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(a) Find the $x$ -intercept(s); (b) Find the vertical asymptotes; (c) Find the horizontal asymptotes. (d) Sketch the graph.$$f(x)=x /(2 x-3)$$

(a) $x=0$(b) $x=3 / 2$(c) $y=1 / 2$(d)

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 4

Limits at Infinity, Infinite Limits and Asymptotes

Derivatives

Oregon State University

Harvey Mudd College

Baylor University

University of Nottingham

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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(a) Find the $x$ -intercep…

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(a) Determine the $x$ -int…

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For the graph of $y=f(x)$<…

the function X divided by two X minus three. We're going to find the X intercept vertical ascent to and horizontal asuntos that we can sketch this graph. So starting with part A let's find the X intercept. So we just need to take our function and set it equal to zero and then solve for X. So X divided by two, X minus three equal to zero. Pretty straightforward. All we have to do is multiply both sides by that denominator and we can see pretty quickly that our X intercept occurs when X is equal to zero. So let's go ahead and plot that on our graph. Part B here. These vertical lasts until it's really we want to pay attention to our denominator. So two X -3. We set that equal to zero will be able to solve for X. And that'll give us that ass and totes. So we have two. X is equal to three, Divide both sides by two and we have X is equal to 3/2 or 1.5. So that will sit right in here. And we can go ahead and sketch a line for us to give ourselves an idea of where that vertical ass and toad is that line which we're not going to want to cross when we start to sketch a graph. And for see we want a horizontal Allison Tolman here, we just need to pay attention to our dominant terms. So in the numerator that's going to be ex that's pretty straightforward, is the only one that we have and in the denominator we have two X. Going ahead and simplify in this, our exes are going to cancel out, but obviously we still have this one right here. So you can see that when y is equal to one half. That gives us our horizontal Assen told. So we can sketch that in here as well. Now to get a better idea of where the actual turning points of our graph are. I like to plug the function into my calculator just to ensure that I actually did this correctly the way of sort of checking my work and more accurately sketching this. But what you'll see is that the graph does follow the basic guidelines we gave ourselves right, it's going to stick to these, assume totes and it's going to cross through that intercept that we found. And then we have another point that sits or another line right up in there so you can see we really are following these general guidelines. We gave ourselves the graphing calculator just tends to be helpful to see where those turning points actually sit.

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