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Chapter 6: Problem 2

The spread of AIDS: The table on the following page shows the cumulative number \(N=N(t)\) of AIDS cases in the United States that have been reported to the Centers for Disease Control and Prevention by the end of the year given. (The source for these data, the U.S. Centers for Disease Control and Prevention in Atlanta, cautions that they are subject to retrospective change.) a. What does \(\frac{d N}{d t}\) mean in practical terms? b. From 1986 to 1992 was \(\frac{d N}{d t}\) ever negative? $$ \begin{array}{|c|c|} \hline t=\text { year } & N=\text { total cases reported } \\ \hline 1986 & 28,711 \\ \hline 1987 & 49,799 \\ \hline 1988 & 80,518 \\ \hline 1989 & 114,113 \\ \hline 1990 & 155,766 \\ \hline 1991 & 199,467 \\ \hline 1992 & 244,939 \\ \hline \end{array} $$

Short Answer

Expert verified
a) \(\frac{dN}{dt}\) is the annual change rate in AIDS cases. b) No, \(\frac{dN}{dt}\) was never negative.

Step by step solution

01

Understand the Derivative

The derivative \( \frac{dN}{dt} \) represents the rate of change of the cumulative number of AIDS cases over time. Practically, it indicates how quickly the number of cases is increasing or decreasing in a particular year. In simpler terms, it's the change in the number of cases per year.
02

Interpret what a Negative Derivative Means

If \( \frac{dN}{dt} \) is negative, it means that the number of cumulative AIDS cases is decreasing during that period. Practically, this would suggest that there are fewer cases being reported year by year, which would be highly unusual for cumulative data that represents totals.
03

Calculate Yearly Differences

Compute the difference in cumulative cases from one year to the next to approximate \( \frac{dN}{dt} \) for each interval: - From 1986 to 1987: \( 49,799 - 28,711 = 21,088 \)- From 1987 to 1988: \( 80,518 - 49,799 = 30,719 \)- From 1988 to 1989: \( 114,113 - 80,518 = 33,595 \)- From 1989 to 1990: \( 155,766 - 114,113 = 41,653 \)- From 1990 to 1991: \( 199,467 - 155,766 = 43,701 \)- From 1991 to 1992: \( 244,939 - 199,467 = 45,472 \)
04

Analyze Yearly Differences

Examine each calculated difference from 1986 to 1992. All computed values are positive, indicating an increase in the number of reported cases year over year throughout this period. Therefore, \( \frac{dN}{dt} \) has never been negative from 1986 to 1992.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivative
In calculus, a derivative represents the rate at which something is changing. For the problem regarding AIDS cases, the derivative \( \frac{dN}{dt} \) specifically describes how the number of reported AIDS cases changes with respect to time. This mathematical concept helps us understand how quickly the situation is developing. For example, if \( \frac{dN}{dt} \) is large, the number of cases is increasing rapidly. Conversely, if it were negative in any scenario, it would imply a decrease in cumulative cases, which is counterintuitive for cumulative data unless cases are being revised downward.
Rate of Change
The rate of change is a synonym for what a derivative measures. It tells us how one quantity changes in relation to another. In the context of this exercise, the rate of change of the number of AIDS cases is crucial for understanding how quickly the epidemic is spreading each year. If you consider the derivative as the speed of change, then each positive difference calculated for consecutive years shows that the spread was only increasing between 1986 and 1992. This steady increase implies a growing epidemic during those years.
Data Analysis
Analyzing data often involves looking at trends over time. In this exercise, you calculate the difference in AIDS cases from year to year to understand the spreading trend. Each calculated difference showed a positive rate of change, meaning each year more cases were being cumulatively reported than the last. A good data analysis practice would include checking these calculations again for accuracy, detecting any anomalies, and understanding the implications of the trend鈥攅specially important in fields like epidemiology.
Epidemiology
Epidemiology involves studying how diseases spread and impact populations. This exercise uses principles of epidemiology by examining and interpreting the data on AIDS cases. By analyzing how the number of reported cases changes over time, we can gain insights into the disease鈥檚 trajectory, effectiveness of interventions, or potential data reporting issues. Understanding the progression through derivatives and rates of change is crucial for public health officials who use such data to plan and allocate resources effectively.

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Most popular questions from this chapter

Looking up: The constant \(g=32\) feet per second per second is the downward acceleration due to gravity near the surface of the Earth. If we stand on the surface of the Earth and locate objects using their distance up from the ground, then the positive direction is up, so down is the negative direction. With this perspective, the equation of change in velocity for a freely falling object would be expressed as $$ \frac{d V}{d t}=-g $$ (We measure upward velocity \(V\) in feet per second and time \(t\) in seconds.) Consider a rock tossed up- ward from the surface of the Earth with an initial velocity of 40 feet per second upward. a. Use a formula to express the velocity \(V=V(t)\) as a linear function. (Hint: You get the slope of \(V\) from the equation of change. The vertical intercept is the initial value.) b. How many seconds after the toss does the rock reach the peak of its flight? (Hint: What is the velocity of the rock when it reaches its peak?) c. How many seconds after the toss does the rock strike the ground? (Hint: How does the time it takes for the rock to rise to its peak compare with the time it takes for it to fall back to the ground?)

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Competition between bacteria: Suppose there are two types of bacteria, type \(A\) and type \(B\), in a place with limited resources. If the bacteria had unlimited resources, both types would grow exponentially, with exponential growth rates \(a\) for type \(A\) and \(b\) for type \(B\). Let \(P\) be the proportion of type \(A\) bacteria, so \(P\) is a number between 0 and 1. For example, if \(P=0\), then there are no type \(A\) bacteria and all are of type \(B\). If \(P=0.5\), then half of the bacteria are of type \(A\) and half are of type \(B\). Each population of bacteria grows in competition for the limited resources, so the proportion \(P\) changes over time. The function \(P\) is subject to the equation of change $$ \frac{d P}{d t}=(a-b) P(1-P) \text {. } $$ Suppose \(a=2.3\) and \(b=1.7\). a. Does \(P\) have a logistic equation of change? b. What happens to the populations if initially \(P=0\) ? c. What happens to \(P\) in the long run if \(P(0)\) is positive (but not zero)? In this case, does it matter what the exact value of \(P(0)\) is?

An investment: You open an account by investing \(\$ 250\) with a financial institution that advertises an APR of \(5.25 \%\), with continuous compounding. What account balance would you expect 1 year after making your initial investment?

Investing in the stock market: You are considering buying three stocks whose prices at time \(t\) are given by \(P_{1}(t), P_{2}(t)\), and \(P_{3}(t)\). You know that \(\frac{d P_{1}}{d t}\) is a large positive number, \(\frac{d P_{2}}{d t}\) is near zero, and \(\frac{d P_{3}}{d t}\) is a large negative number. Which stock will you buy? Explain your answer.
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