What is a tangent line approximation?

Tangent line approximation is used in calculus to approximate the values of a function based on its tangent. Tangent line approximation can be used because any curve examined closely at a particular point begins to bear similarities to a straight line, represented by the tangent.

For example, for the function f(x) with a point x0, tangent line approximation can be used to approximate f near x0 by a simple linear function. The tangent line approximation f(x) is approximately equal to f(x0) + f'(x0)(x - x0). The last term on the right is the equation of a straight line in x. At x = x0, the term agrees with f(x), as it takes the value f(x0) and has the same slope as the tangent to the curve at x0.