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Chapter 2: Problem 94

In calculus the integral of a rational function \(f\) on an interval \([a, b]\) might not exist if \(f\) has a vertical asymptote in \([a, b]\) Find the vertical asymptotes of each rational function. $$f(x)=\frac{6 x-2 x^{2}}{x^{3}+x} \quad[-1,1]$$

Short Answer

Expert verified
The vertical asymptote is at \( x = 0 \) within the interval \([-1, 1]\).

Step by step solution

01

Identify the Rational Function Components

The given rational function is \( f(x) = \frac{6x - 2x^2}{x^3 + x} \). Identify the numerator as \( 6x - 2x^2 \) and the denominator as \( x^3 + x \). For vertical asymptotes, we focus on where the denominator equals zero since division by zero is undefined.
02

Factor the Denominator

Start by factoring the denominator. \( x^3 + x \) can be factored by taking out a common factor of \( x \), giving us \( x(x^2 + 1) \). The factorization helps to find the points where the denominator equals zero.
03

Solve for Zero in the Denominator

Set the denominator equal to zero: \( x(x^2 + 1) = 0 \). This results in two equations: \( x = 0 \) and \( x^2 + 1 = 0 \). Since the second equation \( x^2 + 1 = 0 \) yields no real solutions (as \( x^2 = -1 \) which is impossible for real numbers), the only solution is \( x = 0 \).
04

Determine the Vertical Asymptotes within the Interval

The potential vertical asymptote at \( x = 0 \) should be considered within the interval \([-1, 1]\). Since \( x = 0 \) is within this interval and it makes the denominator zero, \( x = 0 \) is a vertical asymptote for the function on \([-1, 1]\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Functions
Rational functions are a fascinating type of function in mathematics. They're called "rational" because they are the ratio, or fraction, of two polynomials. Generally, a rational function is expressed as \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.These functions are important because they can model a wide range of real-world scenarios, from economics to physics. The behavior of rational functions is largely determined by their numerators and denominators. The numerator can influence the function's zeros, while the denominator reveals critical points like vertical asymptotes.**How Rational Functions Behave**
  • The degree of the numerator and denominator can significantly affect the graph's shape.
  • Vertical asymptotes occur where the denominator is zero (as long as these zeros aren't canceled by factors in the numerator).
  • Horizontal asymptotes depend on the degrees of the polynomials involved.
Denominator Zero
When it comes to rational functions, setting the denominator equal to zero is one of the first steps to identify potential vertical asymptotes. A vertical asymptote happens when a function approaches infinity as the input nears a certain value, which occurs when dividing by zero in a rational expression.In the exercise example, the denominator is \( x^3 + x \). We solve for the values of \( x \) that make this expression zero:- Solving \( x^3 + x = 0 \) involves factoring, which yields \( x(x^2 + 1) = 0 \).- The solutions to \( x(x^2 + 1) = 0 \) are found by setting each factor to zero.**Why Denominator Zero is Essential**
  • A zero denominator indicates a point where the function could potentially have a vertical asymptote.
  • Determining these points helps in sketching the graph and understanding the function's behavior.
  • Vertical asymptotes are key characteristics that influence calculus operations like integration over intervals.
Factoring Polynomials
Factoring polynomials is a crucial step in finding the points where the denominator of a rational function equals zero. This step simplifies complex expressions and uncovers potential asymptotes or intersect points.In our given exercise, the denominator polynomial \( x^3 + x \) was factored by identifying common factors. The factorization of \( x^3 + x \) is done by extracting the common factor \( x \), resulting in:\[ x^3 + x = x(x^2 + 1) \]This step simplifies the polynomial and helps us solve for \( x \) by setting each factor equal to zero. **The Importance of Factoring**
  • It simplifies polynomials, making them easier to work with and understand.
  • Assists in finding solutions to polynomial equations, crucial for understanding asymptotic behavior.
  • It's especially important for determining vertical asymptotes in rational functions.

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