Unemployment The relation u gives the number of people...

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.

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.

Key Concepts

Function

A function is a relation that assigns exactly one output value to each input from its domain. In this context, the relation u is a function because for every value of time t (measured in months after the election), there is a single corresponding value for the number of people unemployed. This helps model situations where each point in time has a unique, determinate rate or figure associated.

Initial Value

The initial value refers to the value of the function at t=0. It provides a starting point or baseline for analyzing changes over time. Here, u(0)=3,000,000 indicates the number of people unemployed at the moment the measurement begins, serving as a critical reference point for subsequent comparisons.

Derivative (Rate of Change)

The derivative of a function represents the rate at which the function's value changes with respect to its independent variable, in this case time. It provides insight into whether the unemployment figure is increasing or decreasing over time. For example, evaluating du/dt or u' at specific times elucidates whether unemployment is rising, falling, or stationary at those moments.

Critical Points

A critical point in the context of a function is a value of t where the derivative is zero, indicating a potential local maximum, minimum, or point of inflection. The statement u'(24)=0 suggests that at 24 months, the number of unemployed people is neither increasing nor decreasing, which could signal a stabilization point in the unemployment trend.

Graphical Representation

Graphical representation involves plotting the function u(t) on a graph with time on the horizontal axis and the number of unemployed people on the vertical axis. This visual tool is essential in illustrating how the unemployment figures change over time, including periods of increase, decrease, or stabilization. A proper graph will label the axes with correct units and reflect key values such as the initial unemployment rate and rates of change at specific times.

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