Linear Approximation Calculator

An online linear approximation calculator helps you to calculate the linear approximations of either parametric, polar, or explicit curves at any given point. The idea behind linearization or local linear approximation is to find a value of the function at the given point and evaluate the derivative to find the slope of entered points. Here’s we know all about how to do linear approximation of different types of curves.

What is Linear Approximation?

In mathematics, use a linear approximation to estimate the value of a general function \(f(x)\) by using linear expressions. This is also known as tangent line approximation, which is the method of determining the line equation that is nearer estimation for entered linear functions at any given value of x. So, the linear approximation calculator approximates the value of the function and finds the derivative of the function to evaluate the derivative to find slope with the help of the linearization formula.

How to do Linear Approximation?

A linear approximation equation can simplify the behavior of complex functions. The point \(x = k\) gives the most accurate linear approximation. As we move farther away from the point \(x = k\), the estimation becomes less accurate. A simple curve linear approximation follows the direction of the curve, but it does not predict the concavity of the curve.

Example: Find the value of \(f(8.3)\) by using the linear approximation at \(x_0 = 3\), where the function is differentiable such that \(f(3)=12\) and \(f'(3)=-2\).

Solution:

By using the linear approximation formula:

\[ L(x) \approx f(x_0) + f'(x_0)(x - x_0) \]

Substituting the given values:

\[ L(x) = f(3) + f'(3)(x-3) \]

\[ L(x) = 12 - 2(x-3) \]

\[ L(x) = 18 - 2x \]

\[ f(8.3) \approx 18 - 2(3.5) \]

\[ f(8.3) = 18 - 7 \]

\[ f(8.3) = 11 \]

Moreover, an Integral Calculator helps you evaluate integrals of functions with respect to the variable involved.

How Linear Approximation Calculator Works?

The online linearization calculator will estimate the values of a given function by using linear approximation formula with the following steps:

Input:

• First, choose the type of linear function for approximation from the drop-down menu.
• Enter a function that requires finding the linear approximation.
• Now, enter a point to find the value function at the given point.
• In some cases, when you select the type parametric from the drop-down then put the value of t in to find x and y coordinates, that corresponds to the t.
• Hit the calculate button.

Output:

• Displays the linear approximation values with step-by-step calculations.
• Evaluate the derivatives at the given point to find a slope.

Reference

From the source of Wikipedia: Period of oscillation, Electrical resistivity, Gaussian optics. From the source of Paul's Notes: Linear Approximations, Linearization of a function, Find the linearization of L(x) of the function.