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…

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21 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.”

      1. 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.

    1. 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.”

      https://en.wikipedia.org/wiki/Convex_function

  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”.

    1. “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?

    1. 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.

  8. I used to think “concave up” and “concave down” were pretty easy. But when I got to college, math professors started insisting on referring to the “concave up” functions as “convex.”

    I made up my own mnemonic,
    “ConVEX functions FLEX UP;
    ConCAVE functions CAVE DOWN.”

    I have yet to see if it helps anyone but me, though.

  9. This strikes me as a needless terminological obstacle to students. The graphs “curve upwards” or “curve downwards”, as any student would instantly know; and the use of jargon words – concave or convex – is merely a way to alienate students from the subject matter. It doesn’t actually add any value; it’s just a superannuated hang-over from the days when learning Latin and ancient Greek was a prerequisite of being allowed to learn anything useful.

    The graph {[x, x^3]: x is real} curves downwards in the left half of the plane and upwards in the right half. How does your jargon describe this ? How about the square root relation, graphed by {[y^2, y]: y is real} – does your jargon describe it as concave to the right ?

    In geometry, convex is a useful term, describing a region within which any two points may be joined by a straight line within the region; convex is somewhat fuzzily defined, but usually as a complement of convex. Overloading concave with the meaning you’re giving it here, describing a line (not a region), is just a recipe for confusing students; the region *above* a line curving up is convex, the region below is concave; but that’s the regions, not the line that separates them. A convex lens is indeed convex (in the geometric sense) and a concave one has faces that curve the opposite way. (A “star”, by which I take it you mean the usual “many points” picture, rather than the actual spherical balls most stars actually are, is “star-shaped”, a proper term of topology, and neither convex nor concave.)

    In general, it’s better to teach the subject matter in terms that eschew jargon whenever natural language expresses the same concept entirely as well. Of course, if the syllabus is specified and the exam is set by folk who insist on familiarity with distracting and unnecessary jargon, derived from dead languages the students aren’t learning, you have to teach it; but it’ll be clearer to students to explain that the graph “curves up” or “curves down” and, as long as it curves consistently in the same direction, pedants and gate-keepers of the path to further education want them to describe it as “concave” in the same direction that it curves. Be on the students’ side, teaching them the meaningful subject-matter in plain terms and coaching them to phrase it in opaque ways only in so far as that’ll get them through exams. Maybe those of them that end up on school boards shall sweep away the needless requirement to use Latin-derived terminology to say what their native language can express just fine.

    1. Yeah, I don’t think I would have agreed with you when I wrote this post, but having taught in the UK where these terms aren’t much used (unlike the US, where they’re ubiquitous) I now pretty much agree.

      FWIW, I think “curves up” and “curves down” are probably clear enough for classroom conversation, but maybe not for written communication (where “curves up” might be mistaken for meaning f ‘ > 0 rather than f ” > 0).

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