Introduction to Mean Value Theorem

The Mean Value Theorem is a fundamental calculus theorem based on the average rate of change of a function. It helps to calculate the average rate of change in a function by using the slope of the tangent line formula. But to apply this theorem, the function should be continuous on a closed interval and differentiable on an open interval. Let’s discuss how the mean value theorem is used to find the slope of the tangent line. 

Understanding of Mean Value Theorem

Mean Value Theorem is a rule defined for a continuous function, i.e., a function that does not undergo any unexpected change or discontinuity. This theorem is named after the mathematician Augustin Louis Cauchy. It is used to find a point where the first derivative is a function's average rate of change. By definition, the Mean Value Theorem states that:

“If f(x) is a continuous function on a closed interval [a, b] and f(a) = f(b), f(x) is differentiable on the open interval (a, b) then there exists at least one point c between the interval where the f’(c) is the average rate of change.” 

Mean Value Theorem Formula

The formula for Mean Value Theorem uses the concept of derivative because it discusses the average rate of change of a function on a specific point. If a function f(x) is differentiable, then by Mean Value Theorem, it must satisfy the following conditions.

• The function f(x) is continuous on the closed interval [a, b].
• The function f(x) is differentiable on the open interval (a, b).
• If f(a) is equal to f(b) then there exists a point c between the interval (a, b) where $f'(c) = \frac{f(b)-f(a)}{b-a}$

It means that the function should be continuous and differentiable to follow the Mean Value Theorem.

How to apply Mean Value Theorem?

Since it is easy to apply this theorem, there is a step-by-step way to apply this rule. These steps can assist you in finding the specific point which is required for a function to be continuous. These steps are:

1. Check whether the function is continuous on the closed interval [a, b]. You can only proceed to the next step if the function is continuous.
2. Calculate the values of f(a) and f(b).
3. Find the derivative of the function as f’(x).
4. Now find the average rate of change by using the formula $f'(c) = \frac{f(b)-f(a)}{b-a}$.
5. After calculating the value of f'(c), use it to find c. 

Let’s understand the application of the Mean Value Theorem in the following example.

Mean Value Theorem Example 1

Verify the Mean Value Theorem for the function $y = x^2+ 8x, a = –1$ and $b = 1$.

To verify the Mean Value Theorem, the function should satisfy the three conditions. For this, we need to calculate f’(x), f(a) and f(b). The function is written as;

$y = x^2+ 8x$

Since the function is a polynomial function, we can use the power rule of derivatives.

$f'(x)=\frac{d}{dx}[x^2+8x]$

$f'(x) = 2x + 8$

Now, 

$f(-1) = f(a) = (-1)^2 +8(-1) = 1-8$

$f(-1) = -7$

And, 

$f(1) = f(b) = 1^2+8(1) = 9$

Now to find c, we use mean value theorem that is,

$f'(c) = \frac{f(b)-f(a)}{b-a}$

$f'(c) = \frac{9-(-7)}{1-(-1)}$

$f'(c) =\frac{9 + 7}{1+1}=\frac{16}{2}=8$

Therefore, f’(c) = 8 and c belongs to (-1, 1). Hence the Mean Value Theorem is satisfied.

Mean Value Theorem Example 2

Calculate the Mean Value Theorem for the function, $f(x) = x^2-4x$ on the interval where a =1 and b =4.

To verify the Mean Value Theorem, the function should satisfy the three conditions. For this, we need to calculate f’(x), f(a) and f(b). The function is written as;

$y = x^2 – 4x$

Since the function is a polynomial function, we can use the power rule of derivatives.

$f‘(x)=\frac{d}{dx}[x^2-4x]$

$f’(x) = 2x – 4 $

Now, 

$f(1) = f(a) = (1)^2 – 4(1) = 1 – 4$

$f(1) = - 3$

And, 

$f(4) = f(b) = 4^2-4(4) = 16-16 = 0$

Now to calculate c, we use mean value theorem that is,

$f'(c) = \frac{f(b)-f(a)}{b-a}$

$f'(c) = \frac{0-(-3)}{4-1}$

$f'(c) = \frac{-3}{3} = -1$

Therefore, $f’(c) = -1$ and c belongs to (1, 4) where f’(c) is the average rate of change of $f(x) = x^2 – 4x$. Hence the Mean Value Theorem is satisfied.

Geometric Interpretation of Mean Value Theorem

If a function f(x) satisfies all three conditions of the Mean Value Theorem then the graphical or geometric representation can be visualised by following some simple steps. 

1. Draw a line segment that connects both points of the interval [a, b] and their corresponding functions values such as f(a) and f(b).
2. Draw a secant line by joining interval points with the corresponding values of f(x) such as f(a) and f(b). This line will not be horizontal because a>b and it will represent the average rate of change in f(x).
3. Now find the slope of the line by using the mean value formula f(b) – f(a) / b-a.
4. Draw a tangent line to the function at the point between the interval (a, b). 
5. Now calculate the average rate of change in f(x) by using derivative. It will be equal to the slope of secant line. 
6. Now compare the tangent line and secant line. Both will be parallel to each other means that the mean value theorem is verified. 

After using the above steps, the geometric representation will be obtained as,

Comparison between Mean Value Theorem and Mean Value Theorem

The comparison between the Mean Value Theorem and Mean Value Theorem can be easily analysed using the following difference table.

Conclusion

Rolle's Theorem is a fundamental theorem in calculus that allows us to discuss the average rate of change of a function over a closed interval. A function satisfies the Mean value Theorem if it satisfies the three conditions. If one of these conditions remains unsatisfied, the function does not verify the Mean Value Theorem.