Question

The numbers (in thousands) of cases \(y\) of HIV/AIDS reported in the years 2001 through 2005 can be modeled by \(y^{2}-1141.6=24.9099 t^{3}-183.045 t^{2}+452.79 t\) where \(t\) represents the year, with \(t=1\) corresponding to 2001. (a) Use a graphing utility to graph the model and describe the results. (b) Use the graph to estimate the year during which the number of reported cases was increasing at the greatest rate. (c) Complete the table to estimate the year during which the number of reported cases was increasing at the greatest rate. Compare this estimate with your answer in part (b).

Short answer

The approximate year during which the reported cases of HIV/AIDS were increasing at the greatest rate can be identified from the graph, and confirmed through calculus. The exact year will depend on the specifics of the graph and calculations, so requires actual completion of the steps.

Step-by-step solution

Graph the model

Use a graphing tool to plot the equation \(y^2 - 1141.6 = 24.9099t^3 - 183.045t^2 + 452.79t\). Note: t ranges from 1 to 5 (2001 to 2005)

Analyze the graph

Observe the graph. Notice how the number of cases (represented by \(y\)) changes with year (represented by \(t\) + 2000). The slope of the curve at a given point indicates the rate of change - the steeper the slope, the faster the increase in cases.

Estimate greatest rate of increase

The point on the graph where the slope is the steepest (i.e., tangent line to the graph is steepest) indicates the greatest rate of increase. Estimate the t-value at this point and add 2000 to get the year.

Calculus computation for year of greatest increase

The exact location of the maximum rate of change can be determined using calculus. First differentiate the function to get its derivative (rate of change function). Then set derivative equal to 0 and solve to find the t when rate of change is maximum. Calculate the t-value, plug in to the original equation and add 2000 to get the year.

Compare estimation and calculation

Compare the estimated year from the graph (part b) with the calculated year, and make final conclusions.

Key concepts

Graphing Utility

When visualizing complex data such as the number of HIV/AIDS cases over time, a graphing utility is an invaluable tool. Through graphical representation, it becomes easier to spot trends, patterns and critical points, such as peaks or troughs in the data. By plotting the given equation, we translate variable interactions into a visual format, turning abstract numbers into a concrete image. For instance, graphing \(y^2 - 1141.6 = 24.9099t^3 - 183.045t^2 + 452.79t\) allows us to see how HIV/AIDS cases have increased or decreased over the years.

In educational settings, graphing utilities like TI-calculators or computer software can aid significantly. They enable students to focus on the behavior of the functions rather than getting bogged down by manual plotting. This promotes a deeper understanding of calculus concepts and their real-world applications. It's pertinent to encourage students to use these tools not just to get a graph but to also interact with it, such as by zooming into different parts to scrutinize the rate of change at specific intervals.

• Use the zoom and trace features to explore the graph closely
• Take screenshots to document various stages of the graph
• Use sliders (if available) to observe how changes in variables affect the graph

Rate of Change

In the context of modeling HIV/AIDS cases, the rate of change refers to how quickly the number of reported cases increases or decreases over time. This concept is a cornerstone of both algebra and calculus, and it provides crucial insights into the dynamics of disease spread. Calculus, specifically through the derivative, offers a mathematical way to quantify this rate.

When we look at our graph, the steepness of the curve at any point directly represents the rate at which HIV/AIDS cases are changing at that year. A steep incline indicates a rapid increase, whereas a decline indicates a decrease. By finding where this curve is steepest, we can determine when the increase in cases was at its highest. This information is critical for public health planning and intervention efforts. It's helpful to remember that:

• The rate of change is positive when cases are increasing
• The rate of change is zero at a peak or trough (no increase or decrease)
• The rate of change is negative when cases are decreasing

Calculus Applications

Calculus, especially differential calculus, is a powerful mathematical tool for understanding real-world phenomena, such as the spread of a disease. By applying calculus to the equation that models HIV/AIDS cases, we're able to garner insights that are not immediately apparent through basic algebraic methods or graph analysis.

For example, once we've graphed the equation using a graphing utility, calculus allows us to find the exact point of maximum increase in reported HIV/AIDS cases by taking the derivative of the model. This derivative represents the rate of change with respect to time and enables us to calculate when this rate is at its peak, providing precision that is critical for epidemiological studies. Key calculus applications here include:

• Finding the derivative of the model to get the rate of change function
• Setting the derivative equal to zero to determine maxima or minima
• Using second-order derivatives for concavity and inflection point analysis

Differential Calculus

Differential calculus delves into the rate at which quantities change, a concept known as the derivative. When modeling diseases like HIV/AIDS, it's essential to understand not just the number of cases, but how quickly these numbers are increasing or decreasing - in other words, the velocity of the epidemic.

The mathematical process of differentiation gives us the ability to extract the function that describes this rate of change from the initial model. Once we have this function, we can apply further calculus techniques to locate the exact value of time, \(t\), where this rate is greatest, providing invaluable information for health policy strategists. It's important for students to grasp the practical steps in this process:

• Take the derivative of the given model to get the rate of change function
• Analyze the derivative to identify critical points where the rate of change is at its maximum or minimum
• Understand that the second derivative can be used to determine the nature of these critical points