Problem 107 - Calculus Early Transcendentals

Question

In the following exercises, two sequences are given, one of which initially has smaller values, but eventually "overtakes" the other sequence. Find the sequence with the larger growth rate and the value of \(n\) at which it overtakes the other sequence. $$a_{n}=e^{n / 2} \text { and } b_{n}=n^{5}, n \geq 2$$

Step-by-Step Solution

Answer

Answer: The sequence \(a_n = e^{\frac{n}{2}}\) has a larger growth rate and overtakes the sequence \(b_n = n^5\) when the value of n is approximately 8.

Step 1: Identify the Growth Rates of Both Sequences

To find the growth rate of each sequence, we can examine their derivatives. For the sequence \(a_n = e^{\frac{n}{2}}\), we use the power rule for derivatives to find: \(\frac{d(a_n)}{dn} = \frac{1}{2}e^{\frac{n}{2}}\) For the sequence \(b_n = n^5\), we use the power rule for derivatives again to find: \(\frac{d(b_n)}{dn} = 5n^4\) Now, we have the growth rates for both sequences.

Step 2: Compare the Growth Rates

To determine which sequence has the larger growth rate, we need to compare the growth rates we found in Step 1. We have: \(\frac{1}{2}e^{\frac{n}{2}}\) and \(5n^4\) We can analyze their behavior by examining these growth rates as \(n \to \infty\). As \(n\) grows, the exponent in the exponential sequence becomes larger, whereas the power of \(n^4\) only affects the rate of growth for the \(n^4\) term. Therefore, eventually, the exponential growth of \(e^{\frac{n}{2}}\) will outpace the polynomial growth of \(n^5\). Thus, the sequence with the larger growth rate is the \(a_n = e^{\frac{n}{2}}\).

Step 3: Determine the Value of \(n\) When One Sequence Overtakes the Other

To find the value of \(n\) when the exponential sequence overtakes the polynomial sequence, we need to solve the inequality: \(e^{\frac{n}{2}}> n^5\) To find the value of \(n\) that satisfies this inequality, we can use numerical methods, as there is no simple analytical solution to this inequality. Trial and error or graphing could be used to find the required value of \(n\). Upon trying different values of \(n\), we can find that when \(n \approx 8\), the inequality holds: \(e^{\frac{8}{2}} \approx 54.6 > 8^5 \approx 32.8\) Therefore, the value of \(n\) at which the \(a_n\) sequence overtakes the \(b_n\) sequence is approximately \(n = 8\).

Key Concepts

Exponential Growth

Exponential growth is a concept that describes a situation where the rate of change of a quantity is directly proportional to the current value of that quantity. In simpler terms, as the quantity grows, its rate of growth accelerates. This type of growth is characterized by the exponential function, such as \(e^{\frac{n}{2}}\) in our sequence \(a_n\).

• Exponential functions grow faster than linear or polynomial functions as \(n\) increases.
• In the function \(e^{\frac{n}{2}}\), \(n\) appears in the exponent, leading to rapid increases in \(a_n\) as \(n\) becomes larger.
• Exponential growth often models real-world scenarios, such as population growth, radioactive decay, and interest compounding.

In our exercise, we see that exponential growth enables the sequence \(a_n\) to eventually surpass the polynomial sequence \(b_n\), illustrating just how powerful exponential functions can be.

Polynomial Growth

Polynomial growth refers to sequences or functions where the variable \(n\) is raised to a power, like in the sequence \(b_n = n^5\). Such functions grow at a rate that depends on the degree of the polynomial, which is the highest exponent of \(n\).

• Polynomials of higher degrees (like \(n^5\)) grow faster than those of lower degrees (such as \(n^2\) or \(n^3\)).
• The growth is not as rapid as exponential growth, since \(n\) only serves as a base, not an exponent.
• Polynomial growth is common in areas such as physics, where certain phenomena need to be approximated over large inputs.

In our scenario, \(b_n = n^5\) begins larger than \(a_n\) but is set to be overtaken due to the accelerating pace of exponential growth. This highlights the limitations of polynomial growth over long intervals when compared to exponential growth.

Sequence Comparison

When comparing sequences, especially polynomial and exponential ones, it's essential to consider their growth rates and behaviors as \(n\) becomes large. This comparison helps us understand which sequence will eventually have larger terms.

• Start by determining the growth rate of each sequence, either via calculus tools like derivatives or by analyzing the function forms.
• Check asymptotic behavior, meaning how sequences behave as they approach infinity.
• Consider real-world contexts: exponential growth quickly outpaces polynomial growth, which is seen when \(a_n\) overtakes \(b_n\) in our example.

By understanding these comparisons, students can solve problems involving long-term behaviors of sequences, predicting when one sequence will surpass another.

Overtaking Sequences

Overtaking sequences occur when one sequence surpasses another as \(n\) increases. Initially, one sequence might have larger values, but due to different growth rates, roles can reverse.

• In our example, \(b_n = n^5\) starts with larger values but is eventually surpassed by the exponentially growing \(a_n = e^{\frac{n}{2}}\).
• Identifying the point of overtaking involves solving inequalities, such as \(e^{\frac{n}{2}} > n^5\).
• Naming the crossover point offers insight into various applications, like when investments might surpass costs or population surpasses sustainability thresholds.

For exercise problems, finding the exact \(n\) where sequences overtake each other may involve numerical methods or graphing, highlighting the practical applications of mathematics in real scenarios.