Answer to Question #197402 in Calculus for Sadeen Khan

Question #197402

Find fx(0,0) and fx(x,y), where (x,y) not = (0,0) for the function f:R^2 is to R defined by

F(x,y)={xy^3/x^2+y^2 if (x,y) is not= (0,0) and (x,y)=(0,0).

Is Fx continuous at (0,0)? Justify your answer.


1
Expert's answer
2021-05-24T17:17:14-0400



fx(0,0)=limh0f(0+h,0)f(0,0)hf_x(0, 0)=\lim\limits_{h\to0}\dfrac{f(0+h, 0)-f(0,0)}{h}

=limh00(h)3(0)2+h20h=0=\lim\limits_{h\to0}\dfrac{\dfrac{0\cdot(h)^3}{(0)^2+h^2}-0}{h}=0



fy(0,0)=limh0f(0,0+h)f(0,0)hf_y(0, 0)=\lim\limits_{h\to0}\dfrac{f(0, 0+h)-f(0,0)}{h}

=limh0h(0)3h2+(0)20h=0=\lim\limits_{h\to0}\dfrac{\dfrac{h\cdot(0)^3}{h^2+(0)^2}-0}{h}=0

If (x,y)(0,0)(x, y)\not=(0, 0)


fx(x,y)=y3(x2+y22x2)(x2+y2)2=y3(y2x2)(x2+y2)2f_x(x, y)=\dfrac{y^3(x^2+y^2-2x^2)}{(x^2+y^2)^2}=\dfrac{y^3(y^2-x^2)}{(x^2+y^2)^2}

Let x=rcosθ,y=rsinθx=r\cos\theta, y=r\sin \theta


lim(x,y)(0,0)fx(x,y)=lim(x,y)(0,0)y3(y2x2)(x2+y2)2\lim\limits_{(x, y)\to(0,0)}f_x(x, y)=\lim\limits_{(x, y)\to(0,0)}\dfrac{y^3(y^2-x^2)}{(x^2+y^2)^2}

=limr0r3sin3θ(r2sin2θr2cos2θ)r4=\lim\limits_{r\to0}\dfrac{r^3 \sin^3\theta(r^2 \sin^2\theta-r^2 \cos^2\theta)}{r^4}

=limr0rcosθ(rsinθ)3(rcosθ)2+(rsinθ)2=\lim\limits_{r\to0}\dfrac{r\cos\theta (r\sin\theta)^3}{(r\cos\theta)^2+(r\sin\theta)^2}


=limr0r2cosθsin3θ=0=f(0,0)=\lim\limits_{r\to0}r^2\cos\theta \sin^3\theta=0=f(0, 0)

=limr0rsin3θ(sin2θcos2θ)=0=fx(0,0)=\lim\limits_{r\to0}r\sin^3\theta(\sin^2\theta- \cos^2\theta)=0=f_x(0, 0)

The function fx(x,y)f_x(x, y) is continuous at (0,0).(0, 0).




Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment