Problem 18 - Calculus

Question

The table gives the average temperature in Wallingford, Connecticut, for the first 10 days in March. (a) Over which intervals was the average temperature increasing? Decreasing? (b) Find a pair of consecutive intervals over which the average temperature was increasing at a decreasing rate. Find another pair of consecutive intervals over which the average temperature was increasing at an increasing rate. $$\begin{array}{c|c|c|c|c|c|c|c|c|c|c} \hline \text { Day } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline^{\circ} \mathrm{F} & 42^{\circ} & 42^{\circ} & 34^{\circ} & 25^{\circ} & 22^{\circ} & 34^{\circ} & 38^{\circ} & 40^{\circ} & 49^{\circ} & 49^{\circ} \\\ \hline \end{array}$$

Step-by-Step Solution

Answer

The temperature increased on days 6-9. It increased at a decreasing rate on days 7-8 and an increasing rate on days 8-9.

Step 1: Identify Intervals

First, list the temperature differences between each day from the provided table. This will help us determine whether the temperature was increasing or decreasing between specific days.

Step 2: Calculate Daily Changes

Subtract the temperature of the previous day from the current day for each interval: - Day 2-1: \(42 - 42 = 0\)- Day 3-2: \(34 - 42 = -8\)- Day 4-3: \(25 - 34 = -9\)- Day 5-4: \(22 - 25 = -3\)- Day 6-5: \(34 - 22 = +12\)- Day 7-6: \(38 - 34 = +4\)- Day 8-7: \(40 - 38 = +2\)- Day 9-8: \(49 - 40 = +9\)- Day 10-9: \(49 - 49 = 0\).

Step 3: Identify Increase and Decrease Intervals

Using the daily changes calculated: - Increasing intervals (positive changes): Day 6-5, Day 7-6, Day 8-7, Day 9-8. - Decreasing intervals (negative changes): Day 3-2, Day 4-3, Day 5-4.

Step 4: Find Increasing at Decreasing Rate

For increasing at a decreasing rate, look for consecutive intervals where the second increase is smaller than the first: - Day 7-6 (increase by 4) -> Day 8-7 (increase by 2).

Step 5: Find Increasing at Increasing Rate

For increasing at an increasing rate, look for consecutive intervals where the second increase is larger than the first: - Day 8-7 (increase by 2) -> Day 9-8 (increase by 9).

Step 6: Conclusion

The average temperature increased over the intervals Day 6-5, Day 7-6, Day 8-7, and Day 9-8. It decreased over the intervals Day 3-2, Day 4-3, and Day 5-4. It increased at a decreasing rate from Day 7-8 and at an increasing rate from Day 8-9.

Key Concepts

Increasing and Decreasing Intervals

Understanding increasing and decreasing intervals in the context of temperature changes involves recognizing patterns and transitions in data over time. When concentrating on the daily temperatures as seen in the table, we first need to calculate the temperature difference between consecutive days. This simple calculation allows us to identify whether the temperature increased, decreased, or remained constant between any two days. Let's break it down:

• If day-to-day temperature difference is positive, this indicates an increase in temperature, hence an increasing interval.
• Conversely, if the difference is negative, it indicates a decrease in temperature, thus a decreasing interval.
• No change in temperature results in a zero difference, indicating a constant interval.

In our exercise, the temperature was found to increase over intervals 6-5, 7-6, 8-7, and 9-8 and it decreased over intervals 3-2, 4-3, and 5-4. Recognizing these transitions helps in understanding how temperature fluctuated throughout the observed period.

Rates of Change

Rates of change give insight into the dynamics of temperature behavior over intervals. It captures not only if a change occurred but also how fast this change was. In the context of average temperatures, rates of change between days can indicate different patterns of increase or decrease. When analyzing the exercise: - **Increasing at a decreasing rate** is observed when the rise in temperature slows down. For example, between Day 7-6 and Day 8-7, the temperature rise reduced from 4 degrees to 2 degrees. - **Increasing at an increasing rate** occurs when the temperature rise accelerates. This is shown between Day 8-7 and Day 9-8, where the increase jumps from 2 degrees to 9 degrees. These insights into rates of change help in predicting and understanding the behavior of temperature variations, reflecting external influences like weather conditions. Consistently acknowledging the rate provides a deeper understanding beyond just identifying whether the temperature is increasing or decreasing.

Temperature Data Interpretation

Interpreting temperature data involves more than just noting if it went up or down. It’s about reading the story the numbers tell. From the temperature table, each interval offers a snapshot of weather conditions which can be interpreted in various ways. Here are key points to consider in data interpretation:

• **Contextual Understanding**: Look at surrounding conditions – colder temperatures could follow a storm, while warmer trends might indicate a transition into spring.
• **Comparative Analysis**: Compare the temperature changes to those of previous months or years to identify trends or anomalies.
• **Predictive Insights**: Use patterns observed (such as rates of change) to predict future trends, aiding in planning and preparedness for weather-related activities.
• **Graphical Representation**: Displaying data through line graphs could visually enhance understanding, highlighting trends and transitions more clearly than just numbers alone.

By carefully interpreting this data, we can extract significant knowledge about climatic behaviors and make informed decisions or hypotheses about future weather patterns. This methodical analysis of temperature data aids not only students learning how to analyze data but also in real-world applications like agriculture and urban planning.