Mean Value Theorem Explanation
In the graph, the tangent line at c (derivative at c) is equal to the slope of [a,b] where a <>.
The Mean Value Theorem is an extension of the Intermediate Value Theorem, stating that between the continuous interval [a,b], there must exist a point c where the tangent at f(c) is equal to the slope of the interval.
This theorem is beneficial for finding the average of change over a given interval. For instance, if a person runs 6 miles in an hour, their average speed is 6 miles per hour. This means that they could have kept that speed the whole time, or they could have slowed down and then sped up (or vice versa) to get that average speed. This theorem tells us that the person was running at 6 miles per hour at least once during the run.
- i) Find (a,f(a)) and (b,f(b))
- ii) Use the Mean Value Theorem
- iii) Find f'(c) of the original function
- iv) Set it equal to the Mean Value Theorem and solve for c.
If we want to find the value of c, we
Mean Value Theorem Examples
Let's do the example from above.
(1) Consider the function f(x) = (x-4)2-1 from [3,6].
First, let's find our y values for A and B.
Now let's use the Mean Value Theorem to find our derivative at some point c.
This tells us that the derivative at c is 1. This is also the average slope from a to b. Now that we know f'(c) and the slope, we can find the coordinates for c. Let's plug c into the derivative of the original equation and set it equal to the result of the Mean Value Theorem.
We have our x value for c, now let's plug it into the original equation.
Let's do another example.
(2) Consider the function f(x) = 1⁄x from [-1,1]
Using the Mean Value Theorem, we get
We also have the derivative of the original function of c
Setting it equal to our Mean Value result and solving for c, we get
c is imaginary! What does this mean? The function f(x) is not continuous over the interval [-1,1], and therefore it is not differentiable over the interval. For the Mean Value Theorem to work, the function must be continous.
Rolle's Theorem is a special case of the Mean Value Theorem. It is stating the same thing, but with the condition that f(a) = f(b). If this is the case, there is a point c in the interval [a,b] where f'(c) = 0.
(3) How many roots does f(x) = x5 +12x -6 have?
We can use Rolle's Theorem to find out. First we need to see if the function crosses the x axis, i.e. if at some point it switches from negative to positive or vice versa.
We can see that as x gets really big, the function approaces infinity, and as x approaches negative infinity, the function also approaches negative infinity.
This means that the function must cross the x axis at least once.
If the function has more than one root, we know by Rolle's Theorem that the derivative of the function between the two roots must be 0.
This is not true. The only way for f'(c) to equal 0 is if c is imaginary. f'(c) is always positive, which means it only has one root.