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Calculadora de Aproximação Linear
Calcular aproximações lineares passo a passo
A calculadora encontrará a aproximação linear para a curva explícita, polar, paramétrica e implícita no ponto determinado, com as etapas mostradas.
Calculadora relacionada: Calculadora de Aproximação Quadrática
Solution
Your input: find the linear approximation to $$$f(x)=\sqrt{x}$$$ at $$$x_0=4$$$.
A linear approximation is given by $$$L(x)\approx f(x_0)+f^{\prime}(x_0)(x-x_0)$$$.
We are given that $$$x_0=4$$$.
Firstly, find the value of the function at the given point: $$$y_0=f(x_0)=2$$$.
Secondly, find the derivative of the function, evaluated at the point: $$$f^{\prime}\left(4\right)$$$.
Find the derivative: $$$f^{\prime}\left(x\right)=\frac{1}{2 \sqrt{x}}$$$ (steps can be seen here).
Next, evaluate the derivative at the given point to find slope.
$$$f^{\prime}\left(4\right)=\frac{1}{4}$$$.
Plugging the values found, we get that $$$L(x)\approx 2+\frac{1}{4}\left(x-\left(4\right)\right)$$$.
Or, more simply: $$$L(x)\approx \frac{1}{4} x+1$$$.
Answer: $$$L(x)\approx \frac{1}{4} x+1 $$$.