Problem 34 - Applied Calculus

Question

Table 1.2 shows the average annual sea level, \(S\), in meters, in Aberdeen, Scotland, \(^{10}\) as a function of time, \(t\) measured in years before 2008. Table 1.2 $$\begin{array}{|c|c|c|c|c|c|c|}\hline t & 0 & 25 & 50 & 75 & 100 & 125 \\\\\hline S & 7.094 & 7.019 & 6.992 & 6.965 & 6.938 & 6.957 \\\\\hline\end{array}$$ (a) What was the average sea level in Aberdeen in \(2008 ?\) (b) In what year was the average sea level 7.019 meters? 6.957 meters? (c) Table 1.3 gives the average sea level, \(S\), in Aberdeen as a function of the year, \(x\). Complete the missing values. Table 1.3 $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & 1883 & ? & 1933 & 1958 & 1983 & 2008 \\\\\hline S & ? & 6.938 & ? & 6.992 & ? & ? \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

(a) 7.094 meters; (b) 1983 for 7.019, 1883 for 6.957; (c) 1883: 6.957, 1933: 6.965, 1983: 7.019, 2008: 7.094.

Step 1: Find Sea Level in 2008

From Table 1.2, the column where \(t = 0\) corresponds to the year 2008. The sea level \(S\) at \(t = 0\) is given as 7.094 meters. Thus, the average sea level in 2008 was 7.094 meters.

Step 2: Determine Years for Specific Sea Levels

To find the years corresponding to specific sea levels, match the sea level values in Table 1.2 with the values given. - For \(S = 7.019\), this occurs at \(t = 25\). Since 2008 corresponds to \(t = 0\), 25 years before 2008 is 1983.- For \(S = 6.957\), this occurs at \(t = 125\). Therefore, 125 years before 2008 is 1883.

Step 3: Complete Table 1.3

Using the data from Table 1.2, fill in the missing values in Table 1.3:- For the year 1883, \(x = 1883\), the sea level is \(S = 6.957\).- For the year 1933, \(t = 75\), look at 75 years before 2008, which corresponds to \(S = 6.965\).- For the year 1983, \(x = 1983\), the sea level is \(S = 7.019\).- For the year 2008, \(x = 2008\), the sea level is \(S = 7.094\). Fill in those values in the table.

Key Concepts

Data Interpretation

Data interpretation is the process of analyzing and making sense of numerical data in order to detect patterns, draw conclusions, or communicate findings. In the context of sea level analysis, the data focuses on changes in sea levels over time.

Looking at Table 1.2, you see the average annual sea levels are listed as a function of the time in years before the year 2008. The numbers tell a story. For example, comparing 2008's sea level with prior years, you can observe changes and trends.

Here's how you can interpret the data given in the exercise:

• First, identify what each column represents. In this case, 't' gives time in years before 2008, and 'S' gives the sea level measurements in meters.
• Next, identify key values such as highest and lowest points, in this example, the 2008 sea level of 7.094 meters seems to be the highest compared to other listed years.
• Finally, recognize patterns or relationships in the data. Here, over the span of 125 years, the sea level appears to show a gradual decrease.

Understanding these basic principles of data interpretation helps you draw logical conclusions and answers to questions like those asked in the exercise.

Time Series

A time series is a sequence of data points recorded at successive points in time. In the sea level analysis exercise, each data point represents an average sea level for a specific year.

Time series data are crucial because they allow you to identify trends and patterns over certain periods. When we look at Table 1.2, each data entry helps to identify how sea levels in Aberdeen have changed over the years leading up to 2008.

Essential features of time series that are evident in this data include:

• Trend: A general direction in which something is developing over the long term. Here, we observe a decline in sea levels.
• Seasonality: Regular, predictable changes recurring over a calendar year. In long-term sea level data, seasonality might be less apparent, but it is essential in areas with greater variability during different periods or months.
• Cycles: Long-term changes that are not of a fixed seasonal frequency. Significant cycles may occur due to climatic changes or human activities over the sampling years.

Time series analysis in exercises like this can be helped by visualizing the data using graphs for a clearer understanding of how sea levels progress year by year.

Applied Calculus

Applied calculus is a branch of mathematics used to solve practical problems by applying calculus concepts to real-world scenarios. In sea level analysis, applied calculus could help to model and predict changes in sea levels.

By using calculus, we can analyze the rate of change in sea levels, which is crucial for understanding how fast the sea level is rising or falling over specified periods. This could involve the use of functions, derivatives, and integrals.

Some ways to leverage applied calculus include:

• Differentiation: Finding the derivative of a function helps us determine whether the sea level is increasing or decreasing at any specific time, and at what rate.
• Integration: Accumulating data over time would help us find the total change in sea levels over a given period. This technique is useful for estimating the cumulative effect of gradual changes.
• Modeling: Creating mathematical models to predict future sea levels based on past data. This helps make informed decisions regarding potential climate impacts and mitigation strategies.

Applied calculus gives us the tools to deeply analyze and solve complex problems seen in exercises like the sea level analysis, providing insights into both the past data and the potential future trends.