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(a) Find the $x$ -intercept(s); (b) Find the vertical asymptotes; (c) Find the horizontal asymptotes. (d) Sketch the graph$$f(x)=\frac{(3 x-2)(2 x+3)^{2}}{x^{2}(5 x-3)}$$

(a) $x=-3 / 2,2 / 3$(b) $x=0,3 / 5$(c) $y=12 / 5$$(d)$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 4

Limits at Infinity, Infinite Limits and Asymptotes

Derivatives

University of Michigan - Ann Arbor

University of Nottingham

Boston College

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 $y=f(x)$<…

given the function that you can see written up top here, we're going to find its X intercepts and ask them to vote so that we can sketch it at the end. So for part A we're gonna be finding these intercepts. We can do that by setting our function equal to zero and you can see if we multiply both sides by that denominator, what we're left with is only our numerator. So these two pieces of top three x minus two and two X plus three squared. So we're going to do is just set each one of those individually equal to zero, three x minus two equals zero and two X plus three squared. But that equal to zero. And then we can just solve for X within each one of these on their own. As you can see that the one on the left hand side is going to give us that X is equal to two thirds. And the second one here will give us X is equal to a negative three halves. So we can go ahead and sketch is now on our graph over here. So we have two thirds which will sit somewhere in here and a negative three halves which is negative 1.5. We'll put that one right there. That was part A let's go ahead and move on to part B now where we want to find a vertical ass until it's and we're finding vertical ass until it's really what we're interested in is our denominator. So we have X squared times five x minus three. We can just set that equal to zero, solve for X. And again, we're gonna split this up to have X squared is equal to zero and five X minus three is equal to zero. Which lets us solve for our as in totes, we can see here that X one equals zero. And over here we've got X will be equal to 3/5. So going ahead and sketching these again, we have X is equal to zero. So that's gonna sit right here on our Y axis. No other one will sit somewhere right in here. Alright, now part C. We want to find our horizontal lesson codes and to do that, we know we need to find the limit of our function as X is approaching infinity and we're doing this limit as X is approaching infinity. What we're specifically interested in is our dominant terms in both the numerator and the denominator. So it's going to be that term with the highest power. So in the numerator, what we can deduce right here is that we're going to end up squaring the two X. So when that gets square, that will give us four X squared. And then that eventually that four X squared will get multiplied by three X. Eventually which would give us four X squared times three X. Gives us 12 X cubed. And that would be our highest term or dominant term then the numerator. And you could factor or multiply that whole numerator out if you wanted. And you would see to that it would be 12 X 12 X cubed. And now on the denominator, if we look at this, we're gonna end up multiplying X squared times five X. Which would give us five X cubed. Looking at this, it's pretty obvious to see that our excuse will cancel out. And that's going to leave us with just 12/5. So we can see we have a horizontal esposito at Y equals 12 5th now 12 50 that's about to just over. So that's going to sit somewhere right in here and now we have the basic outline that we need to graph this or to sketch it, but we don't exactly know where all of the turns occur within this function. So go ahead and graph it in your graphing calculator that we just have a better visual idea, slightly more accurate one and we can go ahead and sketch this now and it looks something like this. Think we right, something like that. That's a little it's not my best sketch, but you get the idea, it does follow the asientos that we identified as well as the intercepts.

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