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Chapter 3: Problem 41

Find the curvilinear asymptote. $$f(x)=\frac{x^{4}-x^{2}-5}{x^{2}-2}$$

Short Answer

Expert verified
The curvilinear asymptote is \( x^2 + 2 \).

Step by step solution

01

Analyze the Degree of Polynomials

Identify the degree of the numerator and the degree of the denominator of the given function \( f(x) = \frac{x^4 - x^2 - 5}{x^2 - 2} \). The degree of the numerator is 4 and the degree of the denominator is 2.
02

Perform Polynomial Division

Since the degree of the numerator (4) is greater than the degree of the denominator (2), perform polynomial long division on \( x^4 - x^2 - 5 \) by \( x^2 - 2 \). Divide the first terms: \( x^4 \) by \( x^2 \) gives \( x^2 \). Multiply the divisor \( x^2 - 2 \) by \( x^2 \) and subtract from the dividend.
03

Continue Polynomial Division

After subtracting, you get a new polynomial: \( 2x^2 - 5 \). Divide \( 2x^2 \) by \( x^2 \) to get \( 2 \). Multiply \( x^2 - 2 \) by \( 2 \), which gives \( 2x^2 - 4 \), and subtract this from \( 2x^2 - 5 \). The remainder is \( -1 \).
04

Write Down the Division Result

Express \( \frac{x^4 - x^2 - 5}{x^2 - 2} \) as the result of the division: \( x^2 + 2 + \frac{-1}{x^2 - 2} \). The quotient \( x^2 + 2 \) forms the curvilinear asymptote of the function.
05

Interpret the Remainder

The remainder \( \frac{-1}{x^2 - 2} \) becomes negligible as \( x \) approaches infinity, thus confirming that the curvilinear asymptote of the function is indeed \( x^2 + 2 \).

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Key Concepts

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

Polynomial Division
Polynomial division is a crucial process, especially when working with rational functions like the one given in this problem. It is similar to the long division of numbers, but instead, we work with algebraic expressions. When the degree of the numerator is higher than the degree of the denominator, we use polynomial division to simplify the expression or find certain characteristics of the function, such as asymptotes.

Here's a simple way to look at it:
  • Identify the terms with the highest powers in both the numerator and denominator.
  • Perform the division of these highest terms, writing down the result.
  • Multiply the entire divisor by this result and subtract from the original polynomial; this gives a new "remainder" polynomial.
  • Repeat this process using the new "remainder" until the degree of the remaining polynomial is less than the degree of the divisor.
By performing these steps, as shown in the solution, we arrive at our quotient and remainder, which is then used to understand the behavior of our function.
Degree of Polynomial
The degree of a polynomial is a key concept in evaluating the behavior of rational functions, especially when determining asymptotes. The degree is simply the highest power of the variable in the polynomial.

Let's break it down:
  • For the numerator, the degree is 4 due to the term \(x^4\).
  • For the denominator, the degree is 2 because of \(x^2\).
Understanding these degrees helps in deciding the approach to take, such as when polynomial division is necessary. Typically, if the numerator's degree exceeds that of the denominator, as in this exercise, it suggests the existence of a curvilinear (or oblique) asymptote. This asymptote can be found by using polynomial division to simplify or restructure the rational function.
Rational Functions
Rational functions are quotients of polynomials. That is, they are defined by taking one polynomial and dividing it by another. These functions can exhibit a wide range of behaviors, including having vertical, horizontal, or curvilinear asymptotes.

Key points about rational functions:
  • The general form is \( \frac{N(x)}{D(x)} \), where \( N(x) \) and \( D(x) \) are polynomials.
  • The degrees of \( N(x) \) and \( D(x) \) influence the type of asymptotes present.
  • If the degree of the numerator is greater than that of the denominator, a curvilinear asymptote may exist, as is the case in this exercise.
Rational functions like \( f(x) = \frac{x^4 - x^2 - 5}{x^2 - 2} \) can have complex structures that reveal interesting asymptotic behavior, especially in large domain values (where \( x \) approaches infinity). Polynomial division helps to identify these asymptotes, providing insights into the function's long-term trends.

