Problem 19 - Applied Calculus

Question

The function \(S=f(t)\) gives the average annual sea level, \(S\), in meters, in Aberdeen, Scotland, \(^{5}\) as a function of \(t,\) the number of years before \(2008 .\) Write a mathematical expression that represents the given statement. The average annual sea level in Aberdeen increased by 1 millimeter from 2007 to 2008.

Step-by-Step Solution

Answer

The expression is \( f(0) = f(-1) + 0.001 \).

Step 1: Identify Variables

The problem involves two main variables: the sea level, denoted as \( S \), and time, denoted as \( t \). Here, \( t \) is the number of years before 2008.

Step 2: Define Function Representation

The function \( S = f(t) \) represents the sea level in meters. Since \( t \) is the number of years before 2008, to represent the year 2008, we set \( t = 0 \).

Step 3: Analyze Given Change

The problem states that the sea level increased by 1 millimeter from 2007 to 2008. In terms of function values, this means \( f(0) = f(-1) + 0.001 \).

Step 4: Convert Units

Remember to convert millimeters to meters since the function \( f(t) \) is in meters. 1 millimeter is equal to 0.001 meters.

Step 5: Write the Expression

Based on the analysis, the change in sea level can be written as: \[ f(0) = f(-1) + 0.001 \] This expression captures the increase in sea level from 2007 (\( t = -1 \)) to 2008 (\( t = 0 \)).

Key Concepts

Function Representation

In calculus and mathematics, functions are used to define relationships between variables. Here, we are dealing with a function representation. The function, denoted as \( S = f(t) \), models the average annual sea level in Aberdeen, Scotland over time.

• \( S \) is the dependent variable, representing the sea level measured in meters.
• \( t \) is the independent variable representing time, defined as the number of years before 2008.

Understanding how to interpret \( f(t) \) is crucial because it tells us exactly what the sea level is for a specific time \( t \). By setting \( t = 0 \), we can derive the sea level for the year 2008. Each different \( t \) adjusts which year's sea level you're referring to, providing a clear representation of change over time.

Units Conversion

Conversion between different units is a key skill in understanding and using mathematical expressions in real-world scenarios. In this exercise, we see an increase given in millimeters, but the function \( f(t) \) uses meters. A quick rule of thumb:

• 1 meter = 1000 millimeters
• 1 millimeter = 0.001 meters

This relationship helps us convert the sea level increase from 1 millimeter to meters, crucial for accurately using and adapting the mathematical function. Without this conversion, the equation \( f(0) = f(-1) + 0.001 \) wouldn't correctly express the situation since the operations need consistent units across all terms.

Sea Level Change

Sea level change is a critical concept, encompassing both increases and decreases in sea level over time. It is affected by various global factors such as melting ice caps and thermal expansion of sea water. Here, we consider a small but telling change.
Between 2007 and 2008, the sea level in Aberdeen rose by 1 millimeter, equivalent to 0.001 meters. Tracking these changes helps scientists and researchers understand climate change patterns and coastal developments.This particular shift is represented in the function by comparing \( f(0) \) and \( f(-1) \), highlighting the importance of accurately capturing even minor changes when modeling real-world phenomena.

Mathematical Expression

Mathematical expressions allow us to succinctly denote changes and conditions within functions. The goal is to represent relationships and changes through symbolic notations, which can be straightforward yet powerful. In the case of our exercise, we've captured the sea level change using the expression:\[ f(0) = f(-1) + 0.001 \]This expression elegantly illustrates the increase in sea level from 2007 to 2008 by 0.001 meters. It's a neat encapsulation of the problem's key data, and it streamlines the information into a tidy mathematical statement. Using clear expressions like this is essential for accurate data representation and manipulation, making complex data easier to work with.