Mean Value Theorem

Mean value theorem example: square root function | AP Calculus AB | Khan Academy
Mean value theorem example: square root function | AP Calculus AB | Khan Academy

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The Mean Value Theorem

The Mean Value Theorem is used for finding a tangent line to a curve that is parallel to the secant line connecting two endpoints of an interval. If f is continuous and differentiable on an interval from a to b, then there exists a number c in that interval such that

Here is an picture of this theorem to help illustrate it better:

As you can see, there has to be at least one point
along the function that has a tangent line with a slope equal to the
secant line through a and b, as long as the function is both continuous and differentiable along that section. Let’s do some examples.

Example 1

Given the following function, find all values along the interval [1, 5] such that the tangent line is parallel to the secant line through x = 1 and x = 5.

Now we know that f ‘ (c) = -1, and because this function satisfies the Mean Value Theorem for the given interval, we know that there is at least one point in the interval that has a slope of -1. Let’s find out which point it is:

We get an answer of positive and negative square root of 5; since the negative value is not in our interval, the answer is the square root of 5. Plugging this back into our original equation to find the point, we get:

Example 2

Given the following function, find all values along the interval [-1, 3] such that the tangent line is parallel to the secant line through x = -1 and x = 3.

Now we know that the f ‘ (c) that we are looking for is .5; let’s find out which x-value has this slope:

Now that we know the x-value of the coordinate is 1, let’s find the y-value to go with it:

To help illustrate what we are actually doing here, this is a graph of the problem we just worked; it has the original function, the secant line through a and b, and the tangent line through the point that we found:

Example 3 – Instantaneous Rate of Change

There are some practical applications to this as well. One such application is that of finding the instantaneous rate of change; let’s find out how to use the Mean Value Theorem for this here.

There are two stationary police radars 6 miles apart on a highway. A car passes the first radar at 50 miles per hour, and 5 minutes later passes the second radar at 50 miles per hour. Prove that the car must have exceeded the speed limit of 65 miles per hour at some point during the 5 minutes.

To do this, we need to use the times that the car passed by the radars as our endpoints. We can let a = 0, since the car passed the first radar at a time of t = 0. For the other endpoint, b, we can use 5 minutes, but need to convert it into hours (as the units are miles per hour):

So, letting s (t) represent the distance traveled by the car, we have s (0) = 0 (at time

t = 0, the car is at a position of 0 miles), and s (1/12) = 6 (at time t = 1/12, the car is at a position of 6 miles). Now for the average velocity of the car over this 6 mile section of highway, using the Mean Value Theorem:

Using the Mean Value Theorem, we proved that at some point along the 6 mile stretch of highway, the car must have been going 72 miles per hour, which is above the speed limit.

Hello! My name is Alex, and I am the creator of CopingWithCalculus.com. I’m not a genius or a math guru; in fact, I struggled with it for several years before becoming proficient. Eventually, I passed the AP Calculus exam in high school, and Calculus II and III in college; because I struggled with it, I understand how necessary clear, concise explanations are. That is why I created this site, and am working on a book to go along with it – to help you cope with calculus!
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