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Newton's method can be used to approximate reciprocals. Suppose you want to approximate $1 / b,$ then let $f(x)=1 / x-b .$ Use this observation to compute (a) $1 / 6,$ (b) $1 / 9,$ (c) $1 / 13$.

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 11

Newton s Method

Derivatives

Oregon State University

University of Michigan - Ann Arbor

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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Approximating reciprocals …

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(a) Apply Newton's me…

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Use Newton's Method t…

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Approximating Reciprocals …

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To approximate the recipro…

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Newton's Method can …

All right. So party is asking us to verify the Newmans myth it. So we're going to do that. So we have a form. A X of M plus one is equal to x event, minus deaf, Uh, except and divided by have prime of x event. So since f of X is equal to one over X minus a we get that have prime of X is equal to negative one over X squared. So this is equal to exurban. Minus one of her ex urban man is a all divided by negative one over AC seven squared, which is equivalent to saying ex urban minus ex urban minus a times X of end squared old about it by negative one. So this becomes X event plus thanks. Event minus any time except end squared, which is two times except and minus a times exhibit and squared, which is except Ed times Tu minus eight times x event. So that's a derivation. And now we're going to go up and coast the calculation for Part B, which is asking us to start with equal seven and find up to eight digits of accuracy. So we're gonna do that right here. As we can see, the approximation becomes your 0.14 to 85714 which is about whatever seven, as it should be now. One thing to note is that we're going to want to half our initial prostate should be somewhere close to the actual answer. So it would probably help to graph this through Dismas or something like that, to get a good estimate for before actually plug in the number. Because, as we can see here, if the initial guess was 0.3, it doesn't converge to 0.14 to 8, and a step goes to some negative and astronomically big number. And that's because the Newmans formula is fairly volatile. If you're not, if you choose an initial guest that is too far from the actual answer, it can at times go off to some wild number. And that's about

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