Two things I tell calculus students (one is the squeeze theorem)

22 Jun

I was subbing for a friend in our math tutoring center the other day and ended up chatting with an undergraduate who was retaking calculus.  She asked if I was a grad student in math, and when I affirmed, she said “wow, you must have memorized so many formulas.”  I laughed.  I told her that math is a lot like cooking.  Yes, you do need to memorize a few basics (how to cut an onion, general measurements like tsp to a TB, etc.), but you certainly don’t need to memorize every recipe you’re going to use.  You should definitely read them through and understand the rough idea of what’s going to happen; the more recipes you read, the better you’ll know how to use various ingredients.  And if you just pick up a cookbook and read a random recipe, maybe you’ll branch out to more exotic ingredients and figure out yourself how to incorporate rutabaga into your existing repertoire.

Hilarious photo from coursera (click for link)

To make the analogy very clear: you should read and understand formulas, proofs, etc. very well, but no one expects you to be a walking textbook.  For a single class or a single exam, yes, you should know the info there.  But the idea is that from studying a theorem really hard for a while, you’ll remember the key idea for much longer than a semester.   Logic is hard, proofs are hard, math is hard.  You have to work really hard at the basics before you can make a perfect souffle.

Another way this analogy works: no one learns to cook by memorizing cookbooks.  You learn to cook by getting your hands dirty in the kitchen, trying out random recipes from the internet, and burning a few more complicated things that you weren’t ready for.  If you’ve never chopped vegetables with your dad in the kitchen as a kid, sure, you start at a disadvantage, but that doesn’t mean you can’t pick up a knife and try.  Use youtube videos, ask friends, cook with friends!  Now replace all the times I said “cook” in this paragraph with “math” and pretend that math is a verb.

Check out this cookbook! There are similar math books

Students (like me) often think we won’t cut it in grad school because we don’t have the experiences of others- didn’t do undergraduate research, take graduate courses while in undergraduate, maybe didn’t even major in math.  But just because you didn’t help your parent as a kid doesn’t mean you can’t cook now, and just because you didn’t focus on math before doesn’t mean you can’t do it now.  You learn to do math by doing math.

So that’s the first thing I tell calculus students, or at least this one that I was talking to last week.

Second thing: she asked me to explain the squeeze theorem to her.  Will do!  My explanation of it involves an old family curse.

So when my little brother was born, someone who was mad at my parents cursed our family.  Luckily they weren’t too mad, so it was a pretty benign curse: I would always be shorter than or the same height as my oldest brother, and I would always be taller than or the same height as my baby brother.  (Another way to say this: I’ll never be taller than my big brother, and my little brother will never be taller than me.  This affects our sibling basketball games, but that’s about as bad as the curse gets).

I think this is pretty good considering I googled ‘basketball’ and ‘basketball hoop’

We grow up, and we always grow according to the curse.  One day when we’re grown ups, someone sees my two brothers and realizes that they’re the same height.  Without even seeing me, they can answer: How tall am I?

… (this is you thinking)…

Yup, I’m the same height as those two!  This is the squeeze theorem, because my brothers’ heights has squeezed mine.

Replace our heights with functions: let’s say my brother’s names are Gerard and Hugo, and indicate their heights at time by g(x) and h(x), respectively.  And I’m f(x).  Since Hugo is always taller than or the same height as me, we have an inequality: $latex f(x) \leq h(x)$ for all time x.  Similarly, $latex g(x) \leq f(x)$.  Putting these two together, we have $g(x) \leq f(x) \leq h(x)$.

The squeeze theorem says that if for some where all three functions have a limit, $\displaystyle \lim_{x\to a} g(x) = \lim_{x\to a} h(x) = L$, then we have forced ourselves into $\lim_{x\to a} f(x) = L$, just as Gerard and Hugo’s heights forced mine to be the same as theirs.

Two things I tell calculus students!  I actually tell calculus students a lot of things (like calculus not using family curses), but these are the two things I told a calculus student last week.