Pattern-matching and hand-holding

Please don't make me think!

You can look in almost any calculus book and find the following statement in the section on power series and polynomial approximations of functions:

If a function f is differentiable through order n in an open interval containing c, the nth-degree Taylor polynomial of f at x = c is given by

f(c) + f′(c)(x − c) + (1/2)f″(c)(x − c)² + ... + (1/n!)f⁽ⁿ⁾(c)(x − c)ⁿ.

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That seems straightforward enough, right?

But you are asking for major trouble if you ask a student to find the third-degree Taylor polynomial for f(x) = sin x at x = 0. You know why? The student will say, “I can't do this problem. You didn't tell me the value of c.”

Look again at the definition of a Taylor polynomial. It contains the statement “x = c.” In stating my problem, I asked for the Taylor polynomial at “x = 0.” So why can't the student discern the exceedingly subtle and mysterious fact that c has been replaced with 0 (and that they should do the same thing in computing the polynomial)?

You'll find the answer if you go to the exercise set in your textbook (doesn't matter which one; they're practically all the same). The Taylor problems are all written in a special way. Let me restate my problem in book-speak:

Find the third-degree Taylor polynomial for f(x) = sin x at c = 0.

See the subtle difference? You're supposed to rub their noses in the value of c. Saying that “x = 0” in lieu of expressly stating that “c = 0” is not enough. If I daresay that I want the polynomial at “x = 0,” I will get travesties that start out with

f(c) + f′(c)(0 − c) + (1/2)f″(c)(0 − c)² + ... + (1/n!)f⁽ⁿ⁾(c)(0 − c)ⁿ,

followed by a plaintive request for the secret value of c. (And indeed I have.)

I've taken several runs at this with different groups of calculus students, but it's no good. Their pattern-matching is extremely rudimentary and not equal to the task of seeing that the c in the statement of the definition of Taylor polynomial corresponds to the zero in my request for the Taylor polynomial for the sine function (or whatever other function and initial value I choose).

As far as I can tell, the calculus books of today differ only in whether they choose to define Taylor polynomials and series in terms of x = c or x = a. Their problem sets are uniformly explicit about the value of c (or a). The same thing is true, for that matter, in the calculus books of yesterday, including early editions of Thomas that go back into the sixties. Since I don't have a copy of one of the Thomas editions from the fifties, I dug out my 1954 third edition of Sherwood & Taylor (who, regrettably, is not the Taylor of Taylor polynomials). They finesse the entire matter by asking for Taylor expansions “in powers of x + 1,” which implies that c = −1, or “in powers of x − 2,” which implies that c = 2.

Hmm. There's an idea.