Problem 67 - Brief Applied Calculus

Question

A patient's temperature at time \(x\) hours is given by a function whose derivative is positive for \(0

Step-by-Step Solution

Answer

The patient's health deteriorates on the first day and improves on the second day.

Step 1: Understand the meaning of the derivative

The derivative of a function gives us the rate of change of that function. If the derivative is positive, the function is increasing, meaning the temperature is rising. If the derivative is negative, the function is decreasing, meaning the temperature is falling.

Step 2: Analyze the first day (0 < x < 24)

For the first day, given by \( 0 < x < 24 \), the derivative is positive. This indicates that the patient's temperature is increasing throughout the day. Hence, the patient's health is likely deteriorating, as an increase in temperature often suggests a rising fever or escalating illness.

Step 3: Analyze the second day (24 < x < 48)

For the second day, indicated by \( 24 < x < 48 \), the derivative is negative. This means that the patient's temperature is decreasing throughout the second day. A declining temperature suggests that the patient is returning to a normal state, indicating improving health.

Key Concepts

Understanding Derivatives

A derivative is a fundamental concept in calculus that represents the rate at which a function changes. In simpler terms, if you imagine tracing a curve on a graph, the derivative tells you how steep the slope is at any given point. For example:

• If the slope is upwards as you move along the curve, the derivative is positive.
• If the slope is downwards, the derivative is negative.

Consider a patient's body temperature as a function over time. Here, the derivative of the temperature function tells us how the temperature is evolving. If the derivative is positive, the temperature is increasing. Conversely, if the derivative is negative, the temperature is decreasing. Understanding the derivative helps make informed judgments about changes, like identifying patterns in health trends.

The Rate of Change

The rate of change might sound like a complex term, but it is simply the speed at which a certain variable changes over time. Consider it similar to how speed measures how quickly a car travels on a highway. In the context of our patient's temperature, this rate is crucial in understanding the patient's condition:

• A high positive rate of change (a steep slope) suggests a rapid increase in temperature, which could be cause for concern, indicating escalating illness.
• A high negative rate of change suggests a quick decrease in temperature, typically a positive sign of recovery.

By looking at the rate at which the temperature changes, doctors and medical staff can make better decisions about treatment and intervention.

Increasing and Decreasing Functions

Functions can be described as increasing or decreasing based on the behavior of their derivatives:

• An increasing function has a positive derivative, which implies that as time progresses, the value of the function (in our case, body temperature) rises.
• A decreasing function has a negative derivative, indicating that the function value is falling over time.

In practical terms, for the patient's temperature:

• During the first day, the function is increasing (derivative is positive), meaning the patient's fever is rising, which usually indicates deteriorating health.
• On the second day, the function is decreasing (derivative is negative), so the temperature lowers, pointing towards better health as the patient recovers.

Understanding whether a function is increasing or decreasing can provide vital information, particularly in medical contexts, helping create effective treatment plans.