Question

The height \(H\), in feet, of a river is recorded \(t\) hours after midnight, April 1. What are the units of \(H^{\prime}(t) ?\)

Short answer

The units of \(H'(t)\) are feet per hour.

Step-by-step solution

Understand the Concept of Derivatives

The derivative of a function describes how the function changes as its input changes. Mathematically, the derivative of a function \(H(t)\), with respect to \(t\), is denoted as \(H'(t)\). It represents the rate of change of \(H\) with respect to \(t\).

Identify Units for the Function \(H(t)\)

The function \(H(t)\) represents the height of the river in feet. Therefore, the units of \(H(t)\) are feet.

Identify Units for the Variable \(t\)

The variable \(t\) represents time measured in hours from midnight. Thus, the units of \(t\) are hours.

Determine the Units for the Derivative \(H'(t)\)

Since \(H'(t)\) is the derivative of \(H(t)\) with respect to \(t\), it shows how height (in feet) changes over time (in hours). This gives \(H'(t)\) units of feet per hour, which is written as \(\frac{\text{feet}}{\text{hour}}\).

Key concepts

Rate of Change

When we talk about the **rate of change**, we're really discussing how one quantity changes in relation to another. Imagine you're watching the river rise; the rate of change tells us how fast or slow this happens. In mathematical terms, this is where derivatives come into play. A derivative helps us find the rate of change of a function with respect to a variable. In our context, \(H'(t)\) is the derivative of the height function \(H(t)\), and it shows the rate at which the river's height rises or falls over time.

For example, if \(H'(t)\) equals 2, it means that every hour, the river's height increases by 2 feet. This tool is super handy in all sorts of real-world applications, where noticing how something changes can help in making predictions or adjustments.

In summary, the rate of change is a way to quantify how quickly one variable affects another in a given scenario, making it a fundamental concept in understanding dynamics in science and everyday life.

Units of Measurement

Units of measurement are like the language of math and physics. They let us know exactly what we're talking about and ensure everyone is on the same page. For functions like \(H(t)\), where height in feet is assessed over time in hours, attaching correct units is crucial.

In our exercise, \(H(t)\) tells us the river's height, so its unit is feet. The variable \(t\), which stands for time since midnight, is expressed in hours. Transforming this into a meaningful derivative, \(H'(t)\) represents how many feet the river rises or falls per hour.

So, while it might seem like a minor detail, the correct units of \("feet per hour"\) provide clarity and avoid confusion, allowing for the consistent application of logic and math in practical settings.

Function Notation

Function notation is a special shorthand used in mathematics to make things clear and straightforward. Function notation lets us know the relationship between variables in a form that is concise and easy to understand. For example, the function \(H(t)\) is a way to convey that the height of the river depends on the time \(t\).

When we see \(H(t)\), we understand that \(H\) is the function name, and \(t\) is the input, i.e., the variable affecting \(H\). The result, or output here, is the river's height. Derivatives add another layer by introducing \(H'(t)\), which implies changes, i.e., how quickly the height changes as time progresses.

By using function notation consistently, we gain a powerful tool that strips down complex relationships into understandable forms, making both interpretation and computation easier. It’s a way of saying a lot with very little writing!