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Most popular questions from this chapter

Because of the combustion of fossil fuels, the concentration of carbon dioxide in the atmosphere is increasing. Research indicates that this will result in a greenhouse effect that will change the average global surface temperature. Assuming a vigorous expansion of coal use, the future amount \(A(t)\) of atmospheric carbon dioxide concentration can be approximated (in parts per million) by $$A(t)=-\frac{1}{2400} t^{3}+\frac{1}{20} t^{2}+\frac{7}{6} t+340$$ where \(t\) is in years, \(t=0\) corresponds to \(1980,\) and \(0 \leq t \leq 60 .\) Use the graph of \(A\) to estimate the year when the carbon dioxide concentration will be 450 .

The average monthly temperatures in "F for two Canadian locations are listed in the following tables. $$\begin{aligned} &\begin{array}{|l|l|l|c|c|} \hline \text { Month } & \text { Jan. } & \text { Feb. } & \text { Mar. } & \text { Apr. } \\ \hline \text { Arctic Bay } & -22 & -26 & -18 & -4 \\ \hline \text { Trout Lake } & -11 & -6 & 7 & 25 \\ \hline \end{array}\\\ &\begin{array}{|l|c|c|c|c|} \hline \text { Month } & \text { May } & \text { June } & \text { July } & \text { Aug. } \\ \hline \text { Arctic Bay } & 19 & 36 & 43 & 41 \\ \hline \text { Trout Lake } & 39 & 52 & 61 & 59 \\ \hline \end{array}\\\ &\begin{array}{|l|c|c|c|c|} \hline \text { Month } & \text { Sept. } & \text { Oct. } & \text { Nov. } & \text { Dec. } \\ \hline \text { Arctic Bay } & 28 & 12 & -8 & -17 \\ \hline \text { Trout Lake } & 48 & 34 & 16 & -4 \\ \hline \end{array} \end{aligned}$$ (a) If January 15 corresponds to \(x=1\), February 15 to \(x=2, \ldots,\) and December 15 to \(x=12,\) determine graphically which of the three polynomials given best models the data. (b) Use the intermediate value theorem for polynomial functions to approximate an interval for \(x\) when an average temperature of \(0^{\circ} \mathrm{F}\) occurs. (c) Use your choice from part (a) to estimate \(x\) when the average temperature is \(0^{\circ} \mathrm{F}\) Trout Lake temperatures (1) \(f(x)=-2.14 x^{2}+28.01 x-55\) (2) \(g(x)=-0.22 x^{3}+1.84 x^{2}+11.70 x-29.90\) (3) \(h(x)=0.046 x^{4}-1.39 x^{3}+11.81 x^{2}-22.2 x+1.03\)

Constructing a storage tank A storage tank for propane gas is to be constructed in the shape of a right circular cylinder of altitude 10 feet with a hemisphere attached to each end. Determine the radius \(x\) so that the resulting volume is \(27 \pi \mathrm{ft}^{3}\). (See Example 8 of Section 2.4 .)

The function \(f\) given by \(f(x)=-0.11 x^{4}-46 x^{3}+4000 x^{2}-76,000 x+760,000\) approximates the total number of preschool children participating in the government program Head Start between 1966 and \(2005,\) where \(x=0\) corresponds to the year 1966 (a) Graph fon the interval \([0,40] .\) Discuss how the number of participants has changed between 1966 and 2005 . (b) Approximate the number of children enrolled in 1986 . (c) Estimate graphically the years in which there were \(500,000\) children enrolled in Head Start.

Find an equation of a rational function \(f\) that satisfies the given conditions. vertical asymptotes: \(x=-3, x=1\) horizontal asymptote: \(y=0\) \(x\) -intercept: \(-1 ; f(0)=-2\) hole at \(x=2\)
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