- #1
Jamin2112
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- 12
Homework Statement
Let f(x,y)=x3 + x2y + y3. Show that f is differentiable at (1,2).
Homework Equations
A function f(x,y) is differentiable at a point (x0,y0) if the partial derivatives of f(x,y) are defined at (x0,y0) and if the following limiting condition holds:
lim(x,y)-->(x0,y0) [ f(x,y) - f(x0,y0) - ∂f/∂x|(x0,y0)(x-x0) - ∂f/∂y|(x0,y0)(y-y0) ] / √( (x-x0)2 + (y-y0)2) = 0.
The Attempt at a Solution
∂f/∂x = 3x2 + 2xy ===> ∂f/∂x|(1,2) defined
∂f/∂y = 2x2 + 3y2 ===> ∂f/∂y|(1,2) defined
But I'm having trouble with making lim(x,y)-->(1,2) = 0.
I got it all the way to
lim(x,y)-->(1,2) = (x3+x2y+y3+22 -7x-13y)/√((x-1)2+(x-2)2).
I need to get rid of the denominator. I thought about rationalizing the denominator, but I don't that'll get me anywhere.
Suggestions welcome.