find f(x,y+k)-f(x,y)/k when
f(x,y)=x^2y+2xy^2
To find the expression f(x,y+k)-f(x,y)/k, we need to substitute the given function f(x,y)=x^2y+2xy^2 into the expression and simplify it.
Let's start by substituting the function f(x,y) into the expression:
f(x, y+k) = (x^2)(y+k) + 2x(y+k)^2
= (x^2)(y+k) + 2x(y^2 + 2y*k + k^2)
= x^2y + x^2k + 2xy^2 + 4xyk + 2xk^2
Next, let's write out f(x, y) using the same function:
f(x, y) = x^2y + 2xy^2
Now, we can subtract f(x, y) from f(x, y+k):
f(x, y+k) - f(x, y) = (x^2y + x^2k + 2xy^2 + 4xyk + 2xk^2) - (x^2y + 2xy^2)
= x^2k + 4xyk + 2xk^2
Finally, divide the expression by k:
(f(x, y+k) - f(x, y))/k = (x^2k + 4xyk + 2xk^2)/k
= x^2 + 4xy + 2k
Therefore, the expression f(x,y+k)-f(x,y)/k simplifies to x^2 + 4xy + 2k.