N阶近似度

The constant, linear and quadratic approximations to a function $f$ about a point $a$ do pretty well, if we stay close to the value of $a$. However, sometimes we may want to go farther - and, by the title of this section and promises in past sections, one might guess that we could.

As mentioned in the section about quadratic approximation, we can find Taylor polynomials $g_0, g_1, ... g_n ...$ that approximate a function $f$ about a point $a$ by matching the first few derivatives at $a$. In previous sections, you found how to match up the first two derivatives of a function $f$ for the polynomial $g_2$.

We'll do one more approximation, then we'll try to give the general formula. For $g_3$, we want: $$\begin{array}[lll]

g_3(a) &=& f(a) \

g_3'(a) &=& f'(a) \ g_3''(a) &=& f''(a) \ g_3'''(a) &=& f'''(a) \end{array}$$

Right! So, as you've noticed, the number in the denominator grows dramatically quickly - as fast as $n!$, in fact. This leads us to the general formula for $g_n$ given as follows: $$g_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x - a)^k$$ where $f^{(k)}(a)$ is the $k$-th derivative of $f$ at $a$, so $f^{(0)}(a) = f(a)$, $f^{(1)}(a) = f'(a)$, etc.

We call the $g_n$s the Taylor polynomials for $f$ at $a$. It turns out that they become dramatically better approximations the further along you go!

Let's have a look at some approximations, with $f = \sin$ (in red) and $a = 0$. First, we note that $g_0$ (in blue) is not a very good approximation. But $g_1$ (in green) stays close a little longer. Since $\sin''(0) = 0$, $g_2 = g_1$. But $g_3$ (in orange) is different, and stays longer still!

One might wonder if this approximation process can be carried out forever, or if these approximations eventually stop getting better. In the next few sections, we'll consider how to analyze the error of these polynomials. Afterwards, in the section on Taylor series, we'll consider what it means to carry this process out "forever" - obtaining what turns out to be a perfect approximation, for many functions.