Linear Approximation Calculator

Calculate linear approximations step by step

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

Related calculator: Quadratic Approximation Calculator

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