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(a) Find the $x$ -intercept(s); (b) Find the vertical asymptotes; (c) Find the horizontal asymptotes. (d) Sketch the graph.$$f(x)=4 x^{2}\left(x^{2}-4\right)$$

(a) $x=0$(b) $x=\pm 2$(c) $y=4$$(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

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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(a) Find the $x$ -intercep…

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Graph the function$$f(…

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(a) Determine the $x$ -int…

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this question is walking us through graphing the function F of x is equal to four X squared divided by x squared minus four by finding its intercepts and ask them to do so starting with the X intercepts, we just need to set this entire function equal to zero and solve for X. Multiplying both sides by the denominator leaves us with four, X squared equal to zero, dividing by four. We have X squared is still equal to zero and obviously the squared of zero is just zero. So we have X is equal to zero for our X intercept and go ahead and plot that on our graph here. Part B finding this vertical assam to it. We need to just take pay attention to our denominator. So X squared minus four and set that equal to zero and then solve for X. So adding affordable sides. Leaves us with X squared is equal to four and then taking the square root. We'll see that we have this vertical ascent toe when x is equal to plus or minus two. Alright, part C. Here we are finding these horizontal asthma totes and for this one we need to take the limit as X is approaching infinity of our functions, so for X squared divided by X squared minus four, and all were truly interested in here are these dominant terms, So we don't need to worry so much about that negative four because at the end of the day, if we're approaching infinity minus four doesn't make a difference. So then we just have four X squared divided by X squared simplify. You can see that our exports are going to cancel out, meaning we're just left with four right here. So our horizontal as sometimes occurs when y is equal to four. Let's go ahead and plot these as I'm told. So when y is equal to four that sets up here and then our vertical ascent occurs when X is equal to plus and minus two, so that's going to sit in here. Yeah. Now, to actually sketch this, obviously our graph is going to fall somewhere within these bounds. We just give ourselves, we don't know exactly where, So go ahead and plug the function, India graphing calculator and it'll give you a better idea of where the curves actually take place and then you'll see pretty quickly that this function does follow these basic outlines that we just gave ourselves. It is going to adhere to the asientos we found as well as this X intercept and your graph looks something similar to that.

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