Problem 55 - Calculus: Early Transcendentals

Question

Let \( H(t) \) be the daily cost (in dollars) to heat an office building when the outside temperature is \( t \) degrees Fahrenheit. (a) What is the meaning of \( H'(58) \)? What are its units? (b) Would you expect \( H'(58) \) to be positive or negative? Explain.

Step-by-Step Solution

Answer

(a) \( H'(58) \) indicates how heating cost changes with temperature; its units are dollars/°F. (b) Expected to be negative since higher temperatures typically lower heating costs.

Step 1: Understand the Function

The function \( H(t) \) represents the daily cost in dollars to heat an office building at an outside temperature of \( t \) degrees Fahrenheit. The goal is to interpret \( H'(58) \), which is the derivative of \( H(t) \) evaluated at \( t=58 \).

Step 2: Interpret the Derivative at a Specific Point

\( H'(58) \) represents the rate of change of the heating cost with respect to the outside temperature when the temperature is 58 degrees Fahrenheit. This means it tells us how the cost of heating changes if the temperature increases by one degree from 58°F.

Step 3: Determine the Units of Measurement

The units of \( H'(58) \) are derived from the function \( H(t) \), which is measured in dollars, and the independent variable \( t \), which is measured in degrees Fahrenheit. Therefore, the units of \( H'(58) \) are dollars per degree Fahrenheit.

Step 4: Sign of the Derivative

Generally, higher outside temperatures lead to lower heating costs because less energy is required to maintain the building's temperature. Therefore, \( H'(58) \) is expected to be negative, indicating that as the temperature increases, the cost decreases.

Step 5: Explain the Expectation

The expectation for \( H'(58) \) to be negative stems from the physical principle that warmer outside temperatures reduce the heating demand, thereby lowering costs. This is typical behavior for heating functions, where cost decreases as outside temperature rises.

Key Concepts

Rate of Change

The rate of change is a fundamental concept in calculus that describes how a quantity changes with respect to another. For the function \( H(t) \), the rate of change is specifically represented by its derivative, \( H'(t) \). This derivative tells us how the cost of heating varies as the outside temperature \( t \) changes.

In our example, \( H'(58) \) stands for the rate of change of the heating cost when the temperature is 58 degrees Fahrenheit. Essentially, it reflects how much the heating cost would increase or decrease if the temperature went up by one degree from 58°F. Understanding this concept helps us predict and analyze behaviors of costs under different temperature scenarios.

Interpretation of Derivatives

Interpreting derivatives involves understanding what they convey about a function. In simple terms, derivatives represent the speed or the rate at which one quantity changes concerning another.

In the context of \( H(t) \), \( H'(58) \) shares valuable insights. It tells us how sensitive heating costs are relative to temperature changes at the specific point of 58 degrees Fahrenheit. That means if the derivative is significant, even a minor temperature shift can lead to considerable changes in heating costs.

This form of interpretation is crucial because it allows us not only to see the current state of change but also to project possible future changes based on the derivative's value.

Units of Measurement in Calculus

Units of measurement in calculus are essential because they provide context to the numerical values obtained from a derivative. For the function \( H(t) \), which measures cost in dollars based on temperature in degrees Fahrenheit, \( H'(t) \)'s units are derived from these quantities.

Specifically, \( H'(58) \) is expressed in "dollars per degree Fahrenheit". This unit indicates the change in heating cost for every degree change in temperature.

Having the right units ensures clarity and accuracy in interpreting results. They allow us to understand practical implications, such as cost efficiencies or inefficiencies related to temperature fluctuations in real-world applications. Keeping track of units in calculus is just as important as the calculations themselves, ensuring we can meaningfully apply our findings.