PPT: Mean Value Theorem | Engineering Mathematics - Civil Engineering (CE) PDF Download

 Page 1


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