Rechner für die quadratische Approximation

Your input: find the quadratic approximation to $$$f(x)=\sqrt{x} + \frac{5}{\sqrt{x}}$$$ at $$$x_0=9$$$.

A quadratic approximation is given by $$$Q(x)\approx f(x_0)+f^{\prime}(x_0)(x-x_0)+\frac{1}{2}f^{\prime \prime}(x_0)(x-x_0)^2$$$.

We are given that $$$x_0=9$$$.

Firstly, find the value of the function at the given point: $$$y_0=f(x_0)=\frac{14}{3}$$$.

Secondly, find the derivative of the function, evaluated at the point: $$$f^{\prime}\left(9\right)$$$.

Find the derivative: $$$f^{\prime}\left(x\right)=\frac{x - 5}{2 x^{\frac{3}{2}}}$$$ (steps can be seen here).

Next, evaluate the derivative at the given point.

$$$f^{\prime}\left(9\right)=\frac{2}{27}$$$.

Now, find the second derivative of the function evaluated at the point: $$$f^{\prime \prime}\left(9\right)$$$.

Find the second derivative: $$$f^{\prime \prime}\left(x\right)=\frac{15 - x}{4 x^{\frac{5}{2}}}$$$ (steps can be seen here).

Next, evaluate the second derivative at the given point.

$$$f^{\prime \prime}\left(9\right)=\frac{1}{162}$$$.

Plugging the found values, we get that $$$Q(x)\approx \frac{14}{3}+\frac{2}{27}\left(x-\left(9\right)\right)+\frac{1}{2}\left(\frac{1}{162}\right)\left(x-\left(9\right)\right)^2$$$.

Simplify: $$$Q(x)\approx \frac{x^{2}}{324} + \frac{x}{54} + \frac{17}{4}$$$.

Answer: $$$Q(x)\approx \frac{x^{2}}{324} + \frac{x}{54} + \frac{17}{4}$$$.