Problem 18 - Calculus for Biology and Medicine

Question

Suppose that you drive from St. Paul to Duluth and you average \(50 \mathrm{mph}\). Explain why there must be a time during your trip at which your speed is exactly \(50 \mathrm{mph}\).

Step-by-Step Solution

Answer

By the Intermediate Value Theorem, there is a moment during the trip when your speed is exactly 50 mph.

Step 1: Understanding Average Speed

The average speed of a trip is calculated by dividing the total distance traveled by the total time taken. In this scenario, your average speed from St. Paul to Duluth is given as \(50 \text{ mph}\).

Step 2: Intermediate Value Theorem

The Intermediate Value Theorem states that for any continuous function \(f(x)\) that takes on values \(f(a)\) and \(f(b)\) at two points \(a\) and \(b\), it also takes on any value \(L\) between \(f(a)\) and \(f(b)\) at some point within \(a\) and \(b\). If we consider the function representing your speed, this theorem can be applied.

Step 3: Applying the Theorem to Speed

Assume your speed relative to time is a continuous function. At the start of your trip, your speed is \(0 \text{ mph}\), and at some point it must exceed \(50 \text{ mph}\) to bring the average to \(50 \text{ mph}\). By the Intermediate Value Theorem, your speed must exactly equal \(50 \text{ mph}\) at least once during your trip.

Key Concepts

Understanding Average Speed

Average speed is a useful concept when analyzing a journey or trip. It answers the question: "If I were to travel uniformly, what would my speed have been?" Calculating average speed is straightforward. You divide the total distance traveled by the total time taken.
For instance, if you travel from St. Paul to Duluth and cover 150 miles in 3 hours, your average speed would be \[\text{Average Speed} = \frac{150}{3} = 50 \text{ mph}.\]

• Averages condense varying speeds into a single number; it's a simplified notion of speed.
• It doesn't specify whether your actual travel speed might have varied significantly during the trip.

Despite the variations in speed along the journey, the integral idea of average speed provides a single representative speed of the overall trip. This is crucial when considering real-world situations where speeds often fluctuate.

Continuous Functions and Their Significance

The notion of a continuous function is fundamental in understanding real-world problems like speed over time. A function is continuous if it has no abrupt changes or breaks. Imagine drawing a curve on a graph without lifting your pen; that depicts a continuous function.
Functions that model real-world phenomena, like speed with respect to time, are typically continuous. This continuity has vital implications. In the context of speed, it means:

• At every moment, there's a well-defined, actual speed.
• The change in speed happens gradually, without any sudden jumps or drops.

Continuity allows us to apply powerful mathematical tools like the Intermediate Value Theorem. When you think about your car journey as a continuous function, it means that all moments, even infinitesimally small ones, are considered in the function's journey from start to finish.

Exploring the Speed Function

A speed function describes how your speed varies over time. Given that speed is a continuous function, it accounts for every possible speed, from start to finish, during a trip.
To understand why your speed is exactly 50 mph at some point during the trip, consider how such functions behave. About a speed function:

• It typically starts at zero (at the beginning of motion) and ends at zero (when the journey is complete or you're stationary).
• The Intermediate Value Theorem can be applied: if the function exceeds or dips below a certain average at any point, it must hit every value in between.

So, if your average speed is 50 mph, and your actual speed varies (sometimes slower, sometimes faster), there must be moments when the instantaneous speed hits 50 mph. This is the essence of how mathematical logic, through continuity and intermediate values, explains real-world scenarios elegantly.