How to Remember Concavity

My Calculus students struggled to remember the difference between the two types of concavity, so I made up a poem for them:

My fiancee, who also teaches Calculus, invented a different mnemonic…

15 thoughts on “How to Remember Concavity

  1. I was about to say something to the effect that I find it weird that people would need a mnemonic to remember concavity, but then it hit me that as I approach 40, I still need to stop and think about left and right. (Yes, I know that I can look at my thumbs outstretched thumbs and forefingers, but then I have to think about which “L” isn’t backwards, so it’s no help. When backseat driving, I tend to point in the direction we need to go and say things like, “No, the other kind of right turn.”

      • Left and right are pretty hard, actually. Especially driving. In cars, my fiancee refers to left as “Ben-ways” and right as “Taryn-ways.” Easier for her.

        Also, I have this fear that by the time I’m old, I’ll have accumulated so many unclosed parentheses I’ll need to spend my final weeks just closing them all.

  2. I go with smile vs. frown. Especially once second derivatives come in, it is easy for most to associate positive/up/smile and negative/down/frown for f”/concavity/shape.

  3. I do it this way: Concave up is a smiley face with plus signs for eyes. Concave down is a frowny face with minus signs for eyes.

  4. Hy, I’m not very skilled at math, so I have a question regarding the subject of this post:
    isn’t “concave down” actually “convex”? And to determin if something actually is concave or convex, one should take into account the viewing point of a concave/convex grapg? For example, if you take a look at a non-flat optic lens (a concave or convex one), from one point of view it will actually be concave, but from the opposite side it will actually be convex.

    • You’re basically right. In geometry, a shape is “convex” if the line connecting two points in the shape always lies inside the shape. (Like a square.) And a shape is “concave” if the line connecting two points sometimes lies OUTSIDE the shape. (Like a star.)

      Here, rather than talking about 2d shapes, we’re talking about graphs of functions, so we have to change the meanings of “concave” and “convex” a little bit. But a “convex function” is actually another way to say “concave up.”

  5. It surprises me that students at your school system need to rote-learn concave up vs. concave down using such terminology. In Hong Kong, where I learnt my basic calculus, we simply call those “minimum point” and “maximum point”.

    • “Minimum” is actually different than “concave up.” A minimum is the lowest value attained by a function (on a given interval); “concave up” means that a function’s first derivative is increasing on an interval.

  6. I notice that the arrows in concave up are pointing up, and the arrows in concave down are pointing down. But I’m curious as to why there needs to be different terms for the two in the first place. Is there some operation where concave down functions needed to be treated differently than concave up?

    • Yeah, good question – there are a few situations where the different behaviors matter.

      One way to think about concavity is as acceleration. Suppose I’m moving forward. If my position function is concave up, then I’m speeding up, whereas if my position function is concave down, then I’m slowing down.

      Concavity also helps us find maxima and minima. Picture a yo-yo bobbing up and down, up and down. We look for a place where its velocity is zero (i.e., it’s momentarily stopped moving). This happens either at the bottom of a bounce, or at the top. And the concavity of it height function will help us determine whether we’re at the bottom (minimum) or the top (maximum).

  7. I always got them backwards, they way that I thought of it was concave up meant it looks like an arrow head pointing up, concave down meant it looked like an arrow head pointing down, however, then I’d have to remember that it is the opposite of what I normally would think. Now, I just remember concave up = positive 2nd derivative, concave down = negative 2nd derivative.

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