Geometry of Functions III: The Mean-Value Theorem
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SAMPLE LESSON
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Our goal here is to describe an extremely important theorem in Calculus - the Mean-Value Theorem. It's importance cannot be overstated as it is used to prove many things in Calculus. The beauty of this theorem is that you can look at a picture and see that it is true. Before we get to the actual theorem we want to prove Rolle's Theorem which is just a special case of the Mean-Value Theorem.
| Rolle's Theorem |
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| If $f(x)$ is differentiable on the open interval $(a,b)$, continuous on the closed interval $[a,b]$, and <Sign in to see all the formulas>, then there is a least one number $e$ in $(a,b)$ such that <Sign in to see all the formulas>. |
While the proof of Rolle's Theorem may be found here, it is better to just look at a picture to see how true it is.
In the picture above note that the slope of the secant line between $a$ and $b$ is zero. The idea is that there is a point between $a$ and $b$ whose derivative is the slope of the secant line between $a$ and $b$. This is the Mean-Value Theorem. Let's state it formally and then show the picture.
| The Mean-Value Theorem |
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If $f(x)$ is differentiable on the open interval $(a,b)$ and continuous on the closed interval $[a,b]$, then there is a least one number $e$ in $(a,b)$ such that<Sign in to see all the formulas> |
In other words, there is a point in $(a,b)$ whose slope is the slope of the secant line between $a$ and $b$.
If you are not required to learn the proof of the Mean-Value Theorem then we recommend not learning it. Understand what the theorem means and keep the picture above in your head. However, if you still want or need to learn the proof then it may be found here.
