The Linear Approximation Calculator is a mathematical tool used to estimate the value of a function at a certain point by approximating it as a linear function. It operates on the principle of linearization, enabling users to predict the value of a function at a specific point by using the function’s derivative at another known point.
Formula of Linear Aproximation Calculator
The formula used by the Linear Approximation Calculator is:
f(x) ≈ f(a) + f'(a)(x – a)
Where:
- f(x) represents the original function to be approximate.
- a is the point where the approximation is made.
- f(a) is the value of the original function at point ‘a’.
- f'(a) denotes the derivative of the original function evaluated at ‘a’.
- x is the value at which the approximate value of f(x) is you require.
Table of General Terms
| Term | Definition |
|---|---|
| Original Function (f) | The primary function for which approximation is done. |
| Point (a) | The specific point at which approximation is made. |
| Derivative (f'(a)) | The rate of change of the original function at ‘a’. |
| Approximated Value | The estimated value of the function at point ‘x’. |
Example of Linear Aproximation Calculator
Let’s consider a quadratic function f(x) = x^2. At point a = 2, f(a) = 4. The derivative of f(x) = x^2 at a = 2 is f'(a) = 4. If we wish to approximate f(3), applying the linear approximation formula:
f(3) ≈ f(2) + f'(2)(3 – 2) = 4 + 4 * (3 – 2) = 8
Hence, the approximate value of f(3) using linear approximation is 8.
Most Common FAQs
A: The accuracy depends on how well the linear function approximates the original function in the vicinity of point ‘a’. Closer the point ‘x’ to ‘a’, higher the accuracy.
A: Linear approximation works well for functions that can be linearly approximate near the chosen point ‘a’. However, for functions with highly nonlinear behavior, the accuracy might vary.