Для начала найдем точку пересечения с биссектрисой первого координатного угла:
Биссектрисой этого угла является прямая y=x
тогда:
![\left \{ {{y=-\sqrt{x}+2} \atop {y=x}} \right. \\x=-\sqrt{x}+2 \\\sqrt{x}=2-x \\x=x^2-4x+4,\ x \geq 0;\ 2-x \geq 0; \ x \leq 2; \ x \in [0;2] \\x^2-5x+4=0 \\D=25-16=9=3^2 \\x_1= \frac{5+3}{2} =4 \notin [0;2] \\x_2= \frac{5-3}{2} =1 \in [0;2]](https://tex.z-dn.net/?f=+%5Cleft+%5C%7B+%7B%7By%3D-%5Csqrt%7Bx%7D%2B2%7D+%5Catop+%7By%3Dx%7D%7D+%5Cright.+%5C%5Cx%3D-%5Csqrt%7Bx%7D%2B2+%5C%5C%5Csqrt%7Bx%7D%3D2-x+%5C%5Cx%3Dx%5E2-4x%2B4%2C%5C+x+%5Cgeq+0%3B%5C+2-x+%5Cgeq+0%3B+%5C+x+%5Cleq+2%3B+%5C+x+%5Cin+%5B0%3B2%5D+%5C%5Cx%5E2-5x%2B4%3D0+%5C%5CD%3D25-16%3D9%3D3%5E2+%5C%5Cx_1%3D+%5Cfrac%7B5%2B3%7D%7B2%7D+%3D4+%5Cnotin+%5B0%3B2%5D+%5C%5Cx_2%3D+%5Cfrac%7B5-3%7D%7B2%7D+%3D1+%5Cin+%5B0%3B2%5D)
y=1
точку пересечения нашли: (1;1)
теперь ищем уравнение нормали:
общая формула:
![y=f(x_0)- \frac{1}{f'(x_0)} *(x-x_0)](https://tex.z-dn.net/?f=y%3Df%28x_0%29-+%5Cfrac%7B1%7D%7Bf%27%28x_0%29%7D+%2A%28x-x_0%29)
в данной задаче:
x0=1
f(1)=-1+2=1
![f'(x)=(-x^{ \frac{1}{2} }+2)'=- \frac{1}{2}*x^{ \frac{1}{2}-1}= - \frac{1}{2}* \frac{1}{\sqrt{x}} =- \frac{1}{2\sqrt{x}} \\f'(1)=- \frac{1}{2} =-0.5](https://tex.z-dn.net/?f=f%27%28x%29%3D%28-x%5E%7B+%5Cfrac%7B1%7D%7B2%7D+%7D%2B2%29%27%3D-+%5Cfrac%7B1%7D%7B2%7D%2Ax%5E%7B+%5Cfrac%7B1%7D%7B2%7D-1%7D%3D+-+%5Cfrac%7B1%7D%7B2%7D%2A+%5Cfrac%7B1%7D%7B%5Csqrt%7Bx%7D%7D+%3D-+%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%7D%7D+%5C%5Cf%27%281%29%3D-+%5Cfrac%7B1%7D%7B2%7D+%3D-0.5)
подставляем значения в уравнение:
![y=1- \frac{1}{-0,5} *(x-1)=1+2x-2=2x-1](https://tex.z-dn.net/?f=y%3D1-+%5Cfrac%7B1%7D%7B-0%2C5%7D+%2A%28x-1%29%3D1%2B2x-2%3D2x-1)
y=2x-1 - искомая нормаль
Ответ: y=2x-1
1.
![y=0,7^{x}](https://tex.z-dn.net/?f=y%3D0%2C7%5E%7Bx%7D+)
<span>
x </span><span><span>-5
</span><span>
-4
</span><span>
-3
</span><span>
-2
</span><span>
-1
</span><span>
0
</span><span>
1
</span><span>
2
</span><span>
3
</span><span>
4
</span><span>
5
</span></span><span>
y </span><span><span><span>5,9
</span><span>
4,2
</span><span>
2,9
</span><span>
2
</span><span>
1,4 </span><span>
1
</span><span>
0,7
</span><span>
0,5
</span><span>
0,3
</span><span>
0,2
</span><span>
0,2
</span></span></span><span>
2.
![y=1^{x}](https://tex.z-dn.net/?f=+y%3D1%5E%7Bx%7D+)
y=1
3.
![y=0^{x}](https://tex.z-dn.net/?f=+y%3D0%5E%7Bx%7D+)
</span>
y=0
10√3 - 4√48 - √75 = 10√3 - 16√3 - 5√3 =-11√3