☰ matematika2

Zadaci za vježbu FUNKCIJE VIŠE VARIJABLA VIŠESTRUKI INTEGRALI

Rješenja zadataka za vježbu

1.

a): $\displaystyle \mathbb{R}^2\backslash\{(0,0)\}$ ,
b): $\displaystyle \mathbb{R}^2 \backslash \{(x,y) \colon y=x^2\}$ ,
c): $\displaystyle \{(x,y) \colon y>-x^2\}$ ,
d): $\displaystyle \mathbb{R}^2$ ,
e): $\displaystyle \{(x,y) \colon y\leq 0, y\leq -2x\}\cup\{(x,y) \colon y\geq 0, y\geq -2x\}$ ,
f): $\displaystyle \{(x,y) \colon -1\leq x\leq 1,-1\leq y\leq 1\}$ ,
g): $\displaystyle \{(x,y) \colon x^2+y^2\leq 1\}\cup\{(x,y) \colon x^2+y^2\geq 4\}$ ,
h): $\displaystyle \bigg(\cup_{k\in\mathbb{Z}}\{(x,y) \colon y\geq 0,2k\pi\leq x\l... ...{l\in\mathbb{Z}} \{(x,y) \colon y\leq 0,(2l+1)\pi\leq x\leq 2(l+1)\pi\} \bigg)$ ,
i): $\displaystyle \{(x,y) \colon x>0,y> x+1\}\cup\{(x,y) \colon x<0,x<y<x+1\}$ .

2.

a): $\displaystyle \frac{\partial f}{\partial x}=\frac{\vert y\vert}{y\sqrt{y^2-x^2}}$ , $\displaystyle \frac{\partial f}{\partial y}=\frac{-x\vert y\vert}{y^2\sqrt{y^2-x^2}}$ ,
b): $\displaystyle \frac{\partial f}{\partial x}= y^z x^{(y^z-1)}$ , $\displaystyle \frac{\partial f}{\partial y}= (x^{y^z} \ln x)zy^{z-1}$ , $\displaystyle \frac{\partial f}{\partial z}=(x^{y^z} \ln x) (y^z \ln y)$ .

3.

a): $\displaystyle \frac{\partial^2 f}{\partial x^2}=\frac{2y-2x^2}{(x^2+y)^2}$ , $\displaystyle \frac{\partial^2 f}{\partial y \partial x}=\frac{\partial^2 f}{\partial x \partial y}=\frac{-2x}{(x^2+y)^2}$ , $\displaystyle \frac{\partial^2 f}{\partial y^2}=\frac{-1}{(x^2+y)^2}$ ,
b): $\displaystyle \frac{\partial^2 f}{\partial x^2}=\frac{-3xy^2}{(x^2+y^2)^{5/2}}$ , $\displaystyle \frac{\partial^2 f}{\partial y\partial x}=\frac{\partial^2 f}{\partial x \partial y}=\frac{2x^2y-y^3}{(x^2+y^2)^{5/2}}$ , $\displaystyle \frac{\partial^2 f}{\partial y^2}=\frac{2xy^2-x^3}{(x^2+y^2)^{5/2}}$ .

4.

$\displaystyle \frac{\partial^3 u}{\partial x^3}=e^{xyz}y^3z^3$ , $\displaystyle \frac{\partial^3 u}{\partial y^3}=e^{xyz}x^3z^3$ , $\displaystyle \frac{\partial^3 u}{\partial z^3}=e^{xyz}x^3y^3$ .

5.

$\displaystyle \frac{\partial^2z}{\partial y \partial x }=\frac{-y^2}{(2xy+y^2)^{3/2}}$ .

6.

$\displaystyle \frac{\partial^3z}{\partial^2 x \partial y }=0$ .

7.

$\displaystyle z(x,y)=2xy+\frac{y^2}{2}+\varphi (x)$ .

8.

$\displaystyle z(x,y)=y^2+xy+1$ .

9.

$\displaystyle dz=\frac{1}{x+y}dx-\frac{x}{y\left( x+y\right) }dy$ .

10.

