3}=1,min{3,2}= 2.) Show that fis not differentiable at any point on the 45-degree line. (Hint: see thelecturenoteormypreviouspostonPiazzaforexamples)

 

4. Solve the maximizationproblem

 

 

max

(x,y)∈A

x2 −y2

 

 

where A = [−1, 1] × [−1, 1]. Is there a point in A such that fx = fy = 0? Is that point a maximum?

 

5. Give an algorithm to solve the minimization problem

 

 

min

(x,y)∈[0,1]×[0,1]

f(x,y)

 

 

wherefisdifferentiableon[0,1]×[0,1].(Aminimizationproblemasksyoutofind the point (x, y) in A that gives the lowest function value than other points inA.)

P(5.u)

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