Please help me understand Rudin Theorem 5.15 I am having trouble understanding the intuition behind the last part of this theorem. I'd appreciate some help understanding the intuition behind the last equation: $f(\beta )  = P (\beta ) + \frac{f^{(n)}(x)}{n!} (\beta - \alpha )^n $. Why are we concerned about the end point $ \beta $, and what is the (intuitive) relationship between $f(\beta ) $ and $ P(\beta )$? 
Here is the theorem typed out:
Suppose f is a real function on [a,b], n is a positive integer, $f^{n-1} $ is continuous on [a,b], $f^{(n)}(t)$ exists for every $t \in (a,b) $. Let $\alpha, \beta $ be distinct points of [a,b], and define 
$ P(t) = \sum^{n-1}_{k=0} \frac{f^{(k)}(\alpha ) }{k!} (t- \alpha )^k  $
Then there exists a point x between $\alpha $ and $\beta $ such that  
$f(\beta )  = P (\beta ) + \frac{f^{(n)}(x)}{n!} (\beta - \alpha )^n $.
 A: Well, you need to understand the idea behind the Taylor's theorem. Roughly it says that if we know the derivative $f^{(n)}$ at a point $a$ then we can use it to calculate the value of $f$ at a nearby point $b$. Normally to show the nearness we write $b = a + h$ and assume that $h$ is small. For example if $f'(a)$ exist then we write $f(a + h) \approx f(a) + hf'(a)$ and if $f''(a)$ exists then $f(a + h) \approx f(a) + hf'(a) + \dfrac{h^{2}}{2!}f''(a)$. The formal justification of such approximations (i.e. calculation of error term) is contained in the precise statement of Taylor's theorem.
In the Rudin's book we have $\alpha$ taking role of $a$ and $\beta$ taking role of $a + h$ so that $h = \beta - \alpha$. The theorem says that $P(\beta)$ is an approximation of $f(\beta)$ and the error term is $\dfrac{f^{(n)}(x)}{n!}(\beta - \alpha)^{n}$.
I hope you do understand the technical aspects of the proof of Taylor's series (basically it is based on Rolle's theorem) and you have issues in the intuitive understanding of it. Let me know if there is something bothering in the proof.
A: You "know" the function $f(x)$ very well at the point $\alpha$. You wonder what the value is at the point $\beta$, so you build the polynomial $P(x)$ with data from $f$ at the point $\alpha$. Then $P(\beta)$ is your guess of the unknown value $f(\beta)$. The last expression gives you a tool to say something about how good your guess is without having to compute the true value $f(\beta)$.
