Linear approximation is a method used in calculus to estimate the value of a function at a certain point by using the value of the function and its derivative at a nearby point. It involves creating a tangent line to the curve at the nearby point and using its slope to approximate the value of the function at the desired point.
To find the linear approximation of a function, you will need to know the value of the function and its derivative at a nearby point. Then, you can use the formula y = f(a) + f'(a)(x-a) where a is the nearby point and x is the desired point, to find the approximate value of the function at x.
Linear approximation is useful because it allows us to estimate the value of a function at a certain point without having to use more complex methods. It is especially helpful when dealing with non-linear functions that are difficult to evaluate directly.
No, linear approximation can only be used for functions that are differentiable. This means that the function must have a well-defined derivative at the nearby point. If the function is not differentiable, then linear approximation cannot be used.
The accuracy of linear approximation depends on how close the nearby point is to the desired point. The closer the two points are, the more accurate the approximation will be. However, as the distance between the two points increases, the accuracy decreases. It is important to choose a nearby point that is as close as possible to the desired point for the most accurate approximation.