Linearization Calculator – Find Linear Approximation
Find the linear approximation of a function at a point.
Table of Contents
How to Use
- Enter your function using standard notation (e.g., x^2, sin(x), exp(x))
- Specify the variable name (default is x)
- Enter the point where you want to linearize the function
- Click calculate to get the linear approximation
What is Linearization?
Linearization is the process of approximating a function near a point using its tangent line. The linear approximation L(x) at point a is given by: L(x) = f(a) + f'(a)(x - a), where f(a) is the function value and f'(a) is the derivative at point a.
This approximation works best for values of x close to a. The further x is from a, the less accurate the approximation becomes.
The Linearization Formula
- L(x) = f(a) + f'(a)(x - a)
- f(a) is the y-coordinate of the point on the curve
- f'(a) is the slope of the tangent line
- (x - a) represents the horizontal distance from the point
Applications of Linearization
- Approximating complex functions with simpler linear ones
- Error estimation in measurements
- Physics: small angle approximations (sin(θ) ≈ θ)
- Engineering: analyzing systems near equilibrium points
- Economics: marginal analysis
Frequently Asked Questions
- When is linearization most accurate?
- Linearization is most accurate when x is very close to the point a. The approximation error grows as you move further from a, especially for functions with high curvature.
- What's the difference between linearization and Taylor series?
- Linearization is the first-order Taylor polynomial—it uses only the function value and first derivative. Taylor series can include higher-order terms for better accuracy over larger intervals.
- Can I linearize any function?
- You can linearize any function that is differentiable at the point of interest. If the function has a discontinuity or corner at that point, linearization isn't possible there.