32. Let \( H(t) \) be the daily cost (in dollars) to heat an office building when the outside temperature is \( t \) degre Fahrenheit. a. What is the meaning of \( H^{\prime}(58) \) ? What are the units? b. Would you expect \( H^{\prime}(58) \) to be positive or negative? Explain

Let's address each part of the problem step by step. ### **a. Meaning and Units of \( H^{\prime}(58) \)** **Meaning:** \( H^{\prime}(58) \) represents the **instantaneous rate of change** of the daily heating cost with respect to the outside temperature when the temperature is **58°F**. In simpler terms, it tells us how sensitive the heating cost is to small changes in the outside temperature around 58°F. **Units:** Since \( H(t) \) measures cost in **dollars** and \( t \) is measured in **degrees Fahrenheit (°F)**, the units of the derivative \( H^{\prime}(58) \) are: \[ \frac{\text{dollars}}{\text{degree Fahrenheit}} \, (\$/°F) \] ### **b. Sign of \( H^{\prime}(58) \)** We **expect \( H^{\prime}(58) \) to be negative**. Here's why: - **Relationship Between Temperature and Heating Cost:** - As the **outside temperature increases**, the need for heating typically **decreases**, leading to a **lower heating cost**. - Conversely, if the temperature were to **decrease**, the heating cost would generally **increase**. - **Implication for the Derivative:** - A **negative derivative** indicates that as temperature \( t \) **increases**, the heating cost \( H(t) \) **decreases**. - Therefore, at \( t = 58°F \), \( H^{\prime}(58) \) being negative aligns with the expectation that warmer outside temperatures reduce heating costs. ### **Summary** - **a.** \( H^{\prime}(58) \) is the rate at which the daily heating cost changes with respect to temperature at 58 °F, measured in dollars per degree Fahrenheit ($⁄°F). - **b.** It is expected to be negative because higher outside temperatures typically lead to lower heating costs.