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Use Newton's method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations.$x^{2}\left(4-x^{2}\right)=\frac{4}{x^{2}+1}$

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$x=-1.94382538$$x=-0.84310820$$x=0.84310820$$x=1.94382538$

Calculus 1 / AB

Chapter 3

Derivatives

Section 8

Linear Approximations and Taylor Polynomials

Missouri State University

Campbell University

Harvey Mudd College

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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Use Newton's method t…

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$23-28$ Use Newton's …

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Use Newton's Method, …

in this trouble were given any question and were asked to use going to Mecca by older whose office separation. So let's start with defining a function that were affects is expert. Well, that's right. That's four x scrape minus X. Afford lunch for or were expert plus one eyes equal to zero. And we What? This function that looks something like this. There are few things that we notice for describe first off, therefore intercepts. And second thing is this. You can see all the powers are even so That means that function is symmetric with respect to t. Why next? So be so only four poles to help. Then be, uh, roots. Undernegative will be the same in magnitude, but opposite in sight. So the intercept are about when? Because it decides 1.9 point agent on the negative side, there to a 0.0.9. And that is important. Eight. So, from this, remember, we need their return for a new cosmetics on this fine new, uh, we find there to that primal thanks to be eight eggs minus for excuse plus eight x over. Expert before squared Now notice methods said that the guest ready a neck situation would be the previous guest minus functional like that. The previous guests divided by the hotel in a second, a previous guest. So let's start with our first does this 0.8 a less black 0.8 for extended this elected function in the day with 3.8. So we find our second guess to be pulling a or two, eh? 7645 And it looks like this. And we find for gas to be about pointing 843108 to 0. And if you plug this thing, be finding four the poor situation to be pointing for everyone here are 8 to 1. And if you like this and we actually find that order approximation is about same as third once on this is definitely one of our rooms. Let's look at the, uh, second rude when nexus point for lane. If you put 0.9 for extend any feel like function of the earth Uh, this point, we find our second guess to be about 1.94 689 Once you're free now, people like this and we find for guys to be 1 49 438 3891 And if it wasn't this, it'd be fine for submission to be about 1.43 age 2538 And we find that the fifth approximation is about Samos, a bird once. All this is definitely also one of our roots. So since we said that dysfunction, it's symmetrical. Suspect Why access and weak inside up roots are plus wing is foreign aid for 3108 to 1 and plus ways both to a negative 1.9438253

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