Linear Approximation Calculator

Linear Approximation Calculator

Select the type of function, enter it, and specify the point. The calculator will quickly determine the linear approximation with detailed step-by-step calculations.

Choose Type:

Enter a point: y = f(x)

Enter a function: x₀

Linear Approximation Calculator

An online Linear Approximation Calculator helps you calculate linear approximations of parametric, polar, or explicit curves at any given point. Linearization or local linear approximation estimates the value of a function at a specific point and evaluates the derivative to find the slope. Here’s everything you need to know about linear approximation of different types of curves.

What is Linear Approximation?

In mathematics, a linear approximation estimates the value of a general function f(x) using a linear expression. This is also called the tangent line approximation, which determines a line equation that closely estimates the function near a given value of x. The calculator uses the linearization formula to approximate the function value and compute the derivative to determine the slope:

Linearization formula:

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

How to Perform Linear Approximation

A linear approximation simplifies the behavior of complex functions. The point x = x₀ gives the most accurate approximation. As you move farther from this point, the approximation becomes less accurate. Linear approximation captures the direction of the curve but does not account for concavity.

Example:

Find the value of f(8.3) using linear approximation with x₀ = 3, where f(3) = 12 and f'(3) = -2.

Solution:

Using the linear approximation formula:

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

Substitute the values:

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

Simplify the expression:

\[ L(x) = 12 - 2(x - 3) = 12 - 2x + 6 = 18 - 2x \]

Evaluate at x = 8.3:

\[ f(8.3) \approx 18 - 2(8.3) = 18 - 16.6 = 1.4 \]

Note: An Online Integral Calculator can help evaluate integrals of functions if needed.

How Linear Approximation Calculator Works

Input:

Select the type of linear function for approximation (explicit, parametric, or polar).
Enter the function for which you want to find a linear approximation.
Enter the point x₀ to approximate the function value.
For parametric functions, enter the parameter value t to find corresponding x and y coordinates.
Click the Calculate button.

Output:

Displays the linear approximation values with step-by-step calculations.
Shows the derivative at the given point to determine the slope of the tangent line.

References

Wikipedia: Linear Approximation
Paul’s Online Math Notes: Linearization of a Function