# Matematika A2a 2008/7. gyakorlat

(Változatok közti eltérés)

Ez az szócikk a Matematika A2a 2008 alszócikke.

1.

$f(x,y)=xy\sin(x^2y)\,$
T = [0,1]×[0,π/2]'
$f(x,y)=xy\,$
T = [0,1]×[0,1]

2.

$f(x,y)=x\sin(x^2)y\,$
T = [0,1]×[0,1]

Mo.

f'x(x,y) = y(sin(x2) + 2x2cos(x2)) = 0
f'y(x,y) = xsin(x2) = 0

3.

$f(x,y)=x^7+sin(y)\cos^3(y)\,$
T = [0,1]×[0,1]

4.

T = [1,e] × [1,2]
$f(x,y)=\frac{\mathrm{ln}^9\,x}{xy}$

5.

T = [-1,1] × [0,π/4]
$f(x,y)=\sin(x^3)\frac{1}{\cos^2 y}$

6.

T = [-1,1] × [0,1]
$f(x,y)=\sin(x^3)\frac{\sin^{2009}(\mathrm{sh}(y))}{\mathrm{ln}\,y}$

7.

T = [a,b] × [c,d]
f(x,y) = g(x)h(y)

téglalapon szeparálható integrandus integrálja szorzattá esik szét:

$\int\limits_{x=0}^b\int\limits_{y=c}^{d}g(x)h(y)\,\mathrm{d}x\,\mathrm{d}y=\int\limits_{x=a}^b g(x)\left(\int\limits_{y=c}^{d} h(y)\,\mathrm{d}y\right)\,\mathrm{d}x=\left(\int\limits_{x=a}^b g(x)\,\mathrm{d}x\right)\cdot\left(\int\limits_{y=c}^{d} h(y)\,\mathrm{d}y\right)$

8.

$T=\{(x,y)\mid 0\leq x\leq 1\;\wedge\;0\leq y\leq x^2\}$
$f(x,y)=x^3\cos(xy)\,$

Mo.

f'x(x,y) = y(sin(x2) + 2x2cos(x2)) = 0
f'y(x,y) = xsin(x2) = 0

3.

$f(x,y)=x^7+sin(y)\cos^3(y)\,$
T = [0,1]×[0,1]

4.

T = [1,e] × [1,2]
$f(x,y)=\frac{\mathrm{ln}^9\,x}{xy}$

5.

T = [-1,1] × [0,π/4]
$f(x,y)=\sin(x^3)\frac{1}{\cos^2 y}$

6.

T = [-1,1] × [0,1]
$f(x,y)=\sin(x^3)\frac{\sin^{2009}(\mathrm{sh}(y))}{\mathrm{ln}\,y}$

7.

T = [a,b] × [c,d]
f(x,y) = g(x)h(y)

téglalapon szeparálható integrandus integrálja szorzattá esik szét:

$\int\limits_{x=0}^b\int\limits_{y=c}^{d}g(x)h(y)\,\mathrm{d}x\,\mathrm{d}y=\int\limits_{x=a}^b g(x)\left(\int\limits_{y=c}^{d} h(y)\,\mathrm{d}y\right)\,\mathrm{d}x=\left(\int\limits_{x=a}^b g(x)\,\mathrm{d}x\right)\cdot\left(\int\limits_{y=c}^{d} h(y)\,\mathrm{d}y\right)$

8.

$T=\{(x,y)\mid 0\leq x\leq 1\;\wedge\;0\leq y\leq x^2\}$
$f(x,y)=x^3\cos(xy)\,$