Problem 3 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

A company is making industrial-size rolls of paper towels. A machine is wrapping paper around a roll at a steady rate. By this we mean that the same number of sheets of paper towels are added to the roll every minute. Let \(D(t)\) be the diameter of the roll of paper towels at time \(t\). Determine the sign of the following. Explain your answers using plain English. (a) \(D(t)\) (b) \(D^{\prime}(t)\) (c) \(D^{\prime \prime}(t)\)

Step-by-Step Solution

Answer

\(D(t) \geq 0\), \(D'(t) > 0\), and \(D''(t) = 0\). The diameter is always non-negative, it increases at a constant positive rate, and the rate of change of the rate of increase is zero.

Step 1: Part (a): Sign of \(D(t)\)

Assuming the process starts from a roll of zero diameter and increases steadily as more sheets are added, the diameter function \(D(t)\) is non-negative for all values of \(t\). That is to say, \(D(t) \geq 0\). This is because the diameter can never be negative, it starts from zero and increases as more paper is wrapped around.

Step 2: Part (b): Sign of \(D'(t)\)

The first derivative \(D'(t)\) of a function denotes the rate of change of the function. In this case, \(D'(t)\) denotes the rate of change of the diameter of the roll with respect to time. Given that 'the same number of sheets of paper towels are added to the roll every minute', the diameter increases at a steady (constant) rate. This means that the first derivative \(D'(t)\) is a positive constant, i.e., \(D'(t) > 0\). Because a constant rate of increase implies a positive derivative.

Step 3: Part (c): Sign of \(D''(t)\)

The second derivative of a function, \(D''(t)\) in this case, signifies the change in the rate of change of the function. In other words, it is the acceleration of the function. As we've established that the diameter of the roll increases at a constant rate (which is signified by a constant \(D'(t)\)), the acceleration or the rate of change of the diameter's rate of change is zero. This is because a constant's rate of change is zero. Therefore, \(D''(t) = 0\).

Key Concepts

Derivatives

In calculus, derivatives are a fundamental concept used to understand how a function changes as its input changes. Simply put, the derivative of a function at a certain point gives us the slope of the tangent line to the function at that point, which represents the function's rate of change at that very moment.

Function rates of change

The rate of change of a function is an essential aspect when studying the behavior of dynamic systems. This concept can be applied to various real-life aspects like the speed of a car, growth populations, or, as in our example, the increase in diameter of a roll of paper towels. When the derivative, often denoted as \( f'(x) \) or \( \frac{df}{dx} \), is positive, the function is increasing; when it's negative, the function is decreasing. If the derivative is zero, the function's output is not changing at that particular point.

Second derivative

The second derivative of a function provides deeper insight into the function's curvature and acceleration. It is the rate of change of the rate of change, which can indicate concavity (where the graph bends upward or downward) and points of inflection (where the graph changes concavity). In physics, this is often related to acceleration, as it represents a change in velocity over time. In our paper roll example, since the diameter increases at a steady rate, the second derivative would be zero, indicating no change in the acceleration of the diameter's growth.