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 $H^{\prime}(58)$ ? What are its units? (b) Would you expect $H^{\prime}(58)$ to be positive or negative? Explain.

Name: Problem 55, Chapter 2: Calculus
Uploaded: 2021-03-13T20:55:42-08:00
Duration: 1 min 12 s
Channel: Heather Eichman
Description: 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.

   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 $H^{\prime}(58)$ ? What are its units?
(b) Would you expect $H^{\prime}(58)$ to be positive or negative? Explain.

Calculus

James Stewart 8th Edition

Chapter 2, Problem 55

Instant Answer

Solved by Expert Heather Eichman

03/13/2021

Step 1

This means it tells us how much the cost to heat the building changes for each degree change in the outside temperature when the temperature is 58 degrees Fahrenheit. Show more…

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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 $H^{\prime}(58)$ ? What are its units? (b) Would you expect $H^{\prime}(58)$ to be positive or negative? Explain.

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Key Concepts

Interpretation of Sign

The sign of the derivative indicates the direction of change. In this scenario, H'(58) is expected to be negative because as the outside temperature increases, the need for heating decreases, leading to lower heating costs.

Physical Units

The derivative's units are derived from the ratio of the units of the output to the input. Here, H(t) is measured in dollars and t in degrees Fahrenheit, so H'(t) has units of dollars per degree Fahrenheit.

Derivative

The derivative represents the instantaneous rate of change of a function. In this case, H'(58) tells us how the daily heating cost changes with a very small change in the outside temperature when it is 58°F.

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