Problem 119 - Algebra and Trigonometry Real Mathematics, Real People

Question

Exploration Use a graphing utility to graph each function. Identify the horizontal asymptote of the graph and determine its relationship to the sum. (a) \(f(x)=6\left[\frac{1-(0.5)^{x}}{1-(0.5)}\right], \sum_{n=0}^{\infty} 6\left(\frac{1}{2}\right)^{n}\) (b) \(f(x)=2\left[\frac{1-(0.8)^{x}}{1-(0.8)}\right], \sum_{n=0}^{\infty} 2\left(\frac{4}{5}\right)^{n}\)

Step-by-Step Solution

Answer

The sum of the series is the horizontal asymptote of the function \(f(x)\). For part \( (a) \), the sum, and hence the horizontal asymptote, is 12. For part \( (b) \), it is 10.

Step 1: Understanding the geometric series

Geometric series is a series with a constant ratio between successive terms. It is given here as \( \sum_{n=0}^{\infty} a(r)^{n} \) where, in part \( (a) \), \( a = 6 \) and \( r = 0.5 \), and in part \( (b) \), \( a = 2 \) and \( r = 0.8 \). The sum of an infinite geometric series when its ratio \( r \) lies between -1 and 1 can be found using the formula \( a / (1 - r) \).

Step 2: Calculating the sum of the series

Apply the formula to calculate the sum of the given series. For part \( (a) \), the sum \( S = 6 / (1 - 0.5) = 12 \). For part \( (b) \), the sum \( S = 2 / (1 - 0.8) = 10 \).

Step 3: Graphing the given function

Use a graphing utility to draw the graph of the function \( f(x) \). The horizontal asymptote of the function can be found by analyzing the graph as \( x \) approaches infinity. See how the function behaves when \( x \) becomes really large.

Step 4: Identifying the horizontal asymptotes and relationship with the sum

By looking at the graph, we see that in both cases, as \( x \) tends to infinity, \( f(x) \) approaches a horizontal line which is the horizontal asymptote. By comparing those lines with our calculated sum of series, we see that the line identifying the horizontal asymptote corresponds to the sum of the series in both parts of the exercise. So, the sum of the given series is the horizontal asymptote of the function \( f(x) \).

Key Concepts

Horizontal Asymptote

When we talk about a horizontal asymptote, we refer to a straight horizontal line that the graph of a function approaches as the independent variable, often represented as \( x \), moves towards positive or negative infinity. You can think of it as a long-term prediction of where the function will settle.

How do you find a horizontal asymptote? A good starting point is to look at the behavior of a function as \( x \) becomes very large or very small. In many cases, for functions involving terms like \( r^x \) where \( |r| < 1\), the terms will become negligible as \( x \) approaches infinity, helping us spot the asymptote easily.

In our exercise, when we graphed \( f(x) = 6\left[\frac{1 - (0.5)^x}{1 - 0.5}\right] \) for part (a) and \( f(x) = 2\left[\frac{1 - (0.8)^x}{1 - 0.8}\right] \) for part (b), we noticed that the horizontal asymptotes occur at \( y = 12 \) and \( y = 10 \), respectively. These values match the sums derived from the geometric series formula. So, the horizontal asymptote represents the sum of the infinite series, showing how crucial this point is in understanding function behavior over an infinite domain.

Infinite Series

An infinite series is a sum of infinitely many terms. It can seem a little intimidating at first, but understanding the concept can help simplify how we view functions that appear complex. Often, these series take the form:\[ \sum_{n=0}^{\infty} a r^n \]where \( a \) represents the initial term, and \( r \) the common ratio.

For a geometric series, if the absolute value of the common ratio \( r \) is less than 1, the series converges to a sum given by \( \frac{a}{1-r} \). This can offer great insights as it allows us to sum what seems like an infinite process into a tidy result.

In our examples, \( a = 6, r = 0.5 \) for part (a) converged to a sum of 12, and \( a = 2, r = 0.8 \) for part (b) summed to 10. These sums represent the constant value toward which the series approaches, and they relate directly to the horizontal asymptotes in the function's graph.

Graphing Functions

Graphing functions is an essential method for visualizing the behavior of mathematical expressions. It allows us to see things like trends, maxima, minima, and asymptotic behavior – all important for a deep understanding of calculus and algebra.

To graph a function accurately:

• Understand the equation and identify key components like the common ratio and constants.
• Select a graphing utility, such as graphing software or a calculator, and set the appropriate range and increment for \( x \) and \( y \) values.
• Observe the graph closely as \( x \) approaches large positive values. Pay attention to horizontal lines the graph seems to approach.

In our exercise, graphing \( f(x) = 6\left[\frac{1-(0.5)^x}{1-0.5}\right] \) and \( f(x) = 2\left[\frac{1-(0.8)^x}{1-0.8}\right] \) demonstrates how geometric series tend to level off, forming horizontal asymptotes at specific values (12 in part (a) and 10 in part (b)).

Graphing is not just about plotting points; it's about interpreting and understanding what those plots represent which provides a powerful insight into the series and their convergence.