- Topics At a Glance
- Second Derivatives via Formulas
- Third Derivatives and Beyond
- Concavity
- Concave Up
- Concave Down
- No Concavity
- Special Points
- Critical Points
- Points of Inflection
- Extreme Points and How to Find Them
- Finding & Classifying Extreme Points
- First Derivative Test
- Second Derivative Test
- Local vs. Global Points
- Using Derivatives to Draw Graphs
- Finding Points
- Finding Shapes
- Connecting the Dots
**In the Real World**- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

Calculators 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*) = x*ln* 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 is correct. Go on, try to tell us that's not useful.