Question

The table records the rate of change of air temperature, H, as a function of time, t, one morning. Where is the global minimum for this function? t (hours since midnight) 5 6 7 8 9 dH/dt (°F/hour) - 5-10 24 Select one: a 5:00 b. 7:00 O c. 8:00 O d. 9:00 O e 6:00

Answer 1

The global minimum for the rate of change of air temperature is at 8:00 AM, where it reaches 24°F/hour, marking the transition from cooling to warming.

Step-by-step explanation:

In the given data, the rate of change of air temperature (\( \frac{dH}{dt} \)) is recorded at different times during one morning. The negative values of the rate of change before 8:00 AM suggest a cooling trend, while the positive values after 8:00 AM indicate a warming trend. The global minimum for the function occurs when the rate of change is at its lowest, which happens at 8:00 AM with a value of 24°F/hour. This implies that at 8:00 AM, the air temperature is changing at the slowest rate, marking the point of transition from cooling to warming.

The rate of change reaching its minimum is a critical point, as it signifies a shift in the temperature dynamics. It's at this moment that the temperature trend undergoes a notable change, providing insight into the specific time when the cooling phase is least pronounced or the warming phase is beginning to accelerate.

The global minimum of a function is typically identified by finding the critical points where the derivative is equal to zero or changes sign. In this case, recognizing the significance of the minimum rate of change helps pinpoint a crucial time during the morning when the temperature behavior undergoes a distinct transformation.

Answer 2

The global minimum for the function occurs at 6:00 AM, which is when the rate of temperature change is at its lowest, with a value of -10 °F/hour.

Step-by-step explanation:

The student is asking where the global minimum for the function that records the rate of change of air temperature, H, as a function of time, t, one morning, is located. The given rates of change (dH/dt) in degrees Fahrenheit per hour at different times are as follows:

• 5:00 AM: -5 °F/hour
• 6:00 AM: -10 °F/hour
• 7:00 AM: 24 °F/hour

To find the global minimum, we look for the lowest value of the derivative, dH/dt, which indicates the temperature is decreasing at the fastest rate. According to the given data, the lowest value of dH/dt is -10 °F/hour at 6:00 AM. Therefore, the global minimum occurs at 6:00 AM.

The correct answer to the question is (e) 6:00. We can see this because the rate of temperature change is most negative at 6:00 AM, indicating that the temperature drop from 5:00 to 6:00 AM is the steepest and that at 7:00 AM, the temperature begins to rise again, given the positive rate of change. Thus, the global minimum rate of temperature change occurs at 6:00 AM.