Problem 47 - Brief Applied Calculus

Question

Suppose that the quantity described is represented by a function \(f(t)\) where \(t\) stands for time. Based on the description: a. Is the first derivative positive or negative? b. Is the second derivative positive or negative? The temperature is dropping increasingly rapidly.

Step-by-Step Solution

Answer

a. The first derivative is negative. b. The second derivative is negative.

Step 1: Understanding the First Derivative

The first derivative of a function, denoted as \(f'(t)\), represents the rate of change of the function with respect to time \(t\). In the context of temperature dropping, a positive derivative indicates the temperature is rising, while a negative derivative indicates it is dropping. Since the temperature is dropping, \(f'(t)\) is negative.

Step 2: Evaluating the Second Derivative

The second derivative, denoted as \(f''(t)\), describes the concavity of the function or the rate of change of the first derivative. If the temperature is dropping increasingly rapidly, the first derivative \(f'(t)\) is becoming more negative, which indicates that \(f''(t)\) is negative, as the function is accelerating in the negative direction.

Key Concepts

First Derivative

In calculus, the first derivative of a function, expressed as \( f'(t) \), is paramount for understanding how a quantity changes over time. Essentially, it gives us the "rate of change" of the function at any given point. Think of it as the slope of the tangent line at a particular point on the function's graph: if the slope is positive, the function is increasing; if negative, it's decreasing.

In the context of temperature, this becomes clear: a positive first derivative means the temperature is rising, while a negative one means it's dropping. For example, if the temperature is dropping at any given moment, the derivative \( f'(t) \) would be negative.

This concept is crucial when studying dynamic systems, such as natural phenomena or even economic trends. Mastering it provides insights into how fast or slow a process is occurring.

Second Derivative

Building on the first derivative, we arrive at the second derivative, denoted \( f''(t) \). It is the derivative of the first derivative, essentially measuring how the rate of change itself is changing. This provides insights into the curvature, or "concavity," of the function.

A negative second derivative suggests the function is concave down, indicating the rate at which the function is decreasing is accelerating. In terms of our temperature example, if the temperature decrease is accelerating, the first derivative \( f'(t) \) is getting more negative, leading \( f''(t) \) to also be negative.

Understanding the second derivative helps in predicting future trends and detecting inflection points where the function's direction could change. It's a powerful tool in fields such as engineering and physics.

Rate of Change

The "rate of change" is a fundamental concept deeply linked to derivatives. It describes how quickly a quantity changes over time. When we speak about the rate of change in calculus, we're typically referring to what the first derivative, \( f'(t) \), tells us.

A slow rate of change means a smaller absolute value of the derivative, while a fast rate of change means a larger absolute value. Returning to our example, if the temperature is dropping at an increasing pace, we're dealing with a situation where the rate of change itself is changing rapidly, indicated by the behavior of the second derivative \( f''(t) \).

Understanding rates of change can provide valuable information in various areas including physics, biology, and economics, helping to model real-world situations and make predictions.