Problem: Use tangent line to find an approximating formula for

for x close to 1.

Solution: The equation of the tangent line at a is

We want approximation around 1, so the natural choice of the point where we make this tangent line is a = 1. We get

With a little bit of luck we can approximate (for x "close" to 1)

Since 1 is the center of this approximation, it makes sense to do things relative to this point, in particular replace x with 1 + h. For h small we then have

How good is this approximation? The tangent line is actually the Taylor polynomial of degree 1 at a = 1. We can consider some x > 0 and use the Lagrange form of remainder to approximate the error. Since we do not know whether the chosen x is larger or smaller than 1, we might have a slight difficulty with notation; to fix it we denote by I the closed interval with endpoints 1 and x.

The maximum in the formula is 1 if x > 1, otherwise it is the other expression. We rewrite it in terms of h:

We see that if h is very close to 0, then the error is negligible compared to the estimate, and thus the estimate is quite good. For instance, if we decide to use the above approximation only for |h| < 0.5, then the error is at most 0.09. With |h| < 0.2 the error is at most 0.007.

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