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2
The Mean Value Theorem
We will see that many of the results depend on one central 
fact, which is called the Mean Value Theorem. But to arrive 
at the Mean Value Theorem we first need the following 
result.
Before giving the proof let’s take a look at the graphs of 
some typical functions that satisfy the three hypotheses.
Page 2


2
The Mean Value Theorem
We will see that many of the results depend on one central 
fact, which is called the Mean Value Theorem. But to arrive 
at the Mean Value Theorem we first need the following 
result.
Before giving the proof let’s take a look at the graphs of 
some typical functions that satisfy the three hypotheses.
3
The Mean Value Theorem
Figure 1 shows the graphs of four such functions.
Figure 1
(c)
(b)
(d)
(a)
Page 3


2
The Mean Value Theorem
We will see that many of the results depend on one central 
fact, which is called the Mean Value Theorem. But to arrive 
at the Mean Value Theorem we first need the following 
result.
Before giving the proof let’s take a look at the graphs of 
some typical functions that satisfy the three hypotheses.
3
The Mean Value Theorem
Figure 1 shows the graphs of four such functions.
Figure 1
(c)
(b)
(d)
(a)
4
The Mean Value Theorem
In each case it appears that there is at least one point          
(c, f (c)) on the graph where the tangent is horizontal and 
therefore f ' (c) = 0. 
Thus Rolle’s Theorem is plausible.
Page 4


2
The Mean Value Theorem
We will see that many of the results depend on one central 
fact, which is called the Mean Value Theorem. But to arrive 
at the Mean Value Theorem we first need the following 
result.
Before giving the proof let’s take a look at the graphs of 
some typical functions that satisfy the three hypotheses.
3
The Mean Value Theorem
Figure 1 shows the graphs of four such functions.
Figure 1
(c)
(b)
(d)
(a)
4
The Mean Value Theorem
In each case it appears that there is at least one point          
(c, f (c)) on the graph where the tangent is horizontal and 
therefore f ' (c) = 0. 
Thus Rolle’s Theorem is plausible.
5
Example 2 
Prove that the equation x
3
 + x – 1 = 0 has exactly one real 
root.
Solution:
First we use the Intermediate Value Theorem to show that 
a root exists. Let f (x) = x
3
 + x – 1. Then f (0) = –1 < 0 and 
f (1) = 1 > 0. 
Since f is a polynomial, it is continuous, so the Intermediate 
Value Theorem states that there is a number c between 0 
and 1 such that f (c) = 0.
Thus the given equation has a root.
Page 5


2
The Mean Value Theorem
We will see that many of the results depend on one central 
fact, which is called the Mean Value Theorem. But to arrive 
at the Mean Value Theorem we first need the following 
result.
Before giving the proof let’s take a look at the graphs of 
some typical functions that satisfy the three hypotheses.
3
The Mean Value Theorem
Figure 1 shows the graphs of four such functions.
Figure 1
(c)
(b)
(d)
(a)
4
The Mean Value Theorem
In each case it appears that there is at least one point          
(c, f (c)) on the graph where the tangent is horizontal and 
therefore f ' (c) = 0. 
Thus Rolle’s Theorem is plausible.
5
Example 2 
Prove that the equation x
3
 + x – 1 = 0 has exactly one real 
root.
Solution:
First we use the Intermediate Value Theorem to show that 
a root exists. Let f (x) = x
3
 + x – 1. Then f (0) = –1 < 0 and 
f (1) = 1 > 0. 
Since f is a polynomial, it is continuous, so the Intermediate 
Value Theorem states that there is a number c between 0 
and 1 such that f (c) = 0.
Thus the given equation has a root.
6
Example 2 – Solution
To show that the equation has no other real root, we use 
Rolle’s Theorem and argue by contradiction. 
Suppose that it had two roots a and b. Then f (a) = 0 = f (b) 
and, since f is a polynomial, it is differentiable on (a, b) and 
continuous on [a, b]. 
Thus, by Rolle’s Theorem, there is a number c between a 
and b such that f '(c) = 0. 
cont’d
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FAQs on PPT: Mean Value Theorem - Engineering Mathematics - Civil Engineering (CE)

1. What is the Mean Value Theorem?
Ans. The Mean Value Theorem states that if a function is continuous on a closed interval [a, b] and differentiable on an open interval (a, b), then there exists at least one point c in the open interval (a, b) where the instantaneous rate of change of the function is equal to the average rate of change of the function over the closed interval [a, b].
2. How is the Mean Value Theorem used in calculus?
Ans. The Mean Value Theorem is often used in calculus to prove the existence of certain values or to help solve problems related to finding extrema or points of inflection. It provides a powerful tool for analyzing the behavior of functions and determining important properties such as the existence of critical points or intervals where the function is increasing or decreasing.
3. Can the Mean Value Theorem be applied to all functions?
Ans. No, the Mean Value Theorem can only be applied to functions that satisfy the necessary conditions, namely being continuous on a closed interval and differentiable on an open interval. If these conditions are not met, then the Mean Value Theorem cannot be applied.
4. How does the Mean Value Theorem relate to the concept of derivatives?
Ans. The Mean Value Theorem is closely related to the concept of derivatives. The theorem states that if a function is differentiable on an open interval, then there exists a point within that interval where the derivative of the function is equal to the average rate of change of the function over a larger interval. In other words, it connects the instantaneous rate of change (represented by the derivative) to the average rate of change over a specific interval.
5. Can the Mean Value Theorem be used to find the exact value of a function at a specific point?
Ans. No, the Mean Value Theorem does not provide the exact value of a function at a specific point. It only guarantees the existence of a point within an interval where the instantaneous rate of change is equal to the average rate of change. To find the exact value of a function at a specific point, additional information or techniques, such as evaluating the function directly or using other calculus methods, may be required.
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