Problem

Use the guidelines of this section to sketch the …

Problem 9 Easy Difficulty

Use the guidelines of this section to sketch the curve.
$$
y=\frac{x}{x-1}
$$

Answer

$y^{\prime \prime}<0$ When $-\infty<x<1$ so $y$ is concave downward on $(-\infty, 1)$ .
$y^{\prime \prime}>0$ When $1<x<\infty$ so $y$ is concave upward on $(1, \infty)$

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Calculus 1 / AB

Essential Calculus Early Transcendentals

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 4

Curve Sketching

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Video Transcript

to find some of the specs for this function. Here we'll start off the domain that includes all real numbers except X cannot equal zero because then we end up with an undefined value. The intercept? Um, it's just one x zero, and that will result in why being zero um, symmetry should have some type of odd symmetry here. And Assam totes. Uh, that's when we'll vertical Assam totes or when you divide by zero. So that happens at X is warm horizontal Assen totes or when you take the limit as X approaches infinity. And that's where you look at the ratio of these two in this case, and that would be 1/1. And so the horizontal listen to it is why most one now in terms of decreasing, increasing, Um, let's go ahead and take the derivative using the quotient rule. So we start off with low D Hi minus Hi, Do you love all over the bottom square? The X's cancel, and so this actually leaves us just with negative one over X minus one square. So, really, the intervals of increasing decreasing just depend on the fact is greater or less than one here. And it turns out that no matter what, the bottom is always positive. And so the tops always negative. So this function is actually just always going to be decreasing, always decreasing. So there's no local Max or men or any of that. And then finally, if we want to find out about the con cavity on, we want to take the second derivative. So that'd be why Double Prime is equal to again. Taking the derivative of this function will do lo de high except the derivative of the top of zero. So that would be zero minus. Hi. De lo would be to X minus one now to the power one times one all over X minus one to the fourth. Let's see how this shapes up. So this is going thio equal. Um, double check here. Okay, so this is Yeah, this is just going to give us too X and then minus two all over X minus one. So the fourth, and so in this case, we have some Okay, a possible change in concave ity at X s one. And then if we verify that by plugging values in here, notice if we plug in something less than Well, let's start with greater than one. We're going to end up with positive values. Actually, let me go ahead and just write. It is a a plus sign instead of, um so positive Con cavity or concave up like that. And then to the left. Negative Sakon cape down. So at this point, we have everything we need. Uh, yeah. So as we begin to plot, one of the first things we could do is just plot, For instance, are vertical aspecto at X s one. We also have a horizontal ASEM toe that why is one so that kind of divides this up a little bit, So it's easier to see. Um, we do end up passing through the origin. 00 Here, let's graphic in blue And so we'll come down con cave down as we knew, um, con cave up and it's always decreasing. So no matter where you are, this craft always decreasing. Onda Gennett goes off in both directions to y equals one

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