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Kalkylator för kvadratisk approximation
Beräkna kvadratiska approximationer steg för steg
Räknaren kommer att hitta den kvadratiska approximationen till den givna funktionen vid den givna punkten, med steg som visas.
Relaterad kalkylator: Kalkylator för linjär approximation
Solution
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: $$$Q(x)\approx \frac{x^{2}}{324} + \frac{x}{54} + \frac{17}{4}$$$.