Question

Напишите уравнения касательной функции f(x) в точке с абцссой Хо
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Answer 1

$f(x) = f(x_0) + f'(x_0)(x - x_0)$

a

$f(x) = ln(x + 1) \\ x_0 = 0$

$f(0) = ln(1) = 0$

$f'(x) = \frac{1}{x + 1} \\$

$f'(0) = \frac{1}{1} = 1 \\$

$f(x) = 0 + 1(x - 0) = x$

б

$f(x) = 2 ln(x) + 1 \\ x_0 = 1$

$f(1) = 0 + 1 = 1$

$f'(x) = \frac{2}{x} \\$

$f'(1) = 2$

$f(x) = 1 + 2(x - 1) = 1 + 2x - 2 = \\ = 2x - 1$

в

$f(x) = 3 ln(x) \\ x_0 = \frac{1}{e} = {e}^{ - 1} \\$

$f( \frac{1}{e} ) = 3 ln(e {}^{ - 1} ) = - 3 \\$

$f'(x) = \frac{3}{x} \\$

$f'( \frac{1}{e} ) = 3e \\$

$f(x) = - 3 + 3e(x - \frac{1}{e} ) = \\ = - 3 + 3ex - 3 = 3ex - 6$

г

$f(x) = log_{2}(x + 1) \\ x_0 = 0$

$f(0) = 0$

$f'(x) = \frac{1}{ ln(2) (x + 1)} \\$

$f'(0) = \frac{1}{ ln(2) } \\$

$f(x) = 0 + \frac{1}{ ln(2) } (x - 0) = \frac{x}{ ln(2) } \\$