Mean value theorem proof for derivatives

[transgalactic]

http://ocw.mit.edu/OcwWeb/Mathematics/18-01Fall-2006/VideoLectures/index.htm

does it consists with theoretical knowledge of solving question like this:

there is a function f(x) which is continues in the borders [a,b]
and derivitable in the borders (a,b), b>a>0
alpha differs 0
prove the there is b>c>a

in that formula:
http://img392.imageshack.us/my.php?image=81208753je3.gif

i never encoutered this kind of questions
and in this MIT calculus course i searched their exams and there is no such question
there are only normal calculus material regarding derivatives ,limits,aproximations
but i can't see this sort of complicated proving questions
??

 

[HallsofIvy]

$$\frac{left|\begin{array}{cc}a^\alpha & b^\alpha \\ f(a) & f(b) \end{array}\right|}{a^\alpha- b^\alpha}= f(c)- c\frac{f'(c)}{\alpha}$$

$$\frac{a^\alpha f(b)- b^\alpha f(a)}{a^\alpha- b^\alpha}= \frac{a^\alpha f(b)- a^\alpha f(a)+ a^\alpha f(a)- b^\alpha f(a)}{a^\alpha- b^\alpha}$$
$$\frac{a^\alpha}{a^\alpha- b^\alpha}(f(b)- f(a))+ f(a)$$

That looks to me like a fairly straight forward variation of the "mean value theorem". I would have no doubt that any Calculus course would include a proof of the mean value theorem but I certainly would not guarantee any course would include that particular variation.

 

[transgalactic]

so these proof questions are about theorems

how much theorems are there in a single variable calculus course
where can i effectively learn them,understand them ?