Mean Value Theorem Example Problems

Understanding and Applying the Mean Value Theorem: Example Problems

The Mean Value Theorem (MVT) is a cornerstone of calculus, bridging the gap between the instantaneous rate of change (derivative) and the average rate of change over an interval. Understanding and applying the MVT requires a solid grasp of derivatives, continuity, and problem-solving strategies. This article will delve into the Mean Value Theorem, providing a comprehensive explanation and working through various example problems to solidify your understanding. We'll explore both simple and more complex applications, demonstrating the theorem's versatility and importance in mathematical analysis.

Introduction to the Mean Value Theorem

The Mean Value Theorem states: If a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number c in the interval (a, b) such that:

f'(c) = (f(b) - f(a)) / (b - a)

This equation essentially says that the instantaneous rate of change at some point c within the interval is equal to the average rate of change over the entire interval. Geometrically, this means there's at least one point on the curve where the tangent line is parallel to the secant line connecting the points (a, f(a)) and (b, f(b)).

The conditions of continuity and differentiability are crucial. If a function fails to meet either condition, the MVT may not apply. Let's explore this further through examples.

Example Problems: Simple Applications

Problem 1: Finding the Value of 'c'

Let's consider the function f(x) = x² + 2x on the interval [1, 3]. Verify if the Mean Value Theorem applies and find the value(s) of c that satisfy the theorem.

Solution:

1. Continuity and Differentiability: f(x) = x² + 2x is a polynomial, which is continuous and differentiable everywhere. Therefore, the MVT applies on the interval [1, 3].

2. Finding f'(x): The derivative is f'(x) = 2x + 2.

3. Applying the MVT: We need to find c such that:

   f'(c) = (f(3) - f(1)) / (3 - 1)

   2c + 2 = ( (3² + 23) - (1² + 21) ) / 2

   2c + 2 = (15 - 3) / 2

   2c + 2 = 6

   2c = 4

   c = 2

Therefore, the value of c that satisfies the Mean Value Theorem is 2, which falls within the interval (1, 3).

Problem 2: Illustrating the Geometric Interpretation

Consider the function f(x) = √x on the interval [1, 4]. Find the value of c and interpret the result geometrically.

Solution:

1. Continuity and Differentiability: f(x) = √x is continuous on [1, 4] and differentiable on (1, 4) (except at x=0, which is outside our interval).

2. Finding f'(x): f'(x) = 1 / (2√x)

3. Applying the MVT:

   f'(c) = (f(4) - f(1)) / (4 - 1)

   1 / (2√c) = (2 - 1) / 3

   1 / (2√c) = 1/3

   2√c = 3

   √c = 3/2

   c = 9/4 = 2.25

Geometrically, this means that the tangent line to the curve y = √x at x = 2.25 is parallel to the secant line connecting the points (1, 1) and (4, 2).

Example Problems: More Complex Applications

Problem 3: Rolle's Theorem as a Special Case

Rolle's Theorem is a special case of the Mean Value Theorem where f(a) = f(b). It states that if f(x) is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists at least one c in (a, b) such that f'(c) = 0.

Let's consider f(x) = x³ - 3x on the interval [-1, 1]. Verify Rolle's Theorem.

Solution:

1. Continuity and Differentiability: f(x) is a polynomial, thus continuous and differentiable everywhere.

2. f(a) = f(b): f(-1) = (-1)³ - 3(-1) = 2 and f(1) = 1³ - 3(1) = -2. Note that f(-1) ≠ f(1), so Rolle's Theorem does not directly apply. However, let's consider the interval [0,1] as an alternative.

3. Applying the modified interval: f(0) = 0 and f(1) = -2. Rolle's Theorem still does not apply since f(0) ≠ f(1).

4. Let's consider a different function: Let's analyze the function f(x) = x³ - 4x. f(0) = 0 and f(2) = 0. On the interval [0,2], f(x) is continuous and differentiable. Therefore, Rolle's theorem applies.

5. Finding f'(x): f'(x) = 3x² - 4

6. Finding c: Set f'(c) = 0:

   3c² - 4 = 0

   c² = 4/3

   c = ±2/√3

Both values of c lie within the interval (0, 2). This demonstrates the existence of at least one point where the tangent line is horizontal.

Problem 4: Applications in Physics

The Mean Value Theorem can be applied to problems in physics, such as calculating average velocity.

Imagine a car accelerating along a straight road. Its position at time t is given by s(t) = t² + 3t meters. What is the car's average velocity between t=1 and t=4 seconds? Find the specific time when the instantaneous velocity equals the average velocity.

Solution:

1. Average Velocity: The average velocity is the change in position divided by the change in time:

   Average velocity = (s(4) - s(1)) / (4 - 1) = (28 - 4) / 3 = 8 m/s

2. Instantaneous Velocity: The instantaneous velocity is the derivative of the position function:

   v(t) = s'(t) = 2t + 3

3. Applying the MVT: We need to find c such that v(c) equals the average velocity:

   2c + 3 = 8

   2c = 5

   c = 2.5 seconds

This means that at t = 2.5 seconds, the car's instantaneous velocity is equal to its average velocity over the interval [1, 4].

Problem 5: Proving Inequalities using MVT

The Mean Value Theorem can be used to prove inequalities. For instance, let's prove that ln(1 + x) < x for x > 0.

Solution:

Let f(x) = ln(1 + x). We want to show that ln(1 + x) < x for x > 0.

1. Applying the MVT: Consider the interval [0, x]. The MVT states there exists a c in (0, x) such that:

   f'(c) = (f(x) - f(0)) / (x - 0)

   1/(1 + c) = (ln(1 + x) - ln(1)) / x

   1/(1 + c) = ln(1 + x) / x

2. Since 0 < c < x, we have 1 < 1 + c, so 1/(1 + c) < 1. Therefore:

   ln(1 + x) / x < 1

   ln(1 + x) < x

This proves the inequality.

Frequently Asked Questions (FAQ)

Q1: What happens if the function is not continuous or differentiable on the given interval?

A1: The Mean Value Theorem does not guarantee the existence of c if the function is not continuous on the closed interval [a, b] or not differentiable on the open interval (a, b). You cannot apply the theorem in such cases.

Q2: Can there be more than one value of 'c' that satisfies the MVT?

A2: Yes, there can be more than one value of c that satisfies the theorem. The theorem only guarantees the existence of at least one such value.

Q3: What is the relationship between the Mean Value Theorem and Rolle's Theorem?

A3: Rolle's Theorem is a special case of the Mean Value Theorem where f(a) = f(b). In essence, Rolle's Theorem deals with situations where the average rate of change is zero.

Conclusion

The Mean Value Theorem is a powerful tool in calculus with wide-ranging applications. It connects the average rate of change with the instantaneous rate of change, providing crucial insights into the behavior of functions. By understanding its conditions and applying it systematically, you can solve a variety of problems, from finding specific points on a curve to proving inequalities and modeling real-world phenomena. The example problems presented here demonstrate the theorem's versatility and importance in mathematical analysis, encouraging further exploration and deeper understanding. Remember to always check the continuity and differentiability conditions before applying the theorem to ensure its validity. Mastering the Mean Value Theorem will significantly enhance your understanding and problem-solving skills in calculus.