Problem 35 - Essential Calculus Early Transcendentals

Question

Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same speed. [Hint: Consider \(f(t)=g(t)-h(t),\) where \(g\) and \(h\) are the position functions of the two runners.

Step-by-Step Solution

Answer

The runners must have the same speed at some point during the race.

Step 1: Problem Introduction

We want to prove that two runners who start and finish a race at the same time have the same speed at some point during the race. The hint suggests considering a function \( f(t) = g(t) - h(t) \), where \( g(t) \) and \( h(t) \) are the position functions of the two runners.

Step 2: Define Function

Define \( f(t) = g(t) - h(t) \). This function represents the difference in positions between the two runners at any time \( t \). Since the runners start and finish at the same time, we know \( f(0) = 0 \) and \( f(T) = 0 \), where \( T \) is the finish time of the race.

Step 3: Apply the Intermediate Value Theorem

By the given conditions, \( f(t) \) starts at 0 and ends at 0. If \( f(t) \) is continuous over the interval from 0 to \( T \), the Intermediate Value Theorem suggests that there exists some time \( t_0 \), \( 0 < t_0 < T \), where \( f'(t_0) = 0 \).

Step 4: Differentiate Function

Differentiate \( f(t) \: f'(t) = g'(t) - h'(t) \). Setting \( f'(t_0) = 0 \) implies \( g'(t_0) = h'(t_0) \), meaning that at time \( t_0 \), both runners have the same speed.

Step 5: Conclusion

By proving that \( f'(t_0) = 0 \) for some time \( t_0 \), we have shown, using the hint and the Intermediate Value Theorem, that the two runners have the same speed at least once during the race.

Key Concepts

Position Function

In the context of a race, position functions are vital. They describe the location of the runners at any given time. Imagine you have two runners, each with their own path, represented by functions \(g(t)\) and \(h(t)\). These are the position functions for each runner. The value of \(g(t)\) at any time \(t\) tells you exactly where the first runner is. Similarly, \(h(t)\) does the same for the second runner.

Why are these important? Because they allow us to analyze and compare the motion of each runner, such as their speed and any relative changes in position. When you look at \(f(t) = g(t) - h(t)\), this function represents the difference in their positions. If \(f(t) = 0\), both runners occupy the same position. When they start and finish the race together, this implies that at those specific times, they must be in the same spot. Position functions give you a way to map their journey and solve problems related to their movements during the race.

Continuous Function

A continuous function is one where you can draw its graph without lifting your pen from the paper. It smoothly connects without abrupt changes, jumps, or gaps. In this race problem, \(f(t) = g(t) - h(t)\) must be continuous. Why? Because both \(g(t)\) and \(h(t)\) are continuous - each runner moves in a smooth motion without suddenly teleporting to a new spot.

The property of continuity is critical when using the Intermediate Value Theorem. This theorem tells us that if a function is continuous on an interval and you start with one value and end with the same or another, there must be at least one point within that interval where the function takes on any value between those two. For \(f(t)\), starting and ending at 0 is key. It shows they start and end together, affirming the Intermediate Value Theorem can be applied to find a moment where both runners have the same speed.

Derivatives

Derivatives in calculus are all about finding the rate at which things change. For runners in a race, this translates to determining their speed - how fast their position changes over time. Mathematically, if \(g(t)\) and \(h(t)\) are position functions, the derivatives \(g'(t)\) and \(h'(t)\) represent their speeds.

When we differentiate \(f(t) = g(t) - h(t)\), we find \(f'(t) = g'(t) - h'(t)\). If \(f'(t) = 0\), their speeds, \(g'(t)\) and \(h'(t)\), must be equal. Thanks to the earlier application of the Intermediate Value Theorem, we discover at some point \(t_0\), \(f'(t_0) = 0\), signifying that both runners have the same speed. Understanding derivatives helps us not only with analysing speed but also with other characteristics of motion, incredibly useful in a variety of physics and engineering applications.