In the Real World
Calculators are good at graphing functions. However, calculators can only show a useful graph if a useful domain and range are specified.
Play with the function
f (x) = xln x
If we graph it on a window with -10 ≤ x ≤ 10 and -10 ≤ y ≤ 10, it looks like this:
This doesn't look like much—a concave up curve that starts a little below the x-axis.
However, if we know that
and therefore f must have a critical point at e-1 ≈ 0.37, we might be tempted to look a little closer.
Here's the graph on a window with -2 ≤ x ≤ 2 and -2 ≤ y ≤ 2:
Knowing where there might be maxima or minima can help you choose the right size of the calculator window, so that you can actually see what the function looks like. In short, mastering all this derivative stuff will keep your calculator in check. Go on, try to tell us that's not useful.