Rekenmachine voor lineaire benadering
Bereken lineaire benaderingen stap voor stap
De rekenmachine bepaalt de lineaire benadering van de expliciete, polaire, parametrische en impliciete kromme in het gegeven punt, met de tussenstappen.
Gerelateerde rekenmachine: Rekenmachine voor kwadratische benadering
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: $$$L(x)\approx \frac{1}{4} x+1 =0.25x+1$$$.