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(a) Determine the $x$ -intercepts, (b) the vertical asymptotes, (c) the horizontal asymptotes and (d) Sketch the graph of the function whose equation is $f(x)=\frac{5 x^{2}-3}{2 x^{5}-3 x+2} \cdot$ You'll need to use your calculator to determine the vertical asymptotes.

(a) $x=\pm \frac{\sqrt{15}}{5}$(b) $x=-1.23297$(c) $y=0$$(d)$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 4

Limits at Infinity, Infinite Limits and Asymptotes

Derivatives

Missouri State University

Harvey Mudd College

Idaho State University

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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For the graph of $y=f(x)$<…

in this problem, we are given the function of F of X. And we're asked to find a few different things. So the first thing we're looking for is X intercepts. So whatever we're hitting an X intercept, that's where Y equals zero. So we can solve this by looking at zero equals five X squared minus three Over two X. to the 5th -3 x plus two. And that's gonna look daunting if you're looking at it this way, what Think about the fact that for why we're zero? We want this to be zero over some number If our denominator zero is just undefined. So really we're looking at 10 equals five X squared minus three. So we'll do some algebra here at R three over then we'll divide five. So we have 3/5 equals X squared. And now we just want to take the square root, We have x equals square root of three over five. And if you put that on your calculator you can get an estimation and this is positive and negative So it's about positive and negative .7746. So that's gonna be your two x in ourselves. And next we're looking for vertical ascent tops and that's where our functions undefined or where the denominator equals zero. So we're looking at zero equals two X. To the fifth minus three X plus two. And you're going to use your calculator to help you solve this year. So we do that. We get x equals native 1.23297. That's our vertical S in tow. And we're also asked for a horizontal a symptom and That one is a little more difficult to remember and to do. So what we do with that? So we have our original function F. Of X. We'll write this out first. And what would do to find a horizontal S into? We take the highest powered variable. So on the top we have five X squared. And in the dominator our highest exponents is the five. So we have to accept the fifth. And we simplify this so we have five over to execute and you think, okay, so if it's bottom heavy or the variable is only left in the denominator, consider the fact that as X gets higher and higher, our denominator gets bigger and bigger. So we approached zero. So this is just me, zero. Why Go? zero is going to be our horizontal ass into now that we have all this information, whereas to graph this function and this can be a little tricky. But with the information we have we can at least get a rough idea of what we're looking at. So we have X intercepts, put one there at 0.77 so we'll call that approximately there and we have that in the negative too. So That's gonna be a zero. Not race these. So we don't get too confused. All right. And we have a vertical as in tow. But why X equals negative 1.23 three. So, we'll put that about here And we have horizontal asientos at y equals zero that's here. Okay? So what we're looking at here, we're not going to want to cross this. So we have some kind of function over on this side of things. And if you want to just plug in a number into our functions, sometimes that can help you. So if you do that, you see it gets more negative. So it's going to kind of like that because we have our ascent at Y equals zero and X equals negative 1.2. And if you graph negative one or plug into your function, you'll see that it's a positive number. So we're going to have something starting up there and we know we cross across the X axis here and we know there's another zero here. So we come up through, but then we still have an awesome tote at Y equals zero. That's going to come back down. It's just going to approach zero like that. So, again, probably not going to be your prettiest sketch of a graph. But with the information we have and plugging a few points, you can get a general idea of what you're looking at.

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