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- 1. Sketch the graph of the level curves of the surface z = y + x^2 when k = −2, k = 0, and k = 2Calculate the value of the limit lim(x,y)→(0,0) 22xy/x^2+y^2 as we approach (0,0) along the path γ(t)=(−t,t).Sketch the contour map of level curves of f(x, y) = 4 − √ x² + y² for the levels k = 4, 3, 2, 1, 0
- Use the limit definition to show that the partial derivatives of F(x,y) = with respect to x and y are/aren’t the same any point(a,b). (i.e show that Fx(a,b) =/≠ Fy(a,b))2) Refer to the following plot of some level curves of f(x, y)= c for c = − 1.5, −1, −0.5, 0, 0.5,1, 1.5draw a contour map of the function f(x,y)=y/(x-1) showing several level curves
- Calculate the are bounded by the curves: A. y = x2 and x2 = y B. y = ln x and x - y - 4 = 0 C. y = e2x and x - 2y + 5 = 0Let f(x, y) = 2x2y + 3xy. Use the (limit) definition of partials derivatives to show that fx(x, y) = 4xy + 3y and fy(x, y) = 2x2 + 3x.Let f (x, y) = 3xy − 6x − 3y + 7. Find the absolute maximum and absolute minimum values of f over the triangular region with vertices (0,0), (3,0) and (0,5).