# Applications of Differentiation – Mean-Value Theorem of Differentiation

Let’s say that you were going on a road trip from New York City to Los Angeles. Let’s say that your average driving speed was 50 miles per hour for the duration of the trip. However, that doesn’t mean that you drove at 50 miles per hour for the entire trip. At some points on the trip, you drove faster, slower, or exactly at the average speed. Therefore, it is safe to conclude that if you averaged 50 miles per hour during your road trip, then at some instant during the trip you were traveling exactly 50 miles per hour.

This is an application of the Mean Value Theorem, but before we look deeper into the Mean Value Theorem, we need to look at a simplified version – Rolle’s Theorem.

 Definition: Rolle’s Theorem If we are given any function F(x) that satisfies the following: The given function F(x) will be continuous on the closed interval [a,b] The given function F(x) is differentiable on the open interval (a,b) F(a) = F(b) Then there will be any number c such that a < c < b and Consider the figure below: For the curve F(x), x = a and x = b are on the same latitude, so their respective y values F(a) and F(b) are also the same. Therefore, the slope of the line between those two points is zero.

There is a x value c in which the slope of the tangent line is also zero.

Geometrically, the Mean Value Theorem is a “slanted” version of Rolle’s Theorem. In Rolle’s Theorem, F(a) and F(b) are at the same height, which means the secant slope at the end points is zero. However, in the Mean Value Theorem, F(a) and F(b) are at different heights, which means that the secant slope will not be zero.

 Definition: Mean Value Theorem (MVT) If we are given any function F(x) that satisfies the following: The given function F(x) will be continuous on the closed interval [a,b] The given function F(x) is differentiable on the open interval (a,b) Then there will be any number c such that a < c < b and You can also think of the Mean Value Theorem in the following ways:

• There is a point on the graph in which the slope of the tangent line through that point is the same as the average rate of change between 2 endpoints on the same graph.
• For a given curve between two endpoints, there is at least one point at which the tangent line to the curve is parallel to the secant line through its endpoints.
• The slope of the tangent line at point c is equal to the slope of the secant line between x = a and x = b (mtangent = msecant).

Below is a visual that demonstrates the Mean Value Theorem:  Example 1:

Determine all the numbers c which satisfy the conclusions of the Mean Value Theorem for the following function: F(x) = x2 + 2x – 3 on the interval of [-1, 2]?

Solution:

Firstly, we will find the first derivative of the function:

F(x) = x2 + 2x – 3

Using the power rule,

F’(x) = 2x + 2

Find the average rate of change on the interval of [a,b].

According to the problem, a = -1 and b = 2. Therefore,

F(a) = F(-1) = (-1)2 + 2(-1) – 3 = 1 – 2 – 3 = -4

F(b) = F(2) = (2)2 + 2(2) – 3 = 4 + 4 – 3 = 5 2x + 2 = 3

Solve for the variable to find the value of c.

Solving for x,

2x = 3 – 2

2x = 1

x = ½

x = c = ½ is the value that will satisfy the conclusion of the Mean-value Theorem.

Below is a graph to demonstrate our results. The blue line is the secant line on the endpoints of [-1,2]. The green line is the tangent line at c = 0.5. At x = c = 0.5, the instantaneous rate of change is equal to the average rate of change over the interval of [-1, 2].

Example 2: Determine all the numbers c which satisfy the conclusions of the Mean Value Theorem for the following function: F(x) = x3+3x2-2 on the interval of [-2, 0]?

Firstly, we will find the first derivative of the function.

F(x) = x3+3x2-2

Using the power rule,

F’(x) = 3x2+6x

Find the average rate of change on the interval of [a,b].

According to the problem, a = -2 and b = 0. Therefore,

F(a) = F(-2) = (-2)3+3(-2)2-2= 2

F(b) = F(0) = (2)3+3(2)2-2= -2 Therefore, the values of c that will satisfy the mean value theorem is x = -1.816 and -0.184

Below is a graph to demonstrate our results.

At c = -1.577 At c = -0.423 For both c values, the instantaneous rate of change was equal to the average rate of change over the interval of [-2, 0] (the slopes are parallel).

Example 3: Determine all the numbers c which satisfy the conclusions of the Mean Value Theorem for the following function: on the interval of [1, 4]?

Firstly, we will find the first derivative of the function. You can either use the quotient rule or break up the rational function into two separate ratios and differentiate each ratio.

We will use the quotient rule:

f = x2-9

f’ = 2x

g = 3x

g’ = 3 x = -2 or 2

Therefore, the values of c that will satisfy the Mean Value Theorem is x = -2 and x = 2.

However, we need to select the c value that falls on the interval of [1, 4]. Therefore, we will pick c = 2.

Below is a graph to demonstrate our results. For c = 2, the instantaneous rate of change was equal to the average rate of change over the interval of [1, 4] (the slopes are parallel).

Example 4: Two police officers who are equipped with a radar are three miles apart on a freeway. As the truck passes the first officer, he is clocked at 55 mph. 2 minutes later, as he passes the second officer, he is clocked at 52 mph. If the speed limit is 55 mph, should the truck driver receive a speeding ticket?

Solution: This shows that even though the truck was clocked at 55mph at the start of the 3 miles and 52mph at the end of the 3 miles, at some point during the three miles, he travelled at 90 mph.

In other words, during the 2 minutes, he sped up from 55 mph to 90 mph and slowed down to 52 mph along the 3 miles.

 Summary of Section Rolle’s Theorem states that any differentiable function that attains equal values at two different points must have at least one point in between where the first derivative is zero: The Mean Value Theorem, which is an extension of Rolle’s Theorem, states that for a given curve between two endpoints, there is at least one point at which the slope of the tangent line to the curve is parallel to the slope of the secant through its endpoints: 