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Explain why $x_{0}=3$ will not produce a root of $f(x)=x^{3}-9 x^{2}+27 x+5$ Determine the root of this function, to six decimal places.

$f^{\prime}(3)=0 ;-0.174802$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 11

Newton s Method

Derivatives

Oregon State University

University of Nottingham

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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given a function F of X equals five times X. To the six plus seven X to the fourth plus X squared plus nine can't have any real zeros because according to the um take cards rule of sign, the possible number of positives. Oh and possible number of negative real zeros can be found. So the possibility of positives because the first term of this function is positive and it never changes. There is no possibility of a positive real zero. For the possibility of negative, you have to consider F. Of negative X. Well, negative X. Of the six is positive. So it doesn't change the sign negative X to the fourth is positive. Doesn't change the sign negative, X squared is positive, doesn't change the sign. And therefore even in the negative we start with positive and never change. There is no possibility for either. So there is no real zero. Also we can't factor out any terms. So that's rolling out zero as being a zero.

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