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Prоblem 1. (10 pоints) Find the limit lim(x,y)→(1,3)x+y-4x+y-2{"versiоn":"1.1","mаth":"lim(x,y)→(1,3)x+y-4x+y-2"} Problem 2. (10 points) Let f(x,y)=x3y+cos(xy){"version":"1.1","mаth":"f(x,y)=x3y+cos(xy)"}. Find fx(x,y){"version":"1.1","mаth":"fx(x,y)"}, fy(x,y){"version":"1.1","math":"fy(x,y)"}, and fy(1,π/2){"version":"1.1","math":"fy(1,π/2)"}. Problem 3. (10 points) Find an equation of the tangent plane to the surface z=2x2+y2-5y{"version":"1.1","math":"z=2x2+y2-5y"} at the point (1,2,-4){"version":"1.1","math":"(1,2,-4)"}. Problem 4. (10 points) Given z=x2+y2{"version":"1.1","math":"z=x2+y2"}, x=2s+3t{"version":"1.1","math":"x=2s+3t"}, and y=s+t{"version":"1.1","math":"y=s+t"}. Find ∂z∂s{"version":"1.1","math":"∂z∂s"} and ∂z∂t{"version":"1.1","math":"∂z∂t"}using the Chain Rule. Problem 5. (10 points) Given x3+y3+z2+6xyz+4=0{"version":"1.1","math":"x3+y3+z2+6xyz+4=0"}. Find ∂z∂x{"version":"1.1","math":"∂z∂x"} and ∂z∂y{"version":"1.1","math":"∂z∂y"} using implicit differentiation. Problem 6. (10 points) Let f(x,y)=xey{"version":"1.1","math":"f(x,y)=xey"} and u→=3/5,4/5{"version":"1.1","math":"u→=3/5,4/5"} a unit vector. Find the gradient ∇f{"version":"1.1","math":"∇f"} and the directional derivative Du→f{"version":"1.1","math":"Du→f"} at the point (1,0){"version":"1.1","math":"(1,0)"}. Problem 7. (15 points) Find and classify all the critical points of the function f(x,y)=x3+y3-3xy+4{"version":"1.1","math":"f(x,y)=x3+y3-3xy+4"}. Here, we need to indicate whether each critical point is a local maximum, local minimum, or a saddle point. Problem 8. (15 points) Find the absolute maximum and minimum values of the function f(x,y)=3x+y{"version":"1.1","math":"f(x,y)=3x+y"} on the domain defined by -1≤x≤1{"version":"1.1","math":"-1≤x≤1"} and -1≤y≤1{"version":"1.1","math":"-1≤y≤1"}. Problem 9. (10 points) Use Lagrange multipliers to find the maximum and minimum values of the function f(x,y)=x2+2y2{"version":"1.1","math":"f(x,y)=x2+2y2"} subject to the constraint x2+y2=1{"version":"1.1","math":"x2+y2=1"}.