The derivative of *f*(*t*) is , thus .
Since , we obtain as an equation for the tangent line at :

One way to see that the tangent line to a function *f*(*x*) at
a given point is the best line approximating the function is to observe that the tangent line is the (only) line
passing through the point and having the same slope as *f*(*x*) at .

So what about about finding the "best'' parabola approximating the function *f*(*x*) near ? We should look for the parabola passing through , which has the same slope (the first derivative) as *f*(*x*) at , **and** which has the same second derivative as *f*(*x*) at !

Let's try it: Consider near . The parabola we are trying to find has the generic form:

Writing the parabola this way, it is easier to compute its derivatives at : *p*'(*x*)=*b* +2 *c* (*x*-1) and *p*''(*x*)=2 *c*. Substituting we obtain:

Recall, we want to find the parabola which has the same derivatives at as *f*(*x*). This yields the conditions:

Now ; and .
Solving for the coefficients and substituting in the formula for *p*(*x*), we obtain

The polynomial *p*(*x*) is called the **second degree Taylor polynomial**
of the function at the point .

The picture below shows *f*(*x*) in black and its second degree Taylor polynomial at in red.

It is not hard to see what the general formula will look like: If we replace by a "general'' above, we obtain:

as the general form of the Taylor polynomial at ; We need that

and consequently

is called the **center** of the Taylor polynomial. Note: The center is fixed, the variable name for the polynomial is *x*. Even if we consider the same function *f*(*x*),
different centers will usually yield different Taylor polynomials (just as a function usually has different tangent lines at various points!).

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Sun Jul 7 22:08:09 MDT 1996

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