Problem 279 - Calculus Volume 1

Question

For the following exercises, find the horizontal and vertical asymptotes. $$ f(x)=\frac{1}{x^{3}+x^{2}} $$

Step-by-Step Solution

Answer

Vertical asymptotes at \(x = 0\) and \(x = -1\); horizontal asymptote at \(y = 0\).

Step 1: Simplify the function

Factor the denominator of the function. Given that \(f(x) = \frac{1}{x^3+x^2}\), the denominator can be factored as \(x^2(x+1)\). So, our function becomes \(f(x) = \frac{1}{x^2(x+1)}\).

Step 2: Find the vertical asymptotes

Vertical asymptotes occur where the denominator equals zero, leading the function to become undefined. Set \(x^2(x+1) = 0\). Solving this gives \(x^2 = 0\) or \(x+1 = 0\). Therefore, the vertical asymptotes are at \(x = 0\) and \(x = -1\).

Step 3: Find the horizontal asymptote

Examine the degrees of the polynomial. The degree of the numerator is 0, and the degree of the denominator is 3. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\). This means that as \(x\) approaches infinity, \(f(x)\) approaches 0.

Key Concepts

Asymptotes

Asymptotes are lines that a graph approaches but never touches. In calculus, we often talk about vertical and horizontal asymptotes. These are crucial for understanding how functions behave as they reach extreme values.

• Vertical Asymptotes: Occur when the function is undefined, typically when the denominator of a rational function equals zero. For example, in the function \(f(x) = \frac{1}{x^2(x+1)}\), the points at which the denominator is zero, \(x = 0\) and \(x = -1\), are vertical asymptotes.
• Horizontal Asymptotes: Describe the behavior of a function as \(x\) approaches positive or negative infinity. In this scenario, the horizontal asymptote is found by comparing the degrees of the polynomials in the numerator and the denominator. If the numerator's degree is less than the denominator's, as it is in \(\frac{1}{x^2(x+1)}\), the horizontal asymptote is \(y = 0\).

Understanding asymptotes allows us to sketch the graph of a function and predict its behavior without requiring a precise point-to-point plot.

Rational Functions

Rational functions are quotients of two polynomials, such as \(f(x) = \frac{1}{x^3 + x^2}\). Like fractions, these functions can be simplified and analyzed to reveal interesting characteristics.

• Rational functions can have complex behaviors because they can involve factors in both their numerators and denominators. This particular rational function simplifies to \(f(x) = \frac{1}{x^2(x+1)}\).
• The degree of a polynomial is the highest power of \(x\) that appears. In the function \(\frac{1}{x^2(x+1)}\), the numerator has a degree of 0, and the denominator has a degree of 3, conveying how rapidly the function changes as \(x\) varies.

These functions are foundational in calculus and help students understand more complex function behaviors, such as those involving limits and asymptotes. Rational functions can be found in various real-world applications such as physics, engineering, and economics.

Function Analysis

Function analysis involves examining the properties and behaviors of functions to gain insights and solve mathematical problems. Here, examining \(f(x) = \frac{1}{x^3+x^2}\) reveals unique features due to the rational structure.

• Finding asymptotes is part of analyzing a function, providing clues on the function's end behavior and points where it isn't defined.
• Simplification and factoring are crucial steps in function analysis, as they reduce complexity and help in identifying critical points such as intercepts and asymptotes.
• Knowing the degrees of the polynomials gives insights into the function's growth rate compared to others, enabling predictions about the graph's shape.

Analysis equips students with the skills to deduce information about a graph and handles advanced calculus tasks involving limits, continuity, and differentiability.