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Kalkylator för linjär approximation
Beräkna linjära approximationer steg för steg
Kalkylatorn bestämmer den linjära approximationen för explicita, polära, parametriska och implicita kurvor vid den givna punkten, med stegvis genomgång.
Relaterad kalkylator: Kalkylator för kvadratisk 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: $$$L(x)\approx \frac{1}{4} x+1 =0.25x+1$$$.