Applying the Mean Value Theorem to Motion and Contextual Problems

Introduction

Key Concepts

Understanding the Mean Value Theorem

The Mean Value Theorem states that for a function $f$ that is continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$, there exists at least one point $c$ in $(a, b)$ where the instantaneous rate of change equals the average rate of change over $[a, b]$. Mathematically, this is expressed as: $$ f'(c) = \frac{f(b) - f(a)}{b - a} $$ This theorem guarantees the existence of such a point $c$, but it does not specify its exact location.

Conditions for Applying MVT

To apply the Mean Value Theorem, the function must satisfy two primary conditions:

• Continuity: The function must be continuous on the closed interval $[a, b]$. Continuity ensures there are no breaks, jumps, or asymptotes within the interval.
• Differentiability: The function must be differentiable on the open interval $(a, b)$. Differentiability implies that the function has a defined tangent (slope) at every point within the interval.

If either condition fails, the Mean Value Theorem cannot be applied.

Applications in Motion Problems

In motion problems, the Mean Value Theorem can relate average velocity to instantaneous velocity. Consider a car that travels from point A to point B over a time interval $[t_1, t_2]$. The average velocity ($v_{avg}$) is given by: $$ v_{avg} = \frac{s(t_2) - s(t_1)}{t_2 - t_1} $$ According to MVT, there exists a time $c$ in $(t_1, t_2)$ where the instantaneous velocity ($v(c)$) equals $v_{avg}$: $$ v(c) = v_{avg} $$ This implies that at some point during the trip, the car's speed was exactly equal to its average speed.

Contextual Problems and Real-World Scenarios

Beyond motion, the Mean Value Theorem applies to various real-world scenarios, such as:

• Economics: Analyzing the rate of change of cost functions to determine marginal costs.
• Medicine: Modeling the rate at which a drug concentration changes in the bloodstream.
• Engineering: Assessing the stress-strain relationships in materials under load.

In each case, MVT helps bridge the gap between average rates and instantaneous rates, providing deeper insights into the system's behavior.

Graphical Interpretation of MVT

Graphically, the Mean Value Theorem states that there exists at least one point where the tangent to the curve $f$ is parallel to the secant line connecting $(a, f(a))$ and $(b, f(b))$. This can be visualized by drawing the secant line between two points on the function and identifying a point where the slope of the tangent line matches the slope of the secant.

Proof of the Mean Value Theorem

The proof of MVT relies on Rolle's Theorem, which is a special case where $f(a) = f(b)$. To prove MVT:

1. Define a new function $g(x) = f(x) - \left( \frac{f(b) - f(a)}{b - a} \right)(x - a) - f(a)$.
2. Show that $g(a) = g(b)$, making $g(x)$ satisfy the conditions of Rolle's Theorem.
3. By Rolle's Theorem, there exists a $c \in (a, b)$ where $g'(c) = 0$.
4. Since $g'(x) = f'(x) - \frac{f(b) - f(a)}{b - a}$, setting $g'(c) = 0$ gives $f'(c) = \frac{f(b) - f(a)}{b - a}$.

This completes the proof, confirming the existence of such a point $c$.

Examples of Applying MVT

Example 1: If a ball is thrown upwards and its height function is $h(t) = -16t^2 + 64t + 80$, find a time $c$ where the instantaneous velocity equals the average velocity over the interval $[1, 3]$ seconds.

First, calculate the average velocity: $$ v_{avg} = \frac{h(3) - h(1)}{3 - 1} = \frac{(-16(9) + 64(3) + 80) - (-16(1) + 64(1) + 80)}{2} = \frac{( -144 + 192 + 80 ) - ( -16 + 64 + 80 )}{2} = \frac{128 - 128}{2} = 0 \text{ ft/s} $$ Next, find $h'(t) = -32t + 64$. Set $h'(c) = 0$: $$ -32c + 64 = 0 \\ c = 2 \text{ seconds} $$ Thus, at $t = 2$ seconds, the instantaneous velocity equals the average velocity.

Example 2: A company's profit over time is given by $P(t) = 50t - 5t^2$. Find a time $c$ in the interval $[2, 5]$ where the instantaneous rate of profit equals the average rate of profit.

Calculate the average rate of profit: $$ \frac{P(5) - P(2)}{5 - 2} = \frac{(50(5) - 5(25)) - (50(2) - 5(4))}{3} = \frac{(250 - 125) - (100 - 20)}{3} = \frac{125 - 80}{3} = \frac{45}{3} = 15 \text{ dollars per unit time} $$ Find $P'(t) = 50 - 10t$. Set $P'(c) = 15$: $$ 50 - 10c = 15 \\ 10c = 35 \\ c = 3.5 \text{ units of time} $$ At $t = 3.5$, the instantaneous rate of profit equals the average rate.

Limitations of the Mean Value Theorem

While the Mean Value Theorem is powerful, it has limitations:

• Applicability: MVT only applies to functions that are continuous on a closed interval and differentiable on the open interval. Functions with discontinuities or sharp corners do not satisfy these conditions.
• Non-Uniqueness: There may be multiple points $c$ that satisfy the theorem, making it sometimes challenging to determine all such points.
• Does Not Provide Exact Values: MVT guarantees the existence of at least one point but does not specify its exact location without additional information.

Understanding these limitations is crucial for correctly applying the Mean Value Theorem in various contexts.

Comparison Table

Summary and Key Takeaways