Linear Approximation Calculator

Function f(x)

Use standard math notation: +, -, *, /, ^, sin(), cos(), etc.

Point a

Point where we know the value

Point x

Point to approximate

Results

Enter function and points to see results

Linear approximation works best when x is close to a. The error increases as the distance between x and a grows.

What is Linear Approximation?

Linear approximation, also known as the tangent line approximation, is a method in calculus used to estimate the value of a function near a point. It uses the tangent line to the function at a known point to approximate function values at nearby points. This technique is based on the idea that for small changes in the input, a function behaves approximately like its tangent line.

The Linear Approximation Formula

The linear approximation of a function f(x) at a point x = a is given by:

L(x) = f(a) + f′(a)(x - a)

Where:

f(a) is the value of the function at point a
f′(a) is the derivative of the function at point a
x is the point at which we want to approximate the function
L(x) is the linear approximation of f(x)

Applications of Linear Approximation

Linear approximation has numerous practical applications:

Estimating function values without complex calculations
Error estimation in numerical methods
Simplifying complex functions for easier analysis
Quick mental calculations in physics and engineering
Approximating solutions to differential equations
Developing algorithms for computer graphics and simulations

How to Use the Linear Approximation Calculator

Enter the function f(x) using standard mathematical notation (e.g., x^2, sin(x), e^x)
Input the point a where you know the function value
Enter the point x where you want to approximate the function
Click "Calculate" to see the results
The calculator will display the linear approximation and the actual function value for comparison

Limitations of Linear Approximation

While linear approximation is a powerful tool, it has some limitations:

The accuracy decreases as you move away from the point of tangency
It works best for functions that are relatively straight near the point of interest
For highly curved functions, the approximation may be poor even for nearby points
It requires the function to be differentiable at the point of tangency

For better approximations of highly nonlinear functions, higher-order methods like Taylor series may be more appropriate.