Problem 6 - Applied Calculus

Question

During an illness a person ran a fever. His temperature rose steadily for eighteen hours, then went steadily down for twenty hours. When was there a critical point for his temperature as a function of time?

Step-by-Step Solution

Answer

The critical point occurs at 18 hours when the temperature stops rising and starts falling.

Step 1: Understand the Problem

We need to determine when there is a critical point in the person's temperature over time. The temperature increases for the first eighteen hours and then decreases for the next twenty hours.

Step 2: Define Critical Point

A critical point in calculus occurs when the derivative of a function is zero or undefined. Since the temperature changes steadily, it suggests a linear increase and decrease, so we are looking for a point where the transition occurs between these changes.

Step 3: Identify Critical Point

In this scenario, a critical point happens when the temperature stops increasing and starts decreasing. This transition occurs precisely at the end of eighteen hours of increasing temperature before it begins to steadily decrease.

Key Concepts

Derivative

In calculus, derivatives are a fundamental concept that measures how a function changes as its input changes. Think of it as the function's rate of change or the slope of the function at any given point. Derivatives help in understanding and predicting the behavior of real-world situations, such as the change in temperature over time in our scenario.

To find a derivative, we calculate the limit of the function's average rate of change as the interval becomes infinitely small. It is represented mathematically as \( f'(x) \). When the derivative is zero or undefined at a specific point, we may have found a critical point.

For the person’s temperature, since it changes steadily, the derivative tells us that there is a constant rate of either increase or decrease. When the derivative changes from positive (temperature rising) to negative (temperature falling), a critical point has been reached.

Linear Function

A linear function is one of the simplest types of functions, characterized by each input corresponding to exactly one output, and its graph being a straight line. The general form of a linear function is \( f(x) = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept.

In the context of the problem, the person's temperature over time is modeled by a linear function during both the increasing and decreasing phases. This means the temperature rises at a constant rate during the first 18 hours and descends at a constant rate over the next 20 hours.

Linear functions are especially useful in real-world applications, like tracking changes over time, because they simplify these changes to a constant rate, allowing for easier predictions and analyses.

Function Analysis

Function analysis in calculus involves understanding the general behavior and properties of a function. This includes determining intervals of increase and decrease, as well as identifying critical points and inflection points.

In our scenario, the function describing the person's temperature demonstrates an interval of increase during the first eighteen hours, followed by an interval of decrease. A key aspect of function analysis is identifying critical points where such transitions occur.

• When a function's derivative changes from positive to negative, it signifies a critical point, such as the peak of the temperature rise in our example.
• Function analysis also considers other aspects like continuity and differentiability, which are intrinsic for predicting the function's behavior accurately over its entire domain.

Understanding the critical point where the temperature stops rising and starts to fall allows us to better comprehend the underlying dynamics of this temperature change over time.