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  3. Rekenmachines: Analyse I
  4. Rekenmachine voor differentiaal- en integraalrekening

Rekenmachine voor kwadratische benadering

Bereken kwadratische benaderingen stap voor stap

De rekenmachine vindt de kwadratische benadering van de gegeven functie bij het gegeven punt, waarbij de stappen worden getoond.

Gerelateerde rekenmachine: Rekenmachine voor lineaire benadering

Enter a function:
Enter a point:
`x_0=`

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