A linear approximation is a method in calculus used to estimate the value of a function at a certain point by using the tangent line to that point on the curve of the function. It is a good approximation for values of x that are close to the given point.
The formula for linear approximation is f(x) ≈ f(a) + f'(a)(x-a), where f(x) is the function, a is the given point, and f'(x) is the derivative of the function.
Linear approximation and linearization are often used interchangeably, but they are slightly different. Linear approximation is used to approximate the value of a function at a specific point, while linearization is used to approximate the behavior of a function near a specific point.
Linear approximation has a variety of applications in mathematics, physics, and engineering. It is commonly used in optimization problems, to estimate the error in numerical methods, and in the construction of mathematical models.
No, linear approximation can only be used for functions that are approximately linear near the given point. For non-linear functions, other methods such as quadratic or cubic approximations may be used.