Предмет: Математика,
автор: Fertypoisufsa
Вычислить приближенно с помощью дифференциала
Ответы
Автор ответа:
0
Используй
![f(x+h) approx f(x) + h f'(x) f(x+h) approx f(x) + h f'(x)](https://tex.z-dn.net/?f=f%28x%2Bh%29+approx+f%28x%29+%2B+h+f%27%28x%29)
1.
![2,09^2 2,09^2](https://tex.z-dn.net/?f=2%2C09%5E2+)
![f(x) = x^{2} , f'(x) = 2x f(x) = x^{2} , f'(x) = 2x](https://tex.z-dn.net/?f=f%28x%29+%3D+x%5E%7B2%7D+%2C+f%27%28x%29+%3D+2x)
x= 2, h=0,09
![f(2,09) approx 2^2 + 0,09*2*2* = 4,36 f(2,09) approx 2^2 + 0,09*2*2* = 4,36](https://tex.z-dn.net/?f=+f%282%2C09%29+approx+2%5E2+%2B+0%2C09%2A2%2A2%2A+%3D+4%2C36)
2.
![sqrt{15,76} sqrt{15,76}](https://tex.z-dn.net/?f=+sqrt%7B15%2C76%7D+)
![f(x) = sqrt{x} , f'(x) = frac{1}{2 sqrt{x} } f(x) = sqrt{x} , f'(x) = frac{1}{2 sqrt{x} }](https://tex.z-dn.net/?f=f%28x%29+%3D++sqrt%7Bx%7D+%2C+f%27%28x%29+%3D++frac%7B1%7D%7B2++sqrt%7Bx%7D+%7D)
x= 16, h=-0,24
![f(15,76) approx sqrt{16} -0,24* frac{1}{2 sqrt{4}} = 4- frac{0,24}{4} = 4 - 0,06 = 3,94 f(15,76) approx sqrt{16} -0,24* frac{1}{2 sqrt{4}} = 4- frac{0,24}{4} = 4 - 0,06 = 3,94](https://tex.z-dn.net/?f=f%2815%2C76%29+approx++sqrt%7B16%7D+-0%2C24%2A+frac%7B1%7D%7B2+sqrt%7B4%7D%7D++%3D+4-+frac%7B0%2C24%7D%7B4%7D+%3D+4+-+0%2C06+%3D+3%2C94)
3.
4,36 + 3,94 = 8,3
![sqrt[3]{8,3} sqrt[3]{8,3}](https://tex.z-dn.net/?f=+sqrt%5B3%5D%7B8%2C3%7D+)
![f(x) = sqrt[3]{x} = x^{ frac{1}{3} } , f'(x) = frac{1}{3} x^{- frac{2}{3}} = frac{1}{3} frac{1}{ (sqrt[3]{x})^2 } f(x) = sqrt[3]{x} = x^{ frac{1}{3} } , f'(x) = frac{1}{3} x^{- frac{2}{3}} = frac{1}{3} frac{1}{ (sqrt[3]{x})^2 }](https://tex.z-dn.net/?f=f%28x%29+%3D++sqrt%5B3%5D%7Bx%7D+%3D+x%5E%7B+frac%7B1%7D%7B3%7D+%7D+%2C+f%27%28x%29+%3D++frac%7B1%7D%7B3%7D++x%5E%7B-+frac%7B2%7D%7B3%7D%7D+%3D+frac%7B1%7D%7B3%7D+frac%7B1%7D%7B+%28sqrt%5B3%5D%7Bx%7D%29%5E2+%7D+)
x= 8, h= 0,3
![f(8,3) approx sqrt[3]{8} + 0,3 * frac{1}{3} frac{1}{ (sqrt[3]{8})^2} = 2 + frac{0,3}{3} frac{1}{ (2)^2} = 2 + frac{0,1}{4} = 2,025 f(8,3) approx sqrt[3]{8} + 0,3 * frac{1}{3} frac{1}{ (sqrt[3]{8})^2} = 2 + frac{0,3}{3} frac{1}{ (2)^2} = 2 + frac{0,1}{4} = 2,025](https://tex.z-dn.net/?f=f%288%2C3%29+approx++sqrt%5B3%5D%7B8%7D+%2B+0%2C3+%2A++frac%7B1%7D%7B3%7D++frac%7B1%7D%7B+%28sqrt%5B3%5D%7B8%7D%29%5E2%7D+%3D+2+%2B+frac%7B0%2C3%7D%7B3%7D+frac%7B1%7D%7B+%282%29%5E2%7D+%3D+2+%2B++frac%7B0%2C1%7D%7B4%7D+%3D+2%2C025+)
1.
x= 2, h=0,09
2.
x= 16, h=-0,24
3.
4,36 + 3,94 = 8,3
x= 8, h= 0,3
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