Taylor Polynomials

The derivative of f(t) is tex2html_wrap_inline189 , thus tex2html_wrap_inline191 . Since tex2html_wrap_inline193 , we obtain as an equation for the tangent line at tex2html_wrap_inline185 :


Second degree Taylor polynomials

One way to see that the tangent line to a function f(x) at a given point tex2html_wrap_inline209 is the best line approximating the function is to observe that the tangent line is the (only) line passing through the point tex2html_wrap_inline211 and having the same slope as f(x) at tex2html_wrap_inline209 .

So what about about finding the "best'' parabola approximating the function f(x) near tex2html_wrap_inline209 ? We should look for the parabola passing through tex2html_wrap_inline211 , which has the same slope (the first derivative) as f(x) at tex2html_wrap_inline209 , and which has the same second derivative as f(x) at tex2html_wrap_inline209 !

Let's try it: Consider tex2html_wrap_inline231 near tex2html_wrap_inline233 . The parabola we are trying to find has the generic form:


Writing the parabola this way, it is easier to compute its derivatives at tex2html_wrap_inline233 : p'(x)=b +2 c (x-1) and p''(x)=2 c. Substituting tex2html_wrap_inline233 we obtain:


Recall, we want to find the parabola which has the same derivatives at tex2html_wrap_inline209 as f(x). This yields the conditions:


Now tex2html_wrap_inline247 ; tex2html_wrap_inline249 and tex2html_wrap_inline251 . 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 tex2html_wrap_inline231 at the point tex2html_wrap_inline233 .

The picture below shows f(x) in black and its second degree Taylor polynomial at tex2html_wrap_inline233 in red.

It is not hard to see what the general formula will look like: If we replace tex2html_wrap_inline233 by a "general'' tex2html_wrap_inline209 above, we obtain:


as the general form of the Taylor polynomial at tex2html_wrap_inline209 ; We need that


and consequently


tex2html_wrap_inline209 is called the center of the Taylor polynomial. Note: The center tex2html_wrap_inline209 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!).

Try it yourself!

Find the quadratic Taylor polynomial for the function tex2html_wrap_inline281 with the center tex2html_wrap_inline283 .

Click here for the answer, or to continue.

Helmut Knaust
Sun Jul 7 22:08:09 MDT 1996

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