# Linear Approximation Calculator

The calculator will find the linear approximation to the explicit, polar, parametric and implicit curve at the given point, with steps shown.

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x_0=( )

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## 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: $$L(x)\approx \frac{1}{4} x+1$$\$.

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