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

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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.
**a.** $$ H^{\prime}(58) $$ represents the rate at which the daily heating cost changes with temperature at 58°F, measured in dollars per degree Fahrenheit ($⁄°F). **b.** $$ H^{\prime}(58) $$ is expected to be negative because higher outside temperatures generally lead to lower heating costs.

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

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