Problem 36 - Calculus Volume 1

Question

Let \(d_{j}\) denote the hours of daylight and \(\delta_{j}\) denote the increase in the hours of daylight from day \(j-1\) to day \(j\) in Fargo, North Dakota, on the jth day of the year. Interpret \(d_{1}+\sum_{j=2}^{365} \delta_{j}\)

Step-by-Step Solution

Answer

The expression equals total daylight hours on the last day of the year.

Step 1: Understand the Variables

We are given two variables: \(d_j\), which represents the hours of daylight on the \(j\)th day, and \(\delta_j\), which denotes the increase in daylight hours from day \(j-1\) to day \(j\). Our goal is to interpret the expression \(d_1 + \sum_{j=2}^{365} \delta_j\).

Step 2: Analyze the Expression

The expression \(d_1 + \sum_{j=2}^{365} \delta_j\) starts with \(d_1\) which is the number of daylight hours on the first day of the year. The summation part \(\sum_{j=2}^{365} \delta_j\) represents the total increase in daylight hours from day 1 to day 365.

Step 3: Interpret the Summation of Increases

\(\sum_{j=2}^{365} \delta_j\) calculates the cumulative increase in daylight hours over the year by adding the changes for each day starting from the second day to the last day of the year.

Step 4: Combine and Interpret the Total

Adding \(d_1\) to the cumulative increase \(\sum_{j=2}^{365} \delta_j\) gives the total daylight hours on the 365th day. This means \(d_1 + \sum_{j=2}^{365} \delta_j = d_{365}\), which is the total hours of daylight on the last day of the year.

Key Concepts

Summation

In calculus and algebra, summation is a way to add up a series of numbers or expressions, represented with the Greek letter sigma (\( \Sigma \) ).
Summation takes a series of values over a set sequence, often following a specific formula. In our exercise, the summation \( \sum_{j=2}^{365} \delta_j \) represents the cumulative addition of the daylight increases from the second day of the year to the last.

• \( \delta_j \) is the change in daylight from one day to the next.
• The summation notation \( \sum_{j=2}^{365} \delta_j \) automatically sums up each daily increase.

Breaking this down helps us understand large series of additions without listing every single operation, especially useful when dealing with many days of a year.

Daylight Analysis

Daylight analysis involves examining how daylight hours change throughout the year.
This analysis considers seasonal variations, where daylight hours normally increase as we move from winter to summer and decrease as we approach winter again.
In this exercise, we look at daylight in Fargo, North Dakota, where each day's daylight is tracked, and the difference between consecutive days is measured as \( \delta_j \).

• This analysis can show patterns like steady increases in spring and decreases in autumn.
• Understanding daylight shifts is crucial for activities like farming and planning events.

Cumulative Increase

The term cumulative increase refers to the total change accumulated over a period of time.
In our problem, the sum of \( \delta_j \) from day 2 to day 365 represents the total accumulation of increase in daylight over the year.

• Cumulative calculations help track overall growth, not just daily variations.
• This helps determine the extent of daylight increase from the beginning to the end of the year.

By understanding cumulative increase, one can visualize trends over time, rather than just looking at singular data points.

Interpret Expression

Interpreting mathematical expressions involves understanding what the components and their summation mean in real-world terms.
In the context of our expression \(d_1 + \sum_{j=2}^{365} \delta_j\), it is used to find the total daylight hours on the last day of the year.

• \(d_1\) covers daylight on the first day.
• The summation \(\sum_{j=2}^{365} \delta_j\) reveals how much more daylight accumulates over the year.
• Together, they equal \(d_{365}\), symbolizing the daylight on December 31st.

This understanding transforms abstract symbols into meaningful data about daylight changes throughout the year.