Problem 39 - Single Variable Calculus

Question

In Exercises 39-42, find the slant asymptotes of the graphs of the function. Then sketch the graph of the function. $$ g(u)=\frac{u^{3}+1}{u^{2}-1} $$

Step-by-Step Solution

Answer

The slant asymptote of the function \(g(u) = \frac{u^3+1}{u^2-1}\) is the line \(y = u\). A rough sketch of the graph would show the function approaching the slant asymptote as the x-value goes to positive or negative infinity, but not crossing it. There are no additional vertical asymptotes or x-intercepts.

Step 1: 1. Perform Polynomial Long Division

We will divide the numerator \(u^3+1\) by the denominator \(u^2-1\): $$ \require{enclose} \begin{array}{r} u \enclose{longdiv}{u^3\phantom{ +0u^2} +0u \phantom{-0}+1} \\[-2pt] u^2-1 \rlap{\;\begin{array}{|@{}l} \\ u^3- u^2u + u^3 + u \\ \\ {-}\phantom{^1}1u^3 \\ \cline{1-1} \\[-8pt] 0u^3 +0u^2 +0u +1 \\ [-6pt] \end{array}} \end{array} $$ The quotient is \(u\).

Step 2: 2. Identify the Slant Asymptote

Since our quotient is \(u\) without any remainder, the slant asymptote of the function \(g(u) = \frac{u^3+1}{u^2-1}\) is the line \(y = u\).

Step 3: 3. Sketch the Graph

To sketch the graph of \(g(u) = \frac{u^3+1}{u^2-1}\), we need to consider the slant asymptote at \(y = u\) and that there are no additional vertical asymptotes or x-intercepts. 1. Draw the line \(y = u\). This line is the slant asymptote. 2. The function will approach the slant asymptote as the x-value goes to positive or negative infinity, but will not cross it. 3. There are no additional points to plot. Here's a rough sketch of the graph: [asy] import graph; size(200); real g(real u) { return (u^3 + 1)/(u^2 - 1); } path slantasymptote = ((-3,-3)--(5,5)); draw(graph(g, -2.5, -1.5, operator ..), red, "g(u)"); draw(graph(g, 1.5, 4.3, operator ..), red); draw(slantasymptote, dashed, "y=u"); limits((-3, -20), (5, 20), Crop); xaxis("\(u\)", Ticks(Label(fontsize(10pt)), new real[]{}, pTick=black+linewidth(1)), above=true); yaxis("\(y\)", Ticks(Label(fontsize(10pt)), new real[]{}, pTick=black+linewidth(1)), above=true); label("Slant Asymptote",(2.5,6),align=N); [/asy]

Key Concepts

Polynomial Long Division

Understanding polynomial long division is crucial for simplifying complex rational expressions and identifying their asymptotic behavior. Imagine dividing one polynomial by another, much like traditional numerical division. Here's how you can approach it:

First, you organize the division in a layout similar to long division with numbers. The goal is to determine how many times the divisor polynomial can