Another introductory calc problem

[Fifty]

Given a function f(x), I'm wondering if the slope of the secant between points a, f(a) and b, f(b) is equal to the average of the derivative at a and b.

Mathematically, I want to know whether:

\frac{f(a) - f(b)}{a - b} = \frac{f'(a) + f'(b)}{2}

is true. I can see that it is for polynomial functions, but I can't seem to prove this relationship for a general function, f(x).

This isn't homework, I'm just curious. Thanks for the help!

 

[Chestermiller]

Have you learned about Taylor series expansions yet?

Chet

 

[Simon Bridge]

You could try to come up with a counter-example.
Also think about what "average" means.

If you have a proof for all polynomials, then can you apply the proof to any arbitrary sum of polynomials?
Can you write an arbitrary function as a sum of polynomials?
[Chester beat me to it]

Notes:
http://homepage.math.uiowa.edu/~idarcy/COURSES/25/2_7.pdf

 

[Fifty]

Have you learned about Taylor series expansions yet?

Chet

Nope. I was literally just introduced to calculus four weeks ago.

EDIT: It's late here, I'm going to have a look at this thread tomorrow (and the comments already posted).

 

[Simon Bridge]

In a nutshell ... any function f(x) can be written as a sum like this:
$$f(x)=a_0+a_1x+a_2x^2+\cdots $$

If you already have a proof that applies to polynomials - you can apply it to that.

 

[Chestermiller]

Nope. I was literally just introduced to calculus four weeks ago.

Your relation is not exact. It is a good approximation, but the difference between the two sides of the relationship increases with increasing distance between the two points a and b.

Chet

 

[Chestermiller]

f(a)=f(b)-(b-a)f'(b)+\frac{(b-a)2}{2}f''(b) + ...
f(b)=f(a)+(b-a)f'(a)+\frac{(b-a)2}{2}f''(a) + ...
Subtracting:
2(f(b)-f(a))=(b-a)(f'(a)+f'(b))-\frac{(b-a)2}{2}(f''(b)-f''(a))+...
Or,
\frac{f(b)-f(a)}{b-a}=\frac{f'(a)+f'(b)}{2}-\frac{(b-a)}{4}(f''(b)-f''(a))+...
A better approximation is:
\frac{f(b)-f(a)}{b-a}≈f'\left(\frac{a+b}{2}\right)
This relation is exact if f is quadratic.

Chet