Problem 82 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Following the lead of the National Wildlife Federation, the Department of the Interior of a South American country began to record an index of environmental quality that measured progress and decline in the environmental quality of its forests. The index for the years 1998 through 2008 is approximated by the function $$ I(t)=\frac{1}{3} t^{3}-\frac{5}{2} t^{2}+80 \quad(0 \leq t \leq 10) $$ where \(t=0\) corresponds to 1998 . Find the intervals where the function \(I\) is increasing and the intervals where it is decreasing. Interpret your results.

Step-by-Step Solution

Answer

The function \(I(t)\) is decreasing on the interval \(0 \leq t < 5\) and increasing on the interval \(5 \leq t \leq 10\). This means that the environmental quality of the forests declined from 1998 to 2003, and there was a recovery or improvement in the environmental quality from 2003 to 2008.

Step 1: Find the First Derivative

The first step is to find the derivative of the given function, \(I(t)\), by applying the basic rules of differentiation. \(I'(t) = \frac{d}{dt} \left(\frac{1}{3} t^{3}-\frac{5}{2} t^{2}+80\right)\) \(I'(t) = \frac{d}{dt} \left(\frac{1}{3} t^{3}\right)-\frac{d}{dt} \left(\frac{5}{2} t^{2}\right)+\frac{d}{dt} \left(80\right)\) Now, use the power rule to differentiate each term: \(\frac{d}{dt} \left(\frac{1}{3} t^{3}\right) = t^{2}\) \(\frac{d}{dt} \left(\frac{5}{2} t^{2}\right) = -5t\) \(\frac{d}{dt} \left(80\right) = 0\) Combine the derivatives to get the first derivative of the function: \(I'(t) = t^{2} - 5t\)

Step 2: Determine the Critical Points

Now, we need to find the critical points of the function. Critical points occur when the derivative is either equal to zero or undefined. In this case, we need to solve the equation \(I'(t) = 0\): \(t^{2} - 5t = 0\) Factor out \(t\): \(t(t - 5) = 0\) Set each factor to zero and solve for \(t\): \(t = 0\) \(t - 5 = 0 \Rightarrow t = 5\) Now we have two critical points: \(t=0\) and \(t=5\) Since we are looking for the intervals for which \(I(t)\) is increasing and decreasing, we will test the intervals formed by the critical points.

Step 3: Analyze Intervals

We found that the critical points are at \(t = 0\) and \(t = 5\). This divides the domain of the function into three intervals: \((0, 5)\), \((5, 10)\). Now, test a point within each interval in the first derivative to determine if the function is increasing or decreasing for that interval. Interval \((0, 5)\): Test \(t=3\) \(I'(3) = 3(3 - 5) = 3(-2) = -6\) The first derivative is negative; thus \(I(t)\) is decreasing in this interval. Interval \((5, 10)\): Test \(t=7\) \(I'(7) = 7(7 - 5) = 7(2) = 14\) The first derivative is positive; thus \(I(t)\) is increasing in this interval. Now that we know on which intervals the function is increasing and decreasing, let's interpret the results.

Step 4: Interpret the Results

We found that the function \(I(t)\) is decreasing for \(0 \leq t < 5\) and increasing for \(5 \leq t \leq 10\). This indicates that from 1998 to 2003 (represented by \( t = 0\) to \(t = 5\)), the environmental quality index showed a decline, suggesting a decline in the environmental quality of the forests during this period. From 2003 to 2008 (represented by \( t = 5\) to \(t = 10\)), the environmental quality index showed an increase, indicating a recovery or improvement in the environmental quality of the forests.

Key Concepts

First Derivative Test

Understanding the behavior of functions is crucial for identifying when a certain quantity is improving or deteriorating, such as the environmental quality of a forest. The first derivative test provides a method to classify critical points -- where the function's derivative is zero or undefined -- and determine whether these points are local maxima, minima, or neither. This is done by examining the sign of the derivative before and after each critical point.

For example, in the environmental quality index function, after finding that the derivative is positive after the critical point at t=5, we infer that the function is increasing and therefore, the environmental quality is improving. Conversely, a negative derivative before this point indicates a declining environmental quality. This test not only helps to chart the course of change but gives us insight into the turning points that are key for analysis in applied fields like environmental studies.

Critical Points in Functions

Critical points in a function are the values where the function's rate of change switches direction or is momentarily static. These points occur when the first derivative equals zero or is not defined. Identifying these points is essential in the study of function behavior. In our environmental quality index function, the critical points at t=0 and t=5 represent moments where the rate of increase or decrease in environmental quality changes.

Spotting critical points is like finding a treasure map's 'X' marks; each one can indicate a potential turning point. In the context of applied mathematics, especially when charting environmental changes, critical points can signal times when policies or natural events made significant impacts on the environmental quality.

Increasing and Decreasing Intervals

Once the critical points of a function have been established, the next step is to determine where the function is increasing or decreasing. These intervals represent periods of growth or decline. In our environmental quality function, we segment the timeline into intervals demarcated by the critical points and test points within these intervals to determine the function's behavior.

For instance, finding a negative first derivative in the interval (0, 5) signifies that the function, and thus the environmental quality index, is decreasing in that time frame. Conversely, a positive first derivative in the interval (5, 10) implies an increasing environmental quality. Understanding these intervals in a real-world context aids policymakers and environmentalists in identifying when to intervene or further analyze specific periods for factors affecting environmental quality.

Applied Mathematics

Applied mathematics is the use of mathematical principles to solve problems in other fields such as science, engineering, business, and economics. In this context, calculus, specifically derivatives and the analysis of functions, is a tool to model and study changes over time. The environmental quality index function is a prime example of applied mathematics, where calculus helps in quantifying changes in the environment.

By applying the derivative test and identifying intervals of increase and decrease, we can extract meaningful information about the state of the environment during specific periods. This allied approach empowers us to use abstract mathematical concepts for tangible insights, leading to more informed decisions in environmental management and conservation.