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Consider $f(x)=(x-1)^{1 / 3},$ obviously, $x=1$ is the root. Try Newton's method with $x_{0}=1.1 .$ What seems to be happening? Try to explain it.

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 11

Newton s Method

Derivatives

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

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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01:40

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03:41

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06:53

in this problem, you're given the function f of X equals X 1/3 minus one and were asked to show the inverse noted by F inverse of X is equal to the cubed root of the quantity X minus one. So in order to do this, our first step is to actually switch the X and Y in the original equation. So when I look at this originally Equation, you might be wondering where the why is and I'd like to remind you that F of X is the same thing as why So I take this left side of why and the X and I physically switch their location. So to look like X equals y to the third minus one toe work towards finding the inverse equation. Now we need to resolve for why equals. Right now it's X equals, So we've got to get the Y term alone. So the white term has a cube on it. But first I need to move this negative one to the other side by addition. So now we have X plus one equals, Why, cubed toe? Undo this cube here, we need to take something called the cubed root and we do that on both sides of the entire quantity on the left side. So on the left we have cube drew of the quantity X plus one. And on the right side, we know that the Cuban ruin the cube. Cancel to give us just why So as I tracked my progress, I noticed that I have a matching part. But I'm missing this f inverse and I instead have a why. So your last and final step is to actually replace why, with f inverse of acts or, in other words, rewrite as f inverse of X equals cubed root of the quantity X plus one. So we can see that we showed it perfectly. Which means that we can put a big check mark here. You have completed the task.

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