Problem 12 - Calculus

Question

In Exercises \(11-16,\) which function dominates as \(x \rightarrow \infty ?\) $$10 e^{0.1 x} \text { or } 5000 x^{2}$$

Step-by-Step Solution

Answer

The function \(10e^{0.1x}\) dominates as \(x \to \infty\).

Step 1: Understand the Problem

We need to compare the growth rates of two functions: \(f(x) = 10 e^{0.1x}\) and \(g(x) = 5000x^2\), as \(x\) approaches infinity. The task is to find which one grows faster or dominates as \(x \to \infty\).

Step 2: Analyze Exponential Growth

The function \(f(x) = 10 e^{0.1x}\) is an exponential function. Exponential functions grow very rapidly, especially as \(x\) becomes large, because it has a constant base \(e\) raised to a variable exponent, \(0.1x\).

Step 3: Analyze Polynomial Growth

The function \(g(x) = 5000x^2\) is a polynomial function of degree 2. Polynomial functions grow at a rate corresponding to their degree. In this case, the growth is quadratic.

Step 4: Compare Rates of Growth

As \(x\) increases, exponential functions ultimately grow faster than polynomial functions. Even though \(g(x)\) has a large coefficient (5000), the exponential nature of \(f(x)\) ensures that it will eventually exceed \(g(x)\) for sufficiently large \(x\).

Step 5: Conclude which Function Dominates

Because exponential functions always eventually outgrow polynomial functions as \(x\to\infty\), in this case, \(10e^{0.1x}\) dominates \(5000x^2\) as \(x \to \infty\).

Key Concepts

Polynomial Growth

In mathematical terms, polynomial growth refers to functions of the form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\). This expression describes a polynomial of degree \(n\), where \(a_n, a_{n-1}, \ldots, a_0\) are constants and \(x\) is the variable. The degree of the polynomial, which is the highest power of \(x\) present, determines the function's growth rate.
- For instance, a degree 2 polynomial like \(g(x) = 5000x^2\) grows quadratically as \(x\) increases. - This means that the value of \(g(x)\) increases at a rate proportional to \(x^2\).
Polynomials of higher degree will grow faster than those of lower degree as \(x\) tends towards infinity. However, compared to other forms of functions such as exponentials, they have limitations in their growth rates over the long term, especially as \(x\) becomes very large.

Comparison of Functions

When comparing functions like polynomial and exponential, we analyze how rapidly each function's output increases as \(x\) becomes very large. Specifically:

• Exponential functions, such as \(10e^{0.1x}\), grow at a rate where the variable \(x\) is an exponent. This means even small increases in \(x\) can drastically increase the function's value.
• Polynomial functions, like \(5000x^2\), grow according to the variable raised to a constant power, which spreads out the rate of growth more evenly as \(x\) becomes larger.

Understanding the growth characteristics of these functions allows us to predict their long-term behavior. As a rule of thumb:- Exponential functions often dominate polynomial functions as \(x\) approaches infinity.- This is because exponentials can increase rapidly, outpacing even high-coefficient polynomials eventually.

Limits as Infinity

The concept of limits as infinity helps us understand how functions behave as their inputs become very large. In formal terms, this is represented by finding \(\lim_{x \to \infty} f(x)\). This helps determine which function grows faster over time.
When we deal with the comparison of an exponential and a polynomial function:

• The limit of an exponential function like \(10e^{0.1x}\) as \(x\to \infty\) is infinity, since exponential growth is unbounded.
• The polynomial \(5000x^2\) also tends towards infinity, but at a much slower pace compared to exponential growth.

Therefore, by evaluating these limits, we conclude that the exponential function grows more rapidly than the polynomial one. As such, in terms of limits, the exponential function will always "dominate" the polynomial function at infinity.