Calculatrice d’approximation linéaire
Calculez des approximations linéaires étape par étape
La calculatrice trouvera l'approximation linéaire pour les courbes explicites, polaires, paramétriques et implicites au point donné, avec les étapes affichées.
Calculatrice associée: Calculatrice d'approximation quadratique
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: $$$L(x)\approx \frac{1}{4} x+1 =0.25x+1$$$.