The temperanure, $$ H $$, in degrees Celsius, of a hos cup of coffee placed on the kifchen counter is given by $$ \boldsymbol{I}=\boldsymbol{f}(\boldsymbol{t}) $$, where $$ \boldsymbol{t} $$ is in minutes since the coffee was put on the counter. a. Is $$ f^{\prime}(t) $$ positive or negurive? $$ \square $$ $$ ? $$ b. What are the units of $$ f^{\prime}(20) $$ ? $$ \square $$ c. Suppose that $$ f^{\prime}(20)=-0.1 $$ and $$ f(20)=75 $$. Fill in the blanks and select the appropriate terms to complete the following statement about the temperature of the coffee in this case. $$ \square $$ minutes after the coffee was put on the counter, its $$ \square $$ ? is $$ \square $$ degrees Celsius and is $$ \square $$ by $$ \square $$ degreses Celsins per minute. d. Suppose that $$ f^{\prime}(\mathbf{2 0})=-0.1 $$. Complete the following notation. d $$ \square $$ $$ \qquad $$ $$ d $$ $$ \square $$ $$ \boldsymbol{t}= $$ $$ \square $$

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Let's break down the problem step by step.

### a. Is $$ f^{\prime}(t) $$ positive or negative?

The function $$ f(t) $$ represents the temperature of the coffee over time. As the coffee cools down, we expect its temperature to decrease. Therefore, the derivative $$ f^{\prime}(t) $$, which represents the rate of change of temperature with respect to time, will be negative.

**Answer:** $$ f^{\prime}(t) $$ is negative.

### b. What are the units of $$ f^{\prime}(20) $$?

The derivative $$ f^{\prime}(t) $$ represents the rate of change of temperature with respect to time. Since temperature $$ H $$ is measured in degrees Celsius and time $$ t $$ is measured in minutes, the units of $$ f^{\prime}(t) $$ will be:

$$
\text{Units of } f^{\prime}(t) = \frac{\text{degrees Celsius}}{\text{minutes}}
$$

**Answer:** The units of $$ f^{\prime}(20) $$ are degrees Celsius per minute.

### c. Fill in the blanks based on $$ f^{\prime}(20) = -0.1 $$ and $$ f(20) = 75 $$.

Given:
- $$ f(20) = 75 $$ degrees Celsius (the temperature of the coffee at $$ t = 20 $$ minutes).
- $$ f^{\prime}(20) = -0.1 $$ (the rate of change of temperature at $$ t = 20 $$ minutes).

Now we can fill in the blanks:

- $$ 20 $$ minutes after the coffee was put on the counter, its **temperature** is $$ 75 $$ degrees Celsius and is **decreasing** by $$ 0.1 $$ degrees Celsius per minute.

**Answer:** 
- $$ 20 $$ minutes after the coffee was put on the counter, its **temperature** is $$ 75 $$ degrees Celsius and is **decreasing** by $$ 0.1 $$ degrees Celsius per minute.

### d. Complete the following notation.

Given $$ f^{\prime}(20) = -0.1 $$, we can express the change in temperature over a small change in time $$ dt $$:

$$
dH = f^{\prime}(20) \cdot dt
$$

Substituting the value of $$ f^{\prime}(20) $$:

$$
dH = -0.1 \cdot dt
$$

This means that for a small change in time $$ dt $$, the change in temperature $$ dH $$ will be $$ -0.1 $$ times that change in time.

**Answer:**
- $$ dH = -0.1 \, dt $$
a. $$ f^{\prime}(t) $$ is negative. b. The units of $$ f^{\prime}(20) $$ are degrees Celsius per minute. c. 20 minutes after the coffee was put on the counter, its temperature is 75 degrees Celsius and is decreasing by 0.1 degrees Celsius per minute. d. $$ dH = -0.1 \, dt $$.

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