Problem 51 - College Algebra and Calculus: An Applied Approach

Question

The resident population \(P\) (in millions) of the United States from 1790 through 2000 can be modeled by \(P=0.00000583 t^{3}+0.005003 t^{2}+0.13776 t+4.658\) \(-10 \leq t \leq 200\), where \(t=0\) corresponds to 1800 (a) Make a conjecture about the maximum and minimum populations in the U.S. from 1790 to 2000 . (b) Analytically find the maximum and minimum populations over the interval. (c) Write a brief paragraph comparing your conjecture with your results in part (b).

Step-by-Step Solution

Answer

The maximum and minimum populations in the U.S. occur at specific points within the interval [-10,200]. These points can be accurately identified by finding the critical points of the given function, calculating the function's value at these points and at the end points of the interval, then comparing to determine the maximum and minimum populations. The conjectures made about these populations should align reasonably well with the analytical results.

Step 1: Identify critical points

The first step is to find the critical points of the function. Critical points occur where the derivative of the function is zero or undefined. The population model is a polynomial, so it's defined everywhere, and we just need to find where its derivative is zero. Let's first find the derivative of the population function \(P(t)=0.00000583 t^{3}+0.005003 t^{2}+0.13776 t + 4.658\). The derivative \(P'(t)=0.00001749t^{2} + 0.010006t + 0.13776\). This is a quadratic equation, so we can find its roots by applying the quadratic formula, \(t = [-b \pm \sqrt{b^{2}-4ac}] / 2a\).

Step 2: Solve for t

Plugging in the coefficients into the quadratic formula gives two solutions. Be sure to regard the limit of t between -10 and 200. Let's substitute these values back into the population function to get the corresponding population for these critical points.

Step 3: Evaluate the function at endpoints and critical points

Evaluate \(P(t)\) at the endpoints (-10 and 200) as well as at the critical points found in the previous step. The maximum and minimum values of \(P(t)\) over the interval occur at either an endpoint or a critical point.

Step 4: Identify the Maximum and Minimum Populations

Now, by comparing the populations calculated from the function evaluated at its critical points and at endpoints of the interval, the highest value will be the maximum population and the lowest will be the minimum population.

Step 5: Comparing Conjectures with Results

Compare the conjectures made with the analytical results. Conjectures should be plausible guesses made before conducting the analysis, based on what is known about the real-world scenario (in this case, the US population). Compare these guesses with the results of the analytical calculations.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving variables raised to whole number powers. In this case, the population of the U.S. is modeled by a cubic polynomial function of time:
- Pass through real-world data frequently, making them invaluable for modeling trends and predicting outcomes.
- Is advantageous because they are continuous and smooth, making them easy to differentiate and analyze.
In our example, the polynomial function is:
\[P(t) = 0.00000583 t^3 + 0.005003 t^2 + 0.13776 t + 4.658\]Each term has a power of the variable \(t\), which correlates to the years since 1800. The coefficients of these terms represent how each power of \(t\) influences the overall population. For instance, the cubic term, while numerically small, signifies that the population growth rate accelerates over time. These models are useful to understand past behaviors and to speculate about future trends based on historical data.

Critical Points

Critical points are found where the derivative of a function is zero or undefined. For polynomial functions, these are particularly important as they often indicate where maximum or minimum values occur. Since our function is defined everywhere, we focus on where the derivative equals zero.When identifying critical points:

• First, take the derivative of the function to find possible changes in population growth rate over time.
• Set the derivative equation to zero and solve for \(t\) to discover critical points.

In this exercise, the derivative is:
\[P'(t) = 0.00001749 t^2 + 0.010006 t + 0.13776\]Using the quadratic formula \[t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]You'll solve for \(t\) to locate points where changes in slope hint at a potential population peak or trough. These critical points help in assessing the US population's changes historically.

Derivative Analysis

Derivative analysis involves studying the rate of change of a function and using it to understand the behavior of the original function. Here, the derivative of the population over time helps identify when the population was growing fastest or slowest.Diving deeper into this analysis:

• The derivative function \(P'(t)\) serves as a tool for locating critical points, where growth may peak or dip, thus affecting population changes.
• This rate of change is crucial for determining when the population's increase was slowing down or accelerating, which is vital for historical analysis and future projections.

By evaluating \(P'(t)\) at found critical points and selecting those in logical bounds (\(-10 \leq t \leq 200\)), we ascertain moments of significant growth shifts. This analysis not only points out critical shifts but also informs realistic frameworks for planning and responding to population changes.

Maximum and Minimum Values

Identifying maximum and minimum values of a function within a particular interval offers insight into extreme values the function can take.
- These points are often derived from evaluating the function at endpoints and any critical points between them.
- The goal is to determine where these extreme values occur based on logical constraints.
To find extrema:

• Check the population at each critical point found.
• Also assess the endpoints of the function's valid interval, here between \(-10\) and \(200\).

The maximum value of \(P(t)\) will give the highest population, while the minimum refers to the lowest within the period observed. In practical application, comparing these calculated extreme values against any conjectures provides perspective on how closely initial guesses align with mathematical analysis, reflecting the reliability of polynomial models in these scenarios.