Sanity Checks are missives on a specific math point in need of clarification. I try to do so using the fewest words possible. Usually, this is still quite a few words.
Behold. I give you a metaphor for Taylor series approximation.
Either LabKitty has gone mad, or this is a demonstration of profound mathematical insight. Or both. Could be both. That is surely a possibility, as surely as there will be slow motion footage of the bikini barrel race.
The point is a Taylor series approximation is a very strange approximation. Most people never give this much thought. Rather, you just wallow in stupid until one day, many years later (when you have a blog, say), the light bulb goes off and you are tasked with spreading the news.
When we say a Taylor series approximates a function, and adding more terms results in a better approximation, we do not mean something like the following:
Rather, we mean the approximation "hangs in there" longer (i.e., for x farther and farther from the point of expansion). For example, here are one-, three-, and six-term Taylor series approximations for cos(x) expanded about x = 0:
If here you are thinking pfft, that's obvious you stupid cat, um, no it isn't. (And, I might add, you are quite the little anger bear. What's up with that?) Especially if you've learned about curve fitting elsewhere -- say, in a statistics course (which many students take before calculus). The first graph series shown above is precisely what is meant when you speak of a good fit versus a poor fit in the least-squares sense. A curve that fits your data exactly at one (or a few) data points is useless. Yet, that is what a Taylor series approximation provides.
Footnote: A Taylor series might not converge at all. Or, it might converge to the wrong function. You work very hard to stay on the bull, but instead you bake a cake. These are separate issues, and ones that shall not concern us here as I could not find a snarky video metaphor for them.
Take home message: A Taylor series approximation -- assuming it converges at all -- starts off accurate at the expansion point and the harder you work the longer it stays accurate. Hence the rodeo metaphor: The rider's goal is to say on the bull as long as possible.
The calculus counterpart of the rodeo clown I shall leave to the reader.
Behold. I give you a metaphor for Taylor series approximation.
Either LabKitty has gone mad, or this is a demonstration of profound mathematical insight. Or both. Could be both. That is surely a possibility, as surely as there will be slow motion footage of the bikini barrel race.
The point is a Taylor series approximation is a very strange approximation. Most people never give this much thought. Rather, you just wallow in stupid until one day, many years later (when you have a blog, say), the light bulb goes off and you are tasked with spreading the news.
When we say a Taylor series approximates a function, and adding more terms results in a better approximation, we do not mean something like the following:
Rather, we mean the approximation "hangs in there" longer (i.e., for x farther and farther from the point of expansion). For example, here are one-, three-, and six-term Taylor series approximations for cos(x) expanded about x = 0:
If here you are thinking pfft, that's obvious you stupid cat, um, no it isn't. (And, I might add, you are quite the little anger bear. What's up with that?) Especially if you've learned about curve fitting elsewhere -- say, in a statistics course (which many students take before calculus). The first graph series shown above is precisely what is meant when you speak of a good fit versus a poor fit in the least-squares sense. A curve that fits your data exactly at one (or a few) data points is useless. Yet, that is what a Taylor series approximation provides.
Footnote: A Taylor series might not converge at all. Or, it might converge to the wrong function. You work very hard to stay on the bull, but instead you bake a cake. These are separate issues, and ones that shall not concern us here as I could not find a snarky video metaphor for them.
Take home message: A Taylor series approximation -- assuming it converges at all -- starts off accurate at the expansion point and the harder you work the longer it stays accurate. Hence the rodeo metaphor: The rider's goal is to say on the bull as long as possible.
The calculus counterpart of the rodeo clown I shall leave to the reader.
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