Calculus Answers

F(x)= Arc sinx² . Arc cosx² + 1/sinx²

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the question is in the photo

compute fxy and fyx for the function f(x,y) =e^x+y sin x+9x^2+2xy at (1,2)

find fx(0,0) and fx (x,y), where (x,y)≠(0,0) for the following function f(x,y)= {xy^3/x^2+y^2, (x,y) ≠(0,0) 0, (x,y)=(0,0) is fx continuous at (0,0)?

find the two repeated limits of the function f(x,y)=(y-x/y+x)(1+x^2/1+y^2) at (0,0).Does the simultaneous limit of f exist as (x,y) tends to (0,0)?

Evaluate triple integration of cos(x^2 + y^2 + z^2)^3/2 dx dy dz , bounded by the sphere x^2 + y^2 + z^2 = 25

Check if the following integrals are independent of path and evaluate the path-independent integrals if they exists.
(i) int[(2x3^y)dx + (x^2 e^y + x -2y)dy] from (0,0) to (1,2).
(ii) int[( x sin xy + y cos xy) dx + (x^2 e^y) + x-2y) dy] from (0,0) to (2,2).

Let f(x,y) = {xy/(x^2 +y^2) , (x,y) not equal to (0,0) , 0 , (x,y) =(0,0)}
(I) show that f(x,y) and f(x,0) are each continuous functions of one variable.
(ii) is f continuous at (0,0)? Give reason for your answer.
(iii) is f differentiable at (0,0)? Give reason.

(1) find the minimum value of the function f(x,y) = x^2 + 2y^2 on the circle x^2 + y^2 =1
(2) let the function f be defined by f(x,y) = 3x^2 y^4 / (x^4 + y^8) (x,y) not equal to (0,0) and 0 , (x,y) = (0,0). Show that f has directional derivatives in all directions at (0,0)