Advance Calculus

  1. Findthepartialderivativesof




x2 −1




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



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



  1. Let




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 or other softwares.)


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


  1. 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)


  1. Solve the maximizationproblem





x2 −y2



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


  1. Give an algorithm to solve the minimization problem








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


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