Problem 51 - Calculus Early Transcendentals: Pearson New International Edition

Question

John traveled 112 miles in 2 hours and claimed that he never exceeded 55 miles per hour. Use the Mean Value Theorem to disprove John's claim. Hint: Let \(f(t)\) be the distance traveled in time \(t\).

Step-by-Step Solution

Answer

John exceeded 55 mph; the average speed was 56 mph using the Mean Value Theorem.

Step 1: Understanding the Mean Value Theorem

The Mean Value Theorem (MVT) states that if a function \( f \) is continuous on a 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 \( f'(c) = \frac{f(b) - f(a)}{b - a} \). In simple terms, there's at least one point where the derivative (instantaneous rate of change) equals the average rate of change over the interval.

Step 2: Setting Up the Problem

Let \( f(t) \) be the distance traveled by John at time \( t \). John's total travel time is from \( t = 0 \) hours to \( t = 2 \) hours, over which he travels 112 miles. So, \( f(2) = 112 \) and \( f(0) = 0 \). The interval \([0, 2]\) is where we apply the Mean Value Theorem.

Step 3: Calculating the Average Rate of Change

The average rate of change of \( f(t) \) over the interval \([0, 2]\) is given by \( \frac{f(2) - f(0)}{2 - 0} \). Substituting the known values: \[ \frac{112 - 0}{2 - 0} = \frac{112}{2} = 56 \text{ miles per hour} \]. This is the average speed over the interval.

Step 4: Application of the Mean Value Theorem

According to the MVT, there must be at least one point \(c\) in \((0, 2)\) such that \(f'(c) = 56\) mph. This means at some moment, John's speed (instantaneous rate of change) must have been exactly 56 mph, contradicting his claim of never exceeding 55 mph.

Step 5: Conclusion and Verification

Since the average speed (56 mph) calculated using MVT exceeds John's claimed maximum speed of 55 mph, the theorem shows that it is impossible for John to have never exceeded 55 mph during his trip.

Key Concepts

Calculus

Calculus is a branch of mathematics that studies how things change. It focuses on two main concepts: differentiation and integration. Differentiate helps us find how a function changes at any instant, while integration is about summing up small parts to find whole areas or volumes. Calculus allows us to analyze dynamic systems and is crucial in fields like physics, engineering, economics, and even biology. In the context of this problem, calculus helps us use the Mean Value Theorem (MVT) to analyze John's travel.

• The Mean Value Theorem is one of the important theorems in calculus. It connects derivatives (instantaneous rates) with the average rates of a function.
• Understanding how a function behaves overall and how it changes at specific points is essential for drawing accurate conclusions about real-world problems.

Instantaneous Rate of Change

The instantaneous rate of change of a function at a given point is essentially its derivative at that point. It tells us how the function is behaving "at that particular instant." In simpler terms, it's like the speedometer reading in a car – it shows how fast the car is traveling at that exact moment.
In John's case, while he was traveling, his speed at any given moment can be described as the instantaneous rate of change of the distance function, \(f(t)\). Using the Mean Value Theorem, we identify a point between his starting and ending times where this rate matches the average rate calculated over the entire trip.

Average Rate of Change

The average rate of change of a function over an interval is like finding the average speed over a certain time period. It tells us how much the function's value has changed over that interval, divided by the time it took.
For John's journey, we calculate the average rate of change of his distance traveled over the 2-hour period. This is done using the formula:
\[\text{Average Rate of Change} = \frac{f(2) - f(0)}{2 - 0}\]This helps us find that John's average speed is 56 miles per hour. The calculation shows that while John says he never went over 55 mph, on average, he had to be driving faster than that. Thus, the average rate of change is a critical step in identifying discrepancies in such claims.

Velocity

Velocity is a vector quantity that refers to "the rate of change of position." It considers not only how fast an object is moving, but also in which direction. In simpler terms, velocity can be thought of as speed with a direction.
In the context of John's travel example, we are more concerned with his speed, which is the magnitude of his velocity. Applying the Mean Value Theorem tells us that at some moment during his journey, John's instantaneous rate of travel (or speed) must have been 56 mph. This indicates his velocity at that moment, as per the theorem, exceeded his claimed maximum speed of 55 mph, although we do not account for possible changes in direction during his trip.