As you’ve probably noticed, different authors use different definitions based on how they plan to present topics later in their texts. So, the same concept seems to have different definitions. A definition in one book is a theorem in another. Students should be aware of this; looking at different definitions and related theorems about the same idea helps their mathematical education.
Here is a thought on exploring how concepts may be defined and learning about concavity are the same time.
- Increasing, concave up
- Increasing concave down
- Decreasing concave up
- Decreasing concave down
Look at the sine or cosine graphs which show all four and the tangent graph that shows only two. Look at other functions as well and identify which parts (intervals) exhibit each shape.
Next, challenge the students, in groups, individually, in class, or for homework to find analytic ways to say what concavity is or to identify it from equations. If they need hints suggest they look at derivatives (first and second) and tangent lines. Don’t limit them – explain that there are other ways. They should try to find several ways.
Hopefully, they will come up with some of these. (I have purposely listed these in a rough preliminary form. That’s how math is done – get an idea and then develop and formalize it)
A function is concave up (down) when:
- The slope (derivative) is increasing (decreasing)
- The second derivative is positive (negative).
- The tangent line lies below (above) the graph
- All (any, every) segments joining points in the interval lie above (below) the graph.
- Others ???
Finally, clean these up. Help the students:
- State them as “if, then” or “if, and only if” statements giving the hypotheses for each.
- Consider how one can be used to imply the others – in either direction.
- Determine which implies the inclusion of endpoints (1 and maybe 3).
- Discuss which the class thinks would make the best definition, and which should become theorems.
- Taking whichever definition your book uses, show how the others can be proved.
The point here is the thinking, forming ideas, doing the mathematics; the understanding of concavity will follow.