Problem 100 - Algebra and Trigonometry

Question

In Exercises \(99-100,\) use a graphing utility to graph the function. Determine the horizontal asymptote for the graph of fand discuss its relationship to the sum of the given series. Function Series $$ f(x)=\frac{4\left[1-(0.6)^{x}\right]}{1-0.6} \quad 4+4(0.6)+4(0.6)^{2}+4(0.6)^{3}+\cdots $$

Step-by-Step Solution

Answer

The sum of the given infinite series is 10 which is the same value as the horizontal asymptote of the given function. This means there's a direct relationship between the sum of the series and the horizontal asymptote of the function.

Step 1: Simplify the Function

Simplify the function \(f(x)=\frac{4\left[1-(0.6)^{x}\right]}{1-0.6}\) to have a manageable format for graphing. This simplifies to \(f(x) = \frac{4 - 4*(0.6)^x}{0.4}\) or \(f(x) = 10 - 10*(0.6)^x \)

Step 2: Calculate the Sum of the Infinite Series

The series given is an infinite geometric series with first term \(a = 4\) and common ratio \(r = 0.6\). The sum, S, of an infinite geometric series is given by the formula \(S = \frac{a}{1-r}\), so replacing \(a\) and \(r\) we get \(S = \frac{4}{1 - 0.6} = 10\)

Step 3: Determine the Horizontal Asymptote

The horizontal asymptote of a function is the value that the function approaches as \(x\) tends towards infinity. For our given function \(f(x) = 10 - 10*(0.6)^x\), the value of \(f(x)\) approaches 10 as \(x\) tends towards infinity. Therefore, the horizontal asymptote for this function is \(y = 10\)

Step 4: Discuss the relationship

The sum of the infinite series, 10, is the same value as the horizontal asymptote of the given function, \(y = 10\). Thus, there is a direct relationship between the sum of the given infinite series and the horizontal asymptote of the function

Key Concepts

Horizontal Asymptote

A horizontal asymptote is a horizontal line that a graph approaches as the input, or x-value, goes to infinity or negative infinity. It is essential to note that the graph might touch the asymptote, but as x moves further away (either positive or negative), the function value gets closer and closer to the asymptote line without actually reaching it.
For the function given in the exercise, \( f(x) = 10 - 10 imes (0.6)^x \), as \( x \) becomes very large, \( (0.6)^x \) becomes very small because 0.6 is a number less than one. Therefore, \( 10 \times (0.6)^x \) approaches 0, meaning that \( f(x) \) approaches 10.
Consequently, the horizontal asymptote for this function is \( y = 10 \). This is the level that the function will appear to stabilize at as \( x \) increases. Understanding horizontal asymptotes helps us determine the long-term behavior of a function.

Infinite Series

An infinite series is a sum of an infinite sequence of terms, which can potentially reach a finite value. In this exercise, we are dealing with an infinite geometric series. Geometric series have constant ratios between consecutive terms, called the common ratio.
To find the sum of an infinite geometric series, we use the formula: \[ S = \frac{a}{1-r} \]

• \( a \): the first term of the series
• \( r \): the common ratio, which must be between -1 and 1 for the series to converge

When the absolute value of the common ratio is less than one, the series converges and the formula can help to find its sum.
In our problem, the series is \( 4 + 4(0.6) + 4(0.6)^2 + 4(0.6)^3 + \cdots \), with \( a = 4 \) and \( r = 0.6 \). Substituting these values into the formula, the sum is \( 10 \). This finite sum is fundamentally linked to the horizontal asymptote of the function derived from these terms.

Common Ratio

The common ratio, symbolized by \( r \), is a fundamental concept in understanding geometric series. It shows how each term in a series relates to the previous one. For a geometric series, the next term is obtained by multiplying the current term by this fixed number.
In our example, the common ratio is \( 0.6 \). This means each subsequent term in the series is 0.6 times the term before it. So, starting with 4, the next term is \( 4 \times 0.6 = 2.4 \), and the process repeats continuously.
The critical point with series converging (reaching a finite sum) is having a common ratio, \( r \), within the range of -1 to 1.
This range ensures that over many terms, the additional amounts being added become exceedingly tiny, allowing the total sum to approach a specific number rather than drift infinitely. This is why the common ratio plays such an essential role in determining the behavior and sum of infinite series.

Graphing Function

Graphing a function gives us a visual representation of its equation, showing how the output (y-value) changes with different inputs (x-values). For the function in this exercise, graphing can help see how \( f(x) = 10 - 10 \times (0.6)^x \) behaves as \( x \) increases.
It begins at \( x = 0 \), where \( f(x) = 10 - 10 \times (0.6)^0 = 0 \). As x grows, \( (0.6)^x \) gets smaller, making the entire \( 10 \times (0.6)^x \) term diminish, drawing \( f(x) \) closer to 10.
Graphing tools can be beneficial in analyzing the graph's shape and verifying the asymptotic value \( y = 10 \).

• Initial steps: Plot several values for \(x\) and calculate \(f(x)\)
• Observe how the graph flattens as \(x\) increases
• Confirm that the curve approaches horizontal line \(y = 10\)

By seeing the function in graphical form, students can better understand the relationship between the series sum and the horizontal asymptote, making abstract concepts more concrete.