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Magazine Chapter 3: Problem 71
Graph the rational function and find all vertical asymptotes, \(x\)- and \(y\)-intercepts, and local extrema, correct to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$y=\frac{x^{5}}{x^{3}-1}$$
Short Answer
Expert verified
Vertical asymptote: \(x = 1\); \(x\)- and \(y\)-intercept: \((0,0)\); Polynomial: \(x^2\).
Step by step solution
01
Factor and Set Up the Function
First, we'll identify any factors that can make the function undefined, as these factors will help us find vertical asymptotes. Given the rational function:\[y = \frac{x^5}{x^3 - 1}.\]Let's factor the denominator. Notice that the denominator is of the form \(x^3 - 1\), which can be factored using the difference of cubes:\[x^3 - 1 = (x-1)(x^2 + x + 1).\]The function is now \(\frac{x^5}{(x-1)(x^2 + x + 1)}.\)
02
Find the Vertical Asymptotes
To find the vertical asymptotes, set the factored denominator equal to zero and solve for \(x\):1. \(x - 1 = 0 \Rightarrow x = 1\).2. \(x^2 + x + 1 = 0 \), which does not have real roots (this is confirmed by the discriminant \(b^2 - 4ac = 1^2 - 4\times1\times1 = -3\), which is negative).So, the vertical asymptote is at \(x = 1\).
03
Find the x- and y-Intercepts
To find the \(x\)-intercepts, set the numerator equal to zero:\[x^5 = 0 \Rightarrow x = 0.\]Thus, \(x = 0\) is an \(x\)-intercept.For the \(y\)-intercept, plug in \(x = 0\) into the function:\[y = \frac{0^5}{0^3 - 1} = 0.\]Therefore, the \(y\)-intercept is \(y = 0\), occurring at the same point as the \(x\)-intercept.
04
Find Local Extrema
To find the local extrema, we'll differentiate the function and find critical points by setting the derivative equal to zero. First, use the quotient rule:\[\frac{dy}{dx} = \frac{(x^{3}-1)\cdot(5x^4) - x^5\cdot(3x^2)}{(x^3 - 1)^2}.\]Simplify the numerator:\[= \frac{5x^7 - 5x^4 - 3x^7}{(x^3 - 1)^2} = \frac{-3x^7 + 5x^4}{(x^3 - 1)^2}.\]Set the numerator (critical point) to zero:\[-3x^7 + 5x^4 = 0 \Rightarrow x^4(-3x^3 + 5) = 0.\]So, \(x = 0\) and critical points when solving \(-3x^3 + 5 = 0\):\[x^3 = \frac{5}{3} \Rightarrow x = \sqrt[3]{\frac{5}{3}} \approx 1.145.\]
05
Long Division for End Behavior
To determine the polynomial with the same end behavior, perform long division of: \(x^5 \div (x^3 - 1)\).1. Divide \(x^5\) by \(x^3\) to get \(x^2\).2. Multiply \(x^2\) by \(x^3 - 1\) to get \(x^5 - x^2\).3. Subtract \(x^5 - x^2\) from \(x^5\) to get \(x^2\).4. Divide \(x^2\) by \(x^3\) to get a remainder of \(x^2\).So, the polynomial with the same end behavior is \(x^2\) because the remainder becomes insignificant as \(x\) approaches infinity.
06
Graph Both Functions
Graph \(y = \frac{x^5}{x^3 - 1}\) and \(y = x^2\). Ensure the viewing window is large enough, for example, from \(-10\) to \(10\) for both \(x\) and \(y\).Verify that as \(x\) approaches infinity or negative infinity, both graphs appear to approach the same paths, confirming their end behavior is similar.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vertical Asymptotes
In rational functions, vertical asymptotes occur where the function is undefined. These are found by setting the denominator equal to zero. For the function \[y = \frac{x^5}{x^3 - 1},\]we factor the denominator using the difference of cubes formula. This gives us:\[x^3 - 1 = (x-1)(x^2 + x + 1).\]Setting each factor to zero, we solve for:
- \(x - 1 = 0 \Rightarrow x = 1.\)
- \(x^2 + x + 1 = 0,\) which has no real solutions due to a negative discriminant.
x-Intercepts
Finding the \(x\)-intercepts of a rational function involves setting the numerator equal to zero and solving for \(x\). In our example,\[x^5 = 0,\]leads to the solution:\(x = 0.\)This means that the graph of the function crosses the x-axis at the point \((0, 0)\). If the numerator is zero, the overall function value is zero, thus creating an \(x\)-intercept. This implies that we need to carefully solve the numerator to find each intercept.
y-Intercepts
The \(y\)-intercepts of a function are determined by setting \(x\) equal to zero and evaluating the function. Here, plugging in \(x = 0\) into our rational function gives:\[y = \frac{0^5}{0^3 - 1} = 0.\]Thus, the \(y\)-intercept is \(y = 0\), which occurs at the point \((0, 0)\), the same as the \(x\)-intercept. Any intersection at the origin implies that both \(x\) and \(y\)-intercepts are zero.
Local Extrema
Local extrema in rational functions are found by taking the derivative and setting it equal to zero to locate critical points. For the function\[\frac{dy}{dx} = \frac{(x^{3}-1)\cdot(5x^4) - x^5\cdot(3x^2)}{(x^3 - 1)^2},\]we simplify the numerator to:\[-3x^7 + 5x^4 = 0,\]resulting in:\(x^4(-3x^3 + 5) = 0.\)From this, we get critical points at \(x = 0\) and \(x = \sqrt[3]{\frac{5}{3}} \approx 1.145.\) Checking these points helps determine local maxima or minima through second derivative tests or the first derivative test. These points can signal changes in the slope of the tangent to the function's curve.
End Behavior
The end behavior of a rational function shows how the function behaves as \(x\) becomes very large or very small. Often, end behavior is revealed by dividing the numerator by the denominator using polynomial long division. In our function\[y = \frac{x^5}{x^3 - 1},\]carrying out the polynomial division gives us\[x^2.\]This solution implies that as \(x\) heads towards positive or negative infinity, the function behaves similarly to \(y = x^2.\) Both graphs in the viewing window confirm that they mirror one another at extremes, highlighting the identical end behaviors of \(y = x^2\) and the original function.
Long Division in Polynomials
Long division in polynomials is applied when simplifying rational expressions to understand their asymptotic behavior. To execute this, divide the numerator by the denominator as if they were numerical long division:
- Divide the leading term of the numerator by the leading term of the denominator.
- Multiply the entire divisor by the quotient term.
- Subtract this result from the original numerator.
- Repeat until no further division is possible.
