Problem 24 - Calculus

Question

A city grew in population throughout the 1980 s and into the early 1990 s. The population was at its largest in \(1995,\) and then shrank until \(2010 .\) Let \(P=f(t)\) represent the population of the city \(t\) years since \(1980 .\) Sketch graphs of \(f(t)\) and \(f^{\prime}(t),\) labeling the units on the axes.

Step-by-Step Solution

Answer

Sketch increasing population till 1995, then decreasing. Derivative crosses zero at 1995.

Step 1: Understand the Problem Context

We are given a function \(f(t)\) representing the population of a city from the year 1980 onward, where \(t\) is the number of years since 1980. The population increased till 1995 (\(t=15\)) and then decreased till 2010 (\(t=30\)). We need to sketch the graph of \(f(t)\) and its derivative \(f'(t)\).

Step 2: Analyze the Graph of f(t)

Since \(f(t)\) describes population, it will start at some positive value in 1980, increase to its maximum in 1995, and then decrease until 2010. The graph should thus start low, rise to a peak at \(t=15\), and then decline, ending still above zero at \(t=30\).

Step 3: Determine the Features of f^{\prime}(t)

\(f^{\prime}(t)\) is the derivative of \(f(t)\), representing the rate of change of population. Between 1980 and 1995, \(f^{\prime}(t) > 0\) since the population is increasing. It will cross zero at 1995 (where the population reaches its peak), and become negative between 1995 and 2010 as the population decreases.

Step 4: Sketch the Graph of f(t)

Draw the graph of \(f(t)\) on axes where the x-axis represents \(t\) in years since 1980 and the y-axis represents population. Start low, reach a peak at \(t=15\), and then decline. Label key points for clarity.

Step 5: Sketch the Graph of f^{\prime}(t)

On new axes, sketch \(f^{\prime}(t)\) with the x-axis as \(t\) and the y-axis as rate of change. Start above the x-axis for \(0 < t < 15\), cross the x-axis at \(t=15\) where the population stops increasing, and drop below the x-axis for \(15 < t < 30\).

Key Concepts

Calculus

Calculus is a branch of mathematics that studies how things change. It’s an essential tool for understanding how functions behave. Here, we're working with a function that models population growth over time, represented as \( P = f(t) \). This function tells us how the population size at different points in time behaves. In calculus, we use derivatives and integrals to analyze these changes.

In this exercise, calculus is used to understand how the population of a city changes over time. The city grows initially but starts to decrease after reaching a peak. Using calculus, specifically differentiation, helps us derive a new function, \( f'(t) \). This derivative tells us the rate at which the population is changing at any point in time.

To summarize:

• Calculus helps us track the evolution of population size with respect to time.
• The derivative \( f'(t) \) gives insight into the population's growth and subsequent decline.

Derivative Interpretation

The derivative, denoted \( f'(t) \), provides a powerful understanding of how the city's population is evolving. It tells us the rate at which the population is increasing or decreasing at any moment. This is crucial, as it highlights not just the population's current state but how it's likely to change soon.

Here’s how we interpret \( f'(t) \):

• When \( f'(t) > 0 \), the population is increasing. This occurs from 1980 to 1995.
• When \( f'(t) = 0 \), the population reaches a maximum, indicating the peak year, 1995.
• When \( f'(t) < 0 \), the population is decreasing, seen from 1995 to 2010.

Using \( f'(t) \), you see exactly where the peak occurs in the population graph. It not only marks the turning point from growth to decline but also quantifies the rate of these changes between years.

Function Analysis

Function analysis allows us to comprehensively understand the behavior of \( f(t) \), the population function. First, we identify the key intervals where the function is either increasing or decreasing.

Through analysis:

• The population function \( f(t) \) increases between 1980 and 1995, peaking at \( t = 15 \).
• Between 1995 and 2010, \( f(t) \) decreases, reflecting the population’s decline.

Maximizing the function involves identifying when the first derivative \( f'(t) = 0 \). This indicates a peak in population growth, marking the transition point where growth converts to decline.

By analyzing such milestones, we get detailed insights into the city’s demographic trends. This helps city planners and researchers forecast population changes and plan for resources.

Graph Sketching

Graph sketching is the visual representation of the underlying mathematical function. It helps us interpret and predict real-world trends, like changes in population.

While sketching the graph for \( f(t) \):

• Begin at a positive value on the \(y\)-axis representing 1980's population.
• Draw an upward curve reaching a maximum in 1995.
• Then, extend a downward curve towards 2010, but keep the curve above the \(x\)-axis since the population remains positive.

For \( f'(t) \), sketch the rate of change graph:

• The plot will stay above the \(x\)-axis from 1980 to just before 1995.
• Cross the \(x\)-axis at \( t = 15 \), where the population reaches a peak.
• Finally, the plot will dip below the \(x\)-axis to 2010, capturing the decline.

Graph sketching turns complex functions into visual diagrams, clarifying trends and shifts for easier understanding.