Problem 52 - Calculus Early Transcendentals: Pearson New International Edition

Question

Translate each statement from the following newspaper column into a statement about derivatives. (a) In the United States, the ratio \(R\) of government debt to national income remained unchanged at around \(28 \%\) up to 1981 , but (b) then it began to increase more and more sharply, reaching \(36 \%\) during \(1983 .\)

Step-by-Step Solution

Answer

\( \frac{dR}{dt} = 0 \) until 1981; \( \frac{dR}{dt} > 0 \), \( \frac{d^2R}{dt^2} > 0 \) after 1981.

Step 1: Understand the Context

In this problem, we need to translate economic statements into mathematical statements related to derivatives. The context involves the ratio \( R \) of government debt to national income over time.

Step 2: Analyze Statement (a)

Statement (a) indicates that the ratio \( R \) stayed constant at \( 28\% \) until 1981. In terms of derivatives, a constant ratio means that the rate of change of \( R \) with respect to time is \( 0 \). Therefore, \( \frac{dR}{dt} = 0 \) for the period leading up to 1981.

Step 3: Analyze Statement (b)

Statement (b) suggests that the ratio \( R \) started to increase more sharply after 1981, reaching \( 36\% \) by 1983. In terms of derivatives, an increasing function corresponds to a positive derivative, and the statement about increasing sharply suggests that the derivative is increasing as well. This can be represented as \( \frac{dR}{dt} > 0 \) for 1981 to 1983, with \( \frac{d^2R}{dt^2} > 0 \) indicating the rate of increase of \( R \) itself is escalating.

Key Concepts

Rate of Change

The concept of the "rate of change" is central when discussing derivatives. Essentially, it tells us how a particular value alters over time. In the discussed exercise, we are looking at the change in the ratio \(R\), which is the government debt compared to national income.
For example:

• A constant rate, where nothing changes, simply means that there is no change in the value you’re examining over time.
• A positive rate indicates an increase, whereas a negative rate indicates a decrease.

In the case of article (a), pre-1981, the "rate of change" was zero, as the government debt compared to national income remained steady at 28%. Nothing was changing during that period. Post-1981, as in article (b), the rate was positive due to the increase in this ratio.

Economic Statements

Economic statements provide qualitative descriptions of how economic variables behave over time. In financial articles or newspapers, these statements might describe trends like increasing debt or stable income.
Converting these narratives into mathematical terms gives us a clearer understanding, as shown in the exercise. This allows us to anticipate potential financial scenarios, which can directly influence economic policies.
In our given exercise, two economic statements were translated into mathematical derivatives:

• The first indicated a constant ratio, meaning there was no change in the deriving economic variable up to 1981.
• The second statement pointed to a sharp increase, meaning the economic variable was changing and potentially escalating post-1981.

Interpreting economic statements mathematically allows both analysts and students to see how economic conditions interact in more explicit and actionable ways.

Mathematical Translation

Mathematical translation involves converting everyday language or non-mathematical descriptions into mathematical terms and equations. This skill is important when working with derivatives because it allows us to see the precise behavior of variables over time.

• For example, a description that something "remained constant" translates to having a derivative of zero because there is no change observed.
• On the other hand, if it "increased sharply," this suggests a positive and increasing derivative.

In the exercise, translation was used to express the unchanged ratio of debt and national income before 1981 as \( \frac{dR}{dt} = 0 \), while the sharp increase later on was described with \( \frac{dR}{dt} > 0 \) and \( \frac{d^2R}{dt^2} > 0 \).
By honing the skills of mathematical translation, you can better understand the underlying message within the data, making predictions and conclusions about economic behaviors.

Constant Ratio

In mathematical analysis, a "constant ratio" indicates that a quantity is not changing over time. This is seen in the context of derivatives as having a derivative equal to zero. When a ratio remains constant, it implies the associated derivative does not have any time variation.

• The exercise provided an example where the ratio of government debt to national income stayed constant at 28%. This constancy translated into \( \frac{dR}{dt} = 0 \).

Recognizing and interpreting constant ratios are important in various fields such as economics, physics, and engineering. It allows analysts to identify stability within systems or variables.
When examining economic reports or data, noting where a constant ratio prevails provides us with insight into periods of economic stability or stasis. Understanding this in terms of derivatives, you ensure you understand the lack of momentum or change during that timeframe.