Theorem.

Question

Theorem.

anonymous · Answer

Am I right if I say:

Given f(x) over the interval [a,b]

1)  Find the slope of the secant.    (The average slope between the two points (a,f(a)) and (b,f(b))      )

2)  Find f '(x)

3)   Set  f'(x)  equal to this slope.

4)   These solutions that we get for step 3, these are the points on f(x)  (provided that they are in the [a,b]  )  that have the same slope, as the slope of secant.

anonymous · Answer

@jdoe0001 @dan815 @sourwing

anonymous · Answer

I can give an example to show what I mean, if necessary.

anonymous · Answer

The mean value theorem states that such an $x \in [a, b]$ exists given that the function is continuous on $[a, b]$, and differentiable on $(a, b)$. What exactly do you mean by the above.

anonymous · Answer

Usually we don't care about the solution set itself. In any proof usually you just use the mean value theorem to establish the existence of such an $x$ and do not care about the set of all such $x$.

anonymous · Answer

I mean that in order to find all of the tangent lines       (to the f(x) that is over [a,b] )              to the curve and have a same slope as the slope of the secant, I should:

1)  Find the slope of the secant
2)  Find f '(x)
3)  Set f'(x) = slope of the secant  (whatever it will be when I find it)
4)  solve for x, in step 3.
5)  After finding the x values at which the tangent to the curve lines would have the same slope as slope of the secant, I would use the point slope formula, to find the lines themselves that are tangent to the curve and have the slope equivalent to the slope of the secant.

anonymous · Answer

If you really want to, but why? If the function is linear then the set of solutions is infinite and equal to the domain of the linear function.

anonymous · Answer

I am saying the steps for finding all the point on the f(x)   (if it is a curve, not a line)  at which the slope of the tangent, would equal to the curve's average slope.

anonymous · Answer

$f'(x) = \frac{f(b) - f(a)}{b - a}$

Yes you may simply solve, but you are not using the mean value theorem you are obtaining an explicit solution. The mean value theorem lets you avoid obtaining an explicit solution to the above problem. It states that a solution DOES exist (given that the function meets the requirements of the theorem).

So just know that you are no longer using the mean value theorem, you are simply solving the above equation.

anonymous · Answer

I see

anonymous · Answer

tnx