Problem 21 - Applied Calculus

Question

For three minutes the temperature of a feverish person has had positive first derivative and negative second derivative. Which of the following is correct? (a) The temperature rose in the last minute more than it rose in the minute before. (b) The temperature rose in the last minute, but less than it rose in the minute before. (c) The temperature fell in the last minute but less than it fell in the minute before. (d) The temperature rose for two minutes but fell in the last minute.

Step-by-Step Solution

Answer

The correct answer is (b).

Step 1: Interpret the Derivative Information

First, analyze what it means to have a positive first derivative and a negative second derivative. A positive first derivative indicates that the temperature is increasing. A negative second derivative implies that the rate of increase is decreasing; in other words, the graph of the temperature is concave down.

Step 2: Applying the Derivative to Analyze Change over Time

With a positive first derivative, we know the temperature has been rising over the past three minutes. However, because the second derivative is negative, the temperature's rate of increase is slowing down. Therefore, the temperature does rise in each of those three minutes, but by a smaller amount each minute.

Step 3: Evaluate Each Answer Option

Now evaluate each option in the context of derivatives: - (a) The temperature increasing more in the last minute contradicts the negative second derivative (the rate of increase slows down). - (b) The temperature rising but less than before aligns with a decelerating positive slope. - (c) The temperature did not fall as the first derivative is positive. - (d) Temperature cannot rise and then fall in this context since the first derivative is positive.

Step 4: Select the Correct Option

Between the given options, option (b) correctly matches the interpretation of a positive first derivative and a negative second derivative. The temperature rose in the last minute, but less than it rose in the minute before.

Key Concepts

Derivatives

In calculus, derivatives serve as fundamental tools to understand how a function changes as its variables change. Essentially, a derivative measures the rate at which a function's value shifts with respect to changes in its input. For a temperature function, if the derivative is positive, it indicates a rise in temperature over time.

• First Derivative: Determines the slope of a function at any given point. A positive first derivative suggests the function is increasing, while a negative one indicates a decrease.
• Second Derivative: Helps in understanding the curvature of the graph. It tells us how the rate of change itself is changing.
  In our case, a negative second derivative means the graph is curving downwards.

Understanding how derivatives work helps in predicting and analyzing the dynamics of changes in various real-world scenarios like temperature shifts.

Concavity

Concavity is all about the direction in which a graph curves. It provides insight into the acceleration or deceleration of a change.

• Concave Up: When a function is concave up, the slope of the tangent line is increasing. Imagine a bowl-shaped curve where the sides curve upwards.
• Concave Down: Conversely, a function is concave down when the slope of the tangent line is decreasing. The graph looks like an upside-down bowl.
  For the feverish person's temperature, the second derivative is negative, indicating concave down behavior.

This concave down posture suggests that, while the temperature keeps rising, the speed of this rise is tapering off.

Rate of Change

The rate of change is a crucial concept in understanding how rapidly or slowly something, like temperature, varies over time. In essence, it shows how one quantity changes in relation to another.

• Positive Rate of Change: Illustrates growth or increase. For this problem, it denotes a rising temperature.
• Decreasing Rate of Change: When the rate of change reduces, it means the growth is slowing down, not speeding up.

Linking it to our scenario, the first derivative implies that temperature is indeed increasing, but the negative second derivative hints at a drop in the rate of this increase, resulting in smaller temperature rises with each passing minute. This nuanced look at rate of change enables us to better comprehend not just the direction, but the momentum of change.