Descibe The and lehovior 4,2x-4 + 20x + 2X
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![## Describe the End Behavior of the following functions:
### 1. Polynomial Function
\[ y = -7x^5 + 20x^4 + 2x - 4 \]
### 2. Rational Function
\[ y = \frac{-5x^3 + 2x + 1}{3x + 2} \]
### Explanation of End Behavior
**1. Polynomial Function:**
The given polynomial function is:
\[ y = -7x^5 + 20x^4 + 2x - 4 \]
To determine the end behavior, focus on the term with the highest degree, which is \( -7x^5 \). As \( x \) approaches infinity (\( x \to \infty \)) or negative infinity (\( x \to -\infty \)), the term \( -7x^5 \) will dominate the behavior of the polynomial.
- As \( x \to \infty \), \( y \to -\infty \) (since \( -7x^5 \to -\infty \)).
- As \( x \to -\infty \), \( y \to \infty \) (since \( -7(-x)^5 \to \infty \)).
**2. Rational Function:**
The given rational function is:
\[ y = \frac{-5x^3 + 2x + 1}{3x + 2} \]
To determine the end behavior of this rational function, examine the degrees of the numerator and denominator.
- The degree of the numerator \( -5x^3 + 2x + 1 \) is 3.
- The degree of the denominator \( 3x + 2 \) is 1.
When the degree of the numerator is greater than the degree of the denominator, the end behavior resembles that of the quotient of the leading terms. Therefore, simplify the highest-degree terms \( \frac{-5x^3}{3x} \) to find the end behavior:
\[ y \approx \frac{-5x^3}{3x} = -\frac{5}{3}x^2 \]
- As \( x \to \infty \), \( y \to -\infty \) (since \( -\frac{5}{3}x^2 \to -\infty \)).
- As \( x \to -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F180e35fb-9b09-4fc5-8a6a-511bee286828%2Fe49584b6-3943-4dc6-8a68-3adf34b124f5%2Fohqbk48_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Describe the End Behavior of the following functions:
### 1. Polynomial Function
\[ y = -7x^5 + 20x^4 + 2x - 4 \]
### 2. Rational Function
\[ y = \frac{-5x^3 + 2x + 1}{3x + 2} \]
### Explanation of End Behavior
**1. Polynomial Function:**
The given polynomial function is:
\[ y = -7x^5 + 20x^4 + 2x - 4 \]
To determine the end behavior, focus on the term with the highest degree, which is \( -7x^5 \). As \( x \) approaches infinity (\( x \to \infty \)) or negative infinity (\( x \to -\infty \)), the term \( -7x^5 \) will dominate the behavior of the polynomial.
- As \( x \to \infty \), \( y \to -\infty \) (since \( -7x^5 \to -\infty \)).
- As \( x \to -\infty \), \( y \to \infty \) (since \( -7(-x)^5 \to \infty \)).
**2. Rational Function:**
The given rational function is:
\[ y = \frac{-5x^3 + 2x + 1}{3x + 2} \]
To determine the end behavior of this rational function, examine the degrees of the numerator and denominator.
- The degree of the numerator \( -5x^3 + 2x + 1 \) is 3.
- The degree of the denominator \( 3x + 2 \) is 1.
When the degree of the numerator is greater than the degree of the denominator, the end behavior resembles that of the quotient of the leading terms. Therefore, simplify the highest-degree terms \( \frac{-5x^3}{3x} \) to find the end behavior:
\[ y \approx \frac{-5x^3}{3x} = -\frac{5}{3}x^2 \]
- As \( x \to \infty \), \( y \to -\infty \) (since \( -\frac{5}{3}x^2 \to -\infty \)).
- As \( x \to -
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