Problem 73 - A Graphical Approach to Precalculus with Limits

Question

Insect Population Suppose that an insect population in millions is modeled by $$ f(x)=\frac{10 x+1}{x+1} $$ where \(x \geq 0\) is in months. (a) Graph \(f\) in the window \([0,14]\) by \([0,14] .\) Find the equation of the horizontal asymptote. (b) Determine the initial insect population. (c) What happens to the population after several months? (d) Interpret the horizontal asymptote.

Step-by-Step Solution

Answer

The horizontal asymptote is \(y = 10\). The initial population is 1 million. The population will stabilize at 10 million.

Step 1: Determine the Horizontal Asymptote

For the function \(f(x) = \frac{10x + 1}{x + 1}\), we need to find the horizontal asymptote. Both the numerator and the denominator are polynomials of degree 1. When the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients. Here, the leading coefficients are 10 (from \(10x\) in the numerator) and 1 (from \(x\) in the denominator). Therefore, the horizontal asymptote is \(y = \frac{10}{1} = 10\).

Step 2: Graph the Function

We graph \(f(x) = \frac{10x + 1}{x + 1}\) in the window \([0,14]\) by \([0,14]\). The graph should start close to \(y = 1\) at \(x = 0\) and approach \(y = 10\) as \(x\) increases. The horizontal asymptote at \(y = 10\) should be shown.

Step 3: Find the Initial Population

To find the initial population, we evaluate \(f(x)\) at \(x = 0\): \[f(0) = \frac{10(0) + 1}{0 + 1} = 1.\] Therefore, the initial population is 1 million insects.

Step 4: Analyze Long-Term Behavior

As \(x\) increases (months pass), \(f(x)\) approaches the horizontal asymptote \(y = 10\). This means that the population will stabilize around 10 million insects after several months.

Step 5: Interpret the Horizontal Asymptote

The horizontal asymptote \(y = 10\) indicates the long-term stable population level. As months go by, the insect population approaches 10 million but never exceeds it.

Key Concepts

Horizontal Asymptote

When we talk about horizontal asymptotes in rational functions, we're referring to what happens to the function's value as the input becomes very large. For the function \(f(x) = \frac{10x + 1}{x + 1}\), both the numerator and the denominator are linear polynomials, each of degree 1. This means, over time, as \(x\) heads towards infinity, the behavior of the function is determined by the ratio of the leading coefficients.

• For the numerator, the leading coefficient is 10 (from \(10x\)).
• For the denominator, the leading coefficient is 1 (from \(x\)).

The horizontal asymptote is found by dividing these coefficients: \( y = \frac{10}{1} = 10 \). Thus, as \(x\) increases, \(f(x)\) approaches the value \(y = 10\), marking the stable state the insect population will tend to stabilize around over time.

Initial Population

The initial population of the insect colony gives us a starting point to understand how the population grows over time. At time zero (\(x = 0\)), the population is represented by \(f(0)\). By substituting 0 in the function, we get:\[f(0) = \frac{10 \times 0 + 1}{0 + 1} = 1.\]This result tells us that the initial population is 1 million insects. It's crucial because it sets the stage for how quickly we can expect the population to grow to its eventual limit. Initially, we see a population of 1 million, but this will change as time progresses.

Long-Term Behavior

Understanding the long-term behavior of a function like this one involves looking at the end behavior, that is, what happens as time (\(x\)) goes to infinity. For \(f(x) = \frac{10x + 1}{x + 1}\), as \(x\) becomes very large, the function approaches the horizontal asymptote of 10.This tells us:

• As time passes, the insect population will grow and eventually level off.
• The long-term prediction is that the population will stabilize around 10 million insects.

These insights offer a glimpse into the natural balance the population will find in its environment, dictated by the coefficients of the rational function.

Rational Functions

Rational functions like \(f(x) = \frac{10x + 1}{x + 1}\) are expressions defined as ratios of polynomial functions. Rational functions can often describe real-world phenomena where changes are proportional to some variables.Their key features include:

• Zeros: Values of \(x\) where the numerator equals zero.
• Vertical asymptotes: Likely points where the function heads to infinity or negative infinity.
• Horizontal asymptotes: What the function approaches as \(x\) becomes very large.

In our case, understanding these features lets us model the behavior of the insect population accurately. We see initial growth that steadies out over a long time, governed by the coefficients of our polynomials and their influence as time approaches infinity.