Answer to Question #105386 in Calculus for Ajay

Question

(1) Verify Euler’s relation for the function f(x,y)=(x^3+y^3)/(x+y).
(2) Check whether the following functions are differentiable at the point given
against them:
i) f(x,y)=y^3+y Sin2x+e^(x+y) at (1,–1)
ii) f(x,y)=|x=1| at (1,0)
(3) Find the range of the function f defined by f(x,y)=10–x^2–y^2 for all (x,y)
for which x^2+y^2 ≤9. Sketch two of its level curve

Answer 1

1.$f(x,y)=\frac{x^3+y^3}{x+y}=\frac{(x+y)(x^2-xy+y^2)}{x+y}=x^2-xy+y^2$

Degree is 2.

According to Euler's relation:

$x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=nf(x,y)$

$x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=x(2x-y)+y(2y-x)=2(x^2-xy+y^2)=nf(x,y)$

(Proved)

3.$f(x,y)=10-x^2-y^2$

$0 \leqslant x^2+y^2 \leqslant9$

$-9 \leqslant -(x^2+y^2) \leqslant0$

$10-9 \leqslant 10-(x^2+y^2) \leqslant0+10$

$1 \leqslant f(x,y)\leqslant10$ is the range of the function.

Level Curve in the image.

2.$f(x,y)=y^3+y Sin2x+e^{x+y}$

$f_x(x,y)=2ycos2x+e^{x+y}$

$f_{xx}(x,y)=-4y Sin2x+e^{x+y}$

$f_y(x,y)=3y^2+ Sin2x+e^{x+y}$

$f_{yy}(x,y)=6y+ e^{x+y}$

Partial derivatives are continuous and differentiable as sin (2x),cos(2x),e^x+y,y²,y³,y are continuous and differentiable.

So,f(x,y) is differentiable at (1,-1).

(ii)$f(x,y)=|x-1| at (1,0)$

$f_y(x,y)=0$

$f_x(1,0)=lim_{h\to0}\frac{f((1+h),0)-f(1,0)}{h}$

$f_x(1,0)=lim_{h\to0}\frac{|h|-0}{h}=lim_{h\to0}\frac{|h|}{h}$

$lim_{h\to0^+}\frac{h}{h}=1$ whereas $lim_{h\to0^-}\frac{-h}{h}=-1$

So,limit does not exist,partial derivative f_x(x,y) does not exist at (1,0) .Thus, f(x,y) is not differentiable.