Rechner für lineare Approximation
Lineare Approximationen Schritt für Schritt berechnen
Der Rechner ermittelt die lineare Approximation an die explizite, polare, parametrische und implizite Kurve am gegebenen Punkt, mit Lösungsschritten.
Verwandter Rechner: Rechner für die quadratische Approximation
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: $$$L(x)\approx \frac{1}{4} x+1 =0.25x+1$$$.