$\displaystyle d^{2}z=-\frac{1}{x}dx^{2}+\frac{2}{y}dx dy-\frac{x}{ y^{2}}dy^{2}$ .

11.

a): $\displaystyle \frac{dz}{dt}=\frac{3-12t^{2}}{\sqrt{1-\left( 3t-4t^{3}\right) ^{2}}}$ ,
b): $\displaystyle \frac{dz}{dt}=\frac{3-\frac{4}{t^{3}}-\frac{1}{2 \sqrt{t}}}{\cos ^{2}\left( 3t+\frac{2}{t^{2}}-\sqrt{t}\right)}$ .

12.

a): $\displaystyle \frac{\partial z}{\partial u}=2u\cos u^{2}-\frac{ 2uv^{2}\left( 1+v^{4}\right) }{\left( v^{2}+u^{2}v^{4}+u^{2}\right) ^{2}}$ , $\displaystyle \frac{\partial z}{\partial v}=\frac{2u^{2}v\left( 1-v^{4}\right) }{\left( v^{2}+u^{2}v^{4}+u^{2}\right) ^{2}}$ ,
b): $\displaystyle \frac{\partial z}{\partial u}=0$ , $\displaystyle \frac{\partial z}{\partial v}=1$ .

13.

$\displaystyle y^{\prime }=-\frac{x}{y}$ , $\displaystyle y^{\prime \prime }=- \frac{x^{2}+y^{2}}{y^{3}}$ .

14.

$\displaystyle \frac{\partial z}{\partial x}=\frac{z}{x\left( z-1\right) }$ , $\displaystyle \frac{\partial z}{\partial y}=\frac{z}{y\left( z-1\right) }$ .

15.

$\displaystyle dz=\frac{\left( yz-1\right) dx+\left( xz-1\right) dy}{1-xy}$ .

16.

$\displaystyle R_{t}\ldots 2x-2y+4z-\pi =0$ , $\displaystyle n\ldots \frac{x-1}{1}=\frac{y-1}{-1}=\frac{z-\frac{\pi }{4}}{2 }$ .

17.

Lokalni minimum je u točki $\displaystyle T=(1,0)$ .

18.

Lokalni maksimum je u točki $\displaystyle T=\left(\frac{\pi}{3},\frac{\pi}{6}\right)$ .

19.

$\displaystyle d=\frac{5\sqrt{6}}{6}$ .

20.

$\displaystyle V=4 \textrm{ m}^2$ .

21.

Lokalni minimum je u točki $\displaystyle T_1=(1,-1,-2)$ , a lokalni maksimum je u točki $\displaystyle T_2=(1,-2,6)$ .

22.

$\displaystyle f_{maks}=125, f_{min}=-75$ .

23.

Minimum je u točki $\displaystyle T_1=\left(\frac{4}{5},\frac{3}{5}\right)$ , a maksimum je u točki $\displaystyle T_2=\left(-\frac{4}{5},-\frac{3}{5}\right)$ .

24.

$\displaystyle P=\frac{3\sqrt{3}}{4}R^2$ .

25.

Najmanje udaljena točka je $\displaystyle T_1=\left(\frac{4}{\sqrt{5}},-\frac{3}{\sqrt{5}}\right)$ , a najviše udaljena točka je $\displaystyle T_2=\left(-\frac{4}{\sqrt{5}},\frac{3}{\sqrt{5}}\right)$ .

26.

Minimum je u točki $\displaystyle T_1=\left(-\frac{1}{3},\frac{2}{3},-\frac{2}{3}\right)$ , maksimum je u $\displaystyle T_2=\left(\frac{1}{3},-\frac{2}{3},\frac{2}{3}\right)$ .

27.

Nema ekstrema u točki $\displaystyle T_1=\left(-\frac{1}{4},\frac{1}{4},\frac{1}{4}\right)$ .

28.

$\displaystyle T=\left(\frac{2P}{3a},\frac{2P}{3b},\frac{2P}{3c}\right)$ .

Zadaci za vježbu FUNKCIJE VIŠE VARIJABLA VIŠESTRUKI INTEGRALI