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I've talked a bit about how the approximation $\sin(x) \approx x$ when $x$ is small is important in physics. That is mentioned in this section, which is one reason why I wanted to have a look at 2.9:
More generally, we frequently use the linear function representing the tangent line at a point to make certain approximations.
That linear function is called the linearization of $f$ at $a$:
Differentials versus increments
The increment is the true change in the function value; the differential approximates the true change: One way to remember the difference:
$dy \approx \Delta y$
We want the increment, but may settle for the easily computed differential.
We have hopefully deduced these:
Rephrased: If $f$ has a local max or min at $c$, then $c$ is a critical number of $f$.