Problem 38 - The Calculus with Analytic Geometry

Question

Suppose \(s=f(t)\) is an equation of motion of a particle moving in a straight line where \(f\) satisfies the hypothesis of the mean-value theorem. Show that the conclusion of the mean-value theorem assures us that there will be some instant during any time interval when the instantaneous velocity will equal the average velocity during that time interval.

Step-by-Step Solution

Answer

The MVT ensures there is a point where the instantaneous velocity equals the average velocity in any time interval.

Step 1 - State the Mean Value Theorem

The Mean Value Theorem (MVT) states that 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 point \(c\) in \((a, b)\) such that \[ f'(c) = \frac{f(b) - f(a)}{b - a} \]

Step 2 - Define the Given Motion Function

Let \(s = f(t)\) represent the position of the particle as a function of time \(t\). By the hypothesis, \(f\) is continuous and differentiable, satisfying the conditions needed for the MVT.

Step 3 - Apply the Mean Value Theorem to the Motion Function

Apply the MVT on the interval \([t_1, t_2]\). According to the MVT, there exists a point \(c\) in \((t_1, t_2)\) such that \[ f'(c) = \frac{f(t_2) - f(t_1)}{t_2 - t_1} \]

Step 4 - Interpret the Result

In the context of motion, \(f'(c)\) represents the instantaneous velocity of the particle at time \(c\), and \[ \frac{f(t_2) - f(t_1)}{t_2 - t_1} \] represents the average velocity over the interval \([t_1, t_2]\).

Step 5 - Conclude with Mean Value Theorem Assurances

Thus, the Mean Value Theorem assures that there is at least one instant \(c\) during any time interval \([t_1, t_2]\) when the instantaneous velocity \(f'(c)\) equals the average velocity \[ \frac{f(t_2) - f(t_1)}{t_2 - t_1} \].

Key Concepts

Instantaneous Velocity

Instantaneous velocity is a measure of how fast something is moving at a specific moment in time. This differs from average velocity, which considers the total distance covered over a period of time. To compute instantaneous velocity, you need the derivative of the position function with respect to time. Essentially, it is the slope of the tangent line to the position vs. time graph at a particular point. When you hear \( f'(c) \), this represents the instantaneous velocity at a specific time \( c \).

Average Velocity

Average velocity gives an overall picture of how fast an object is moving over a given time interval. It’s calculated by taking the change in position (displacement) and dividing it by the time interval. For example, if a particle moves from position \( f(t_1)\) to \( f(t_2)\) over the time interval \[ t_1, t_2 \], then the average velocity is \ \[ \frac{f(t_2) - f(t_1)}{t_2 - t_1} \ \]. This value tells you how much ground is covered, on average, for each unit of time.

Continuous Function

A continuous function is a function you can draw without lifting your pen off the paper. Mathematically, a function \( f \) is continuous on an interval if, for every point \( c \) in that interval, the limit as you approach \( c \) is equal to the function's value at \( c \) ( \[ \text{lim}_{x \to c} f(x) = f(c) \ \] ). For the Mean Value Theorem to apply, your function must be continuous; this ensures there are no breaks or jumps.

Differentiable Function

A differentiable function is one that has a derivative at every point in its domain. In simpler terms, it means that you can find the slope of the tangent line at any point on the curve. Differentiability is a stronger condition than continuity: all differentiable functions are continuous, but not all continuous functions are differentiable. For the Mean Value Theorem, your function should not only be continuous but also differentiable, so you can find \( f'(x) \) within the interval.