Problem 15 - Calculus

Question

In Exercises \(11-16,\) which function dominates as \(x \rightarrow \infty ?\) $$20 x^{4}+100 x^{2}+5 x \text { or } 25-40 x^{2}+x^{3}+3 x^{5}$$

Step-by-Step Solution

Answer

The function \(3x^5 + x^3 - 40x^2 + 25\) dominates as \(x \rightarrow \infty\).

Step 1: Identify Leading Terms

First, identify the leading term of each function as the variable approaches infinity. For the first function, \(20x^4 + 100x^2 + 5x\), the leading term is \(20x^4\). For the second function, \(25 - 40x^2 + x^3 + 3x^5\), the leading term is \(3x^5\).

Step 2: Compare Degrees of Leading Terms

Compare the degrees of the leading terms identified in Step 1. The degree of \(20x^4\) is 4, while the degree of \(3x^5\) is 5. Since 5 is greater than 4, \(3x^5\) will grow faster than \(20x^4\) as \(x\) approaches infinity.

Step 3: Determine the Dominating Function

Since \(3x^5\) has a higher degree than \(20x^4\), the function \(3x^5 + x^3 - 40x^2 + 25\) will dominate \(20x^4 + 100x^2 + 5x\) as \(x\) approaches infinity.

Key Concepts

Leading Term

In mathematics, specifically polynomials, the leading term is crucial for understanding how a function behaves as its variables grow larger. - The leading term is the term in the polynomial with the highest degree. This term dictates the end behavior of the polynomial.- For example, in the polynomial \(20x^4 + 100x^2 + 5x\), the leading term is \(20x^4\). This term is dominant because it has the highest degree compared to the other terms.- Similarly, for the polynomial \(25 - 40x^2 + x^3 + 3x^5\), the leading term is \(3x^5\).Understanding which term is dominant helps in predicting how the polynomial will grow compared to others.

Degrees of Polynomials

The degree of a polynomial is a key factor in analysing functions. It is defined as the highest power of the variable present in the polynomial.- Each term in a polynomial has a degree, determined by the exponent of the variable in that term.- For the polynomial \(20x^4 + 100x^2 + 5x\), the term \(20x^4\) has the highest degree, which is 4.- In contrast, the polynomial \(25 - 40x^2 + x^3 + 3x^5\) has a term \(3x^5\) with a degree of 5.This concept is essential when comparing polynomials, as a higher degree often indicates faster growth as \(x\) increases.

Limit as x Approaches Infinity

The limit of a function as \(x\) approaches infinity helps determine how the function behaves at large values of \(x\).- When comparing functions, understanding their limits as \(x\) grows can reveal which function grows faster and eventually dominates.- For polynomials, the leading term heavily influences the limit behavior. It determines the polynomial's growth rate as it heads towards infinity.- For instance, between \(20x^4\) and \(3x^5\), \(3x^5\) has a higher degree and thus dominates because it grows faster as \(x\) goes to infinity.Evaluating the limit by focusing on leading terms is a reliable approach to understanding which function will eventually become more significant.