ОДЗ:
сos(y-x)>0
4-x²-y²≥0
-π/2 ≤y-x≤π/2 или х - (π/2) ≤ y ≤ x + (π/2)
Область ограничена сверху прямой у=х+(π/2) синий цвет на рис.,
снизу прямой у=х-(π/2) - зеленый цвет на рис.
х²+y²≤4 - внутренность круга с центром в точке (0;0) радиусом 2.
Область определения часть полосы, заключенная внутри круга. Розовый цвет.
![z`_x=\frac{1}{cos(y-x)} \cdot (cos(y-x))`_x+ \frac{1}{2 \sqrt{4-x^2-y^2} } \cdot(4-x^2-y^2)`_x= \\ \\ = \frac{1}{cos(y-x)} \cdot (-sin (y-x))\cdot(y-x)`_x+ \frac{1}{2 \sqrt{4-x^2-y^2} } \cdot(-2x)= \\ \\ =\frac{-sin(y-x)}{cos(y-x)} \cdot(-1)- \frac{2x}{2 \sqrt{4-x^2-y^2} }=\\ \\ =tg(y-x)-\frac{x}{ \sqrt{4-x^2-y^2} }](https://tex.z-dn.net/?f=z%60_x%3D%5Cfrac%7B1%7D%7Bcos%28y-x%29%7D+%5Ccdot+%28cos%28y-x%29%29%60_x%2B+%5Cfrac%7B1%7D%7B2+%5Csqrt%7B4-x%5E2-y%5E2%7D+%7D+%5Ccdot%284-x%5E2-y%5E2%29%60_x%3D+%5C%5C++%5C%5C+%3D+%5Cfrac%7B1%7D%7Bcos%28y-x%29%7D+%5Ccdot+%28-sin+%28y-x%29%29%5Ccdot%28y-x%29%60_x%2B+%5Cfrac%7B1%7D%7B2+%5Csqrt%7B4-x%5E2-y%5E2%7D+%7D+%5Ccdot%28-2x%29%3D+%5C%5C++%5C%5C+%3D%5Cfrac%7B-sin%28y-x%29%7D%7Bcos%28y-x%29%7D+%5Ccdot%28-1%29-+%5Cfrac%7B2x%7D%7B2+%5Csqrt%7B4-x%5E2-y%5E2%7D+%7D%3D%5C%5C++%5C%5C+%3Dtg%28y-x%29-%5Cfrac%7Bx%7D%7B+%5Csqrt%7B4-x%5E2-y%5E2%7D+%7D)
![z`_y=\frac{1}{cos(y-x)} \cdot (cos(y-x))`_y+ \frac{1}{2 \sqrt{4-x^2-y^2} } \cdot(4-x^2-y^2)`_y= \\ \\ = \frac{1}{cos(y-x)} \cdot (-sin (y-x))\cdot(y-x)`_y+ \frac{1}{2 \sqrt{4-x^2-y^2} } \cdot(-2y)=\\ \\ =\frac{-sin(y-x)}{cos(y-x)} \cdot 1- \frac{2y}{2 \sqrt{4-x^2-y^2} }=\\ \\ =-tg(y-x)-\frac{y}{ \sqrt{4-x^2-y^2} }](https://tex.z-dn.net/?f=z%60_y%3D%5Cfrac%7B1%7D%7Bcos%28y-x%29%7D+%5Ccdot+%28cos%28y-x%29%29%60_y%2B+%5Cfrac%7B1%7D%7B2+%5Csqrt%7B4-x%5E2-y%5E2%7D+%7D+%5Ccdot%284-x%5E2-y%5E2%29%60_y%3D+%5C%5C++%5C%5C+%3D+%5Cfrac%7B1%7D%7Bcos%28y-x%29%7D+%5Ccdot+%28-sin+%28y-x%29%29%5Ccdot%28y-x%29%60_y%2B+%5Cfrac%7B1%7D%7B2+%5Csqrt%7B4-x%5E2-y%5E2%7D+%7D+%5Ccdot%28-2y%29%3D%5C%5C++%5C%5C+%3D%5Cfrac%7B-sin%28y-x%29%7D%7Bcos%28y-x%29%7D+%5Ccdot+1-+%5Cfrac%7B2y%7D%7B2+%5Csqrt%7B4-x%5E2-y%5E2%7D+%7D%3D%5C%5C++%5C%5C+%3D-tg%28y-x%29-%5Cfrac%7By%7D%7B+%5Csqrt%7B4-x%5E2-y%5E2%7D+%7D)
![dz=z`_xdx+z`_ydy \\ \\ dz=(tg(y-x)-\frac{x}{ \sqrt{4-x^2-y^2} })dx+(-tg(y-x)-\frac{y}{ \sqrt{4-x^2-y^2} })dy](https://tex.z-dn.net/?f=dz%3Dz%60_xdx%2Bz%60_ydy+%5C%5C++%5C%5C+dz%3D%28tg%28y-x%29-%5Cfrac%7Bx%7D%7B+%5Csqrt%7B4-x%5E2-y%5E2%7D+%7D%29dx%2B%28-tg%28y-x%29-%5Cfrac%7By%7D%7B+%5Csqrt%7B4-x%5E2-y%5E2%7D+%7D%29dy)
2) Методом выделения полного квадрата x^4-7x^2+12=0
(x^2)^2-2*x^2*3.5+3.5^2-3.5^2+12=0
(x^2-3.5)^2=12.25-12=0.25
x^2-3.5=-0.5 x^2=3 x1=-V3 x2=V3
x^2-3.5=0.5 x^2=4 x3=-2 x4=2
14a3b÷21a2b2=2a÷3b
я делила на 7
(x-5)²-x(x+3)=12
(x-5)(x-5)-x(x+3)=12
x²-10x+25-x²-3x=12
-13x=12-25
-13x=-13
x=1
X^2+7x+10=0
d=49-4*10=9;3
x1=-2
x2=-5
меньший корень -5