Problem 22 - Calculus Early Transcendentals

Question

Determine whether the statement is true or false. Explain your answer. A 50 -foot horizontal metal beam is supported on either end by concrete pillars and a weight is placed on the middle of the beam. If \(f(x)\) models how many inches the center of the beam sags when the weight measures \(x\) tons, then the units of the rate of change of \(y=f(x)\) with respect to \(x\) are inches/ton.

Step-by-Step Solution

Answer

True, the rate of change units are inches/ton.

Step 1: Understand the Function

The function given is \(f(x)\), which represents the sag in inches at the center of the beam when a weight \(x\) in tons is placed on it.

Step 2: Rate of Change Definition

The rate of change of a function \(y = f(x)\) with respect to \(x\) is the derivative \(f'(x)\). It represents how much \(y\) changes for a small change in \(x\).

Step 3: Determine the Units of Derivative

Since \(f(x)\) has units of inches and \(x\) has units of tons, the derivative \(f'(x)\) is measured in \(\text{inches per ton}\) because derivatives express the change in \(f(x)\) (inches) per unit change in \(x\) (tons).

Step 4: Conclusion

Because the derivative \(f'(x)\) is measured in inches per ton, the statement claiming that the units of the rate of change of \(y = f(x)\) with respect to \(x\) are inches/ton is true.

Key Concepts

Rate of Change

The rate of change in a mathematical context usually refers to how a quantity changes over time or across some variable. When considering a function like \( f(x) \), which tells us how the beam sags as weight increases, the rate of change describes how much the sagging changes when we add a unit of weight. The rate of change is often represented by the derivative of the function, noted as \( f'(x) \). This derivative gives us insight into the behavior of the function, especially how quickly it increases or decreases.

• If \( f'(x) \) is positive, the sag increases as the weight increases.
• If \( f'(x) \) is negative, the sag decreases with more weight.
• If \( f'(x) = 0 \), the sag doesn't change with added weight.

Thinking of rate of change in terms of derivatives is crucial as it provides dynamic insights- showing accurately and at any given point how sensitive a situation is to changes.

Units of Measurement

Units of measurement are essential for understanding how different quantities relate. They standardize how we talk about and calculate the size, amount, or extent of something. In the context of our beam problem, two key units come into play: inches and tons.

• Inches: This unit measures how much the beam sags under weight. It's a direct physical change.
• Tons: Represents the weight applied to the center of the beam, a factor causing physical change.

To find the rate of change or derivative \( f'(x) \), we calculate how the sag measured in inches changes per unit of weight in tons. Therefore, the units are expressed as inches per ton, indicating the change in sag per additional ton of weight placed on the beam. Proper understanding of units enables precise calculations and makes communication through mathematical expressions clearer.

Functions

Functions are fundamental constructs in mathematics used to represent relationships between quantities. Here, the function \( f(x) \) is a model showing how the sag of the beam (the output) responds to the weight applied (the input). A function maps each input to exactly one output, making it a powerful tool for modeling scenarios like the sagging beam.

• The input, \( x \), is the independent variable (weight in tons).
• The output, \( f(x) \), is the dependent variable (sag in inches).

In practical situations, functions allow for the prediction of outcomes. Knowing how a beam reacts to different weight loads helps us design safer and more efficient structures. The derivative of a function, like \( f'(x) \), further elucidates how these changes happen, providing deeper insight into the sensitivity of the beam's response in engineering applications.