Problem 14 - Calculus Concepts

Question

Unemployment The relation \(u\) gives the number of people unemployed in a country \(t\) months after the election of a new president. a. Is \(u\) a function? Why or why not? b. Interpret the following facts about \(u(t)\) in statements describing the unemployment situation: $$ \text { i. } u(0)=3,000,000 $$ ii. \(u(12)=2,800,000\) iii. \(u^{\prime}(24)=0\) iv. \(\left.\frac{d u}{d t}\right|_{t=36}=100,000\) v. \(u^{\prime}(48)=-200,000\) c. On the basis of the information in part \(b\), sketch a possible graph of the number of people unemployed during the first 48 months of the president's term. Label numbers and units on the axes.

Step-by-Step Solution

Answer

a. Yes, u is a function. b. Describes unemployment changes over time. c. Graph begins at 3,000,000, decreases to 2,800,000, stabilizes, slightly increases, then decreases sharply.

Step 1: Determine if u is a function

A relation is a function if every input has exactly one output. In this case, for each month \( t \), there is a specific number of unemployed people, \( u(t) \). Therefore, \( u \) is a function because it assigns exactly one unemployment number to each month \( t \).

Step 2: Interpret u(0)=3,000,000

The statement \( u(0)=3,000,000 \) means that at the time of the election (0 months after), there were 3,000,000 people unemployed.

Step 3: Interpret u(12)=2,800,000

The statement \( u(12)=2,800,000 \) indicates that 12 months after the election, the number of unemployed people decreased to 2,800,000, suggesting an improvement in employment numbers one year into the presidency.

Step 4: Interpret u'(24)=0

The derivative \( u'(24)=0 \) implies that at 24 months after the election, the rate of change of unemployment was zero, meaning that the unemployment rate stabilized and was not increasing or decreasing.

Step 5: Interpret du/dt |t=36 = 100,000

The information \( \left.\frac{du}{dt}\right|_{t=36}=100,000 \) suggests that at 36 months, the unemployment rate was increasing at the rate of 100,000 people per month.

Step 6: Interpret u'(48)=-200,000

The derivative \( u'(48)=-200,000 \) shows that at 48 months, the unemployment rate was decreasing at the rate of 200,000 people per month, indicating a rapid improvement or reduction in unemployment.

Step 7: Sketch the graph

To sketch a graph: start at the point (0, 3,000,000). Draw a downward trend to (12, 2,800,000) indicating a decrease in unemployment. At month 24, show a flat line reflecting \( u'(24)=0 \). Then, from 24 to 36 months, show a curve that starts to increase, reflecting \( \left.\frac{du}{dt}\right|_{t=36}=100,000 \). Finally, after month 36, draw a sharp decline down to month 48 reflecting \( u'(48)=-200,000 \). Label the x-axis as "Months after election" and the y-axis as "Number of unemployed people."

Key Concepts

Functions

In calculus, a function is a fundamental concept where each input is associated with exactly one output. In practical terms, functions describe relationships like the one between time and unemployment numbers in our problem. Here, the relationship is expressed as \( u(t) \), where \( t \) represents time in months after the president's election, and \( u(t) \) represents the number of unemployed people at that time.

Every month \( t \), corresponds to one specific unemployment count \( u(t) \), ensuring this relationship is a function. Understanding this helps us model scenarios by predicting or describing specific outputs with known inputs. Functions provide a clear mathematical framework to analyze real-world phenomena like changes in unemployment figures over time.

Derivatives

Derivatives are a key tool in calculus that describe the rate at which a quantity changes. When we take the derivative of a function, we're essentially looking at how that function behaves as its input changes. In the context of unemployment modeling, derivatives can offer insight into how unemployment numbers are increasing or decreasing at specific points in time.

For instance, \( u'(24) = 0 \) tells us that 24 months after the election, the unemployment rate remained constant; it wasn't rising or falling. Meanwhile, \( \left.\frac{du}{dt}\right|_{t=36} = 100,000 \) indicates that unemployment was rising by 100,000 individuals per month at month 36. On the other hand, \( u'(48) = -200,000 \) suggests that, at 48 months, unemployment was decreasing by 200,000 each month, representing a recovery or improvement.

• This detailed rate of change analysis helps economists and policymakers make informed decisions based on how quickly unemployment trends are shifting.

Graph Interpretation

Interpreting graphs based on given points and derivatives can help visualize complex data easily. In this exercise, plotting \( u(t) \) on a graph provides a clear picture of unemployment over time.

We start plotting at \( (0, 3,000,000) \), the initial unemployment. By moving to \( (12, 2,800,000) \), a steady decline is shown as a downward slope, indicating improved employment. At \( t=24 \), the slope flattens, representing stable unemployment numbers per year.

The departure from a flat line after 24 months towards \( t=36 \) shows increasing unemployment (rise by 100,000 each month). Finally, a sharp downward slope from months 36 to 48, corresponding to \( u'(48) = -200,000 \), illustrates rapid improvement as unemployment rates drop significantly.

• This visual approach enhances understanding and conveys underlying trends and potential future shifts in unemployment figures.

Unemployment Modeling

Unemployment modeling involves using mathematical functions to represent employment trends in a given region over time. It's a valuable tool for predicting future unemployment patterns and assessing economic policies' impacts.

The function \( u(t) \), representing unemployment, offers valuable insights when reviewed at various points over time. At the core, it helps stakeholders understand current states (e.g., \( u(0) = 3,000,000 \)) and predict future conditions based on derivative insights (like \( u'(24) \) or \( u'(48) \)).

• Modeling unemployment aids in evaluating policy effectiveness, foreseeing economic challenges, and steering social policies. With these models, decision-makers can prepare for and mitigate adverse economic impacts by adjusting strategies appropriately.

Using a combination of historical data analysis and mathematical predictions, unemployment modeling can inform both long-term planning and immediate policy adjustments for better socio-economic outcomes.