Question

How do we know when a linear approximation is truly depicting a good approximation of the curve of a nonlinear function near some base point?

I know that there are two ways to understand the formula for linear approximation. One that involves the definition of the derivative whereas, the other includes a tangent line that approximates the curve f(x) near the point of tangency.

Suppose the base point is changed by a small amount delta x resulting in a specific change delta y. If delta y/ delta x is way larger than the value of the derivative at the base point, should we still accept the secant line joining the base point and the second point as a good approximation to the curve? How close should delta x be to zero to say that the secant line is a 'good' approximation to the curve?

Answer 1

Think of the approximation in a different way:

If we have a linear (or quadratic, cubic or whatever) approximation of some function `y=f(x)` then what have we really got. Hopefully if the approximation is well constructed it should be identical to the truncated Taylor (or more likely Maclaurin) series, thus if we have the following Maclaurin Series:

`y= a + bx + cx^2 + dx^3 + ....`

So, hopefully our linear approximation is the same as:

`y= a + bx`

If not, then we do not have a very good approximation!

If we add higher order terms then we get an even better approximation.

The issue of "how good is a linear approximation", is then really a case of how big is the remainder of the series when truncated.

For small `x` these errors are generally quite small, and the values of `x` we can actually use will be determined by the radius of convergence of the power series.

The actual value of the truncation error is an entire new topic in Mathematics and way beyond the context of this question:

https://en.wikipedia.org/wiki/Taylor%27s_theorem

will provide some insight