Ensino Superior ⇒ derivadas parciais

[camilasfcosta]

Seja g : R² → R uma função diferenciável em (0, 0), com g (x, y) = 0 somente no ponto (x, y) = (0, 0). Considerar
f (x, y) =[tex3]\begin{cases}
\frac{tan²(g(x,y))}{g(x,y)} se (x,y) \neq (0,0)\\
0 se(x,y) =(0,0)
\end{cases}[/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]\begin{cases}
\frac{tan²(g(x,y))}{g(x,y)} se (x,y) \neq (0,0)\\
0 se(x,y) =(0,0)
\end{cases}[/tex3]

(a) Calcular as derivadas parciais de f em (0, 0), em termo das derivadas parciais de g .
(b) Mostrar que f é diferenciável em (0, 0).(usar a regra da cadeia)

[rcompany]

[tex3]
\begin{array}{rl}f\text{ contínua em }(0,0):\ \lim\limits_{x,y\to 0}\dfrac{\tan^2(g(x,y))}{g(x,y)}\!\!\!\!\!&=\lim\limits_{u\to 0}\dfrac{\tan^2(u)}{u}\\
&=\lim\limits_{u\to0}\tan(u)\cdot\dfrac{\tan u}{u}\\
&=0\times1\quad\quad\text{já que }\lim\limits_{u\to0}\tan(u)=0\text{ e }\lim\limits_{u\to0}\dfrac{\tan u}{u}=1\\
&=f(0,0)\end{array}\\[60pt]
\begin{array}{rl}\dfrac{\partial f(x,y) }{\partial x}&=\dfrac{\dfrac{2\tan(g(x,y))}{\cos^2(g(x,y))}\dfrac{\partial g(x,y)}{\partial x}g(x,y)+\tan^2(g(x,y))\dfrac{\partial g(x,y)}{\partial x}}{[g(x,y)]^2}\\
&=\dfrac{\partial g(x,y)}{\partial x}\cdot\dfrac{1}{\cos^2(g(x,y)}\left(\dfrac{2\tan(g(x,y))}{g(x,y)}+\dfrac{\sin^2(g(x,y))}{[g(x,y)]^2}\right)
\end{array}\\[24pt]
\text{e por simetria:}\\
\begin{array}{rl}\dfrac{\partial f(x,y) }{\partial y}&=\dfrac{\partial g(x,y)}{\partial y}\cdot\dfrac{1}{\cos^2(g(x,y)}\left(\dfrac{2\tan(g(x,y))}{g(x,y)}+\dfrac{\sin^2(g(x,y))}{[g(x,y)]^2}\right)
\end{array}\\[48pt]
\lim\limits_{u\to 0}\dfrac{\sin u}{u}=1,\ \lim\limits_{u\to 0}\dfrac{\tan u}{u}=1,\ \lim\limits_{x,y\to 0}g(x,y)=0\text{ e }\lim\limits_{y\to0}\dfrac{\partial g(0,y)}{\partial y=}=\lim\limits_{x\to0}\dfrac{\partial g(x,0)}{\partial x}=\dfrac{\mathrm{d}g(x,y)}{\mathrm{d}(x,y)}(0,0)\\
\text{e então }\\
\lim\limits_{y\to 0}\dfrac{\partial f(x,y)}{\partial x}=\lim\limits_{x\to 0}\dfrac{\partial f(x,y)}{\partial y}=3\dfrac{\mathrm{d}g(x,y)}{\mathrm{d}(x,y)}(0,0)




[/tex3]

Código: Selecionar tudoSHIFT+Click na eq = ZOOM

[tex3]
\begin{array}{rl}f\text{ contínua em }(0,0):\ \lim\limits_{x,y\to 0}\dfrac{\tan^2(g(x,y))}{g(x,y)}\!\!\!\!\!&=\lim\limits_{u\to 0}\dfrac{\tan^2(u)}{u}\\
&=\lim\limits_{u\to0}\tan(u)\cdot\dfrac{\tan u}{u}\\
&=0\times1\quad\quad\text{já que }\lim\limits_{u\to0}\tan(u)=0\text{ e }\lim\limits_{u\to0}\dfrac{\tan u}{u}=1\\
&=f(0,0)\end{array}\\[60pt]
\begin{array}{rl}\dfrac{\partial f(x,y) }{\partial x}&=\dfrac{\dfrac{2\tan(g(x,y))}{\cos^2(g(x,y))}\dfrac{\partial g(x,y)}{\partial x}g(x,y)+\tan^2(g(x,y))\dfrac{\partial g(x,y)}{\partial x}}{[g(x,y)]^2}\\
&=\dfrac{\partial g(x,y)}{\partial x}\cdot\dfrac{1}{\cos^2(g(x,y)}\left(\dfrac{2\tan(g(x,y))}{g(x,y)}+\dfrac{\sin^2(g(x,y))}{[g(x,y)]^2}\right)
\end{array}\\[24pt]
\text{e por simetria:}\\
\begin{array}{rl}\dfrac{\partial f(x,y) }{\partial y}&=\dfrac{\partial g(x,y)}{\partial y}\cdot\dfrac{1}{\cos^2(g(x,y)}\left(\dfrac{2\tan(g(x,y))}{g(x,y)}+\dfrac{\sin^2(g(x,y))}{[g(x,y)]^2}\right)
\end{array}\\[48pt]
\lim\limits_{u\to 0}\dfrac{\sin u}{u}=1,\ \lim\limits_{u\to 0}\dfrac{\tan u}{u}=1,\ \lim\limits_{x,y\to 0}g(x,y)=0\text{ e }\lim\limits_{y\to0}\dfrac{\partial g(0,y)}{\partial y=}=\lim\limits_{x\to0}\dfrac{\partial g(x,0)}{\partial x}=\dfrac{\mathrm{d}g(x,y)}{\mathrm{d}(x,y)}(0,0)\\
\text{e então }\\
\lim\limits_{y\to 0}\dfrac{\partial f(x,y)}{\partial x}=\lim\limits_{x\to 0}\dfrac{\partial f(x,y)}{\partial y}=3\dfrac{\mathrm{d}g(x,y)}{\mathrm{d}(x,y)}(0,0)




[/tex3]