Understanding the Mean Value Theorem in Calculus

[courtrigrad]

Why is it that if you have $$\frac{f(x_1) - f(x_2)}{x_1-x_2} = f'(\xi)$$ then $$\xi = x_1 + \theta(x_2-x_1)$$ where $$0<\theta<1$$?

Thanks

 

[learningphysics]

What does the mean value theorem say about what values $$\xi$$ can take?

 

[courtrigrad]

It says that $$x<\xi<x+h$$

 

[vincentchan]

in you problem $$x_{1} = x$$ and $$x_{2} - x_{1} = h$$
therefore, $$\xi = x_1 + \theta(x_2-x_1)= x + \theta h$$ where $$0<\theta<1$$
does it make sense now?

 

[courtrigrad]

yep. thanks a lot