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Given $f(x)=x^{2}-4 x+3,$ suppose you take $x_{0}=2,$ what happens why? Suppose you take $x_{0}>2$ what happens, what if $x_{0}<2 ?$

$f^{\prime}(2)=0$$x_{0}>2$=larger root,$x_{0}< 2$=smaller root

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 11

Newton s Method

Derivatives

Campbell University

Baylor University

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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In this problem we are given that the derivative of function f is equal to 2 x and we are asked to find the valet function at 2. So we're asked to find f of 2 point now, given the original of this function, let's find the function after their fate is equal to 2 x. We know that f should be of the 4 x spread, plus some constant co, depending on the value of the functions for part, a b and c this c. This constant will change. So, let's start with part a we know that equals 0 is equal to 0. So 0, this is equal to 0 square plus c. So from this we defined c to be 0, so f of x in the space is equal to x. Scrattan f of 2 is equal to 2 straight, and that is equal to 4 point now in part b, we are given that f of 1 is equal to 0, so this is 1 squared plus c, then c is equal to negative 1 and f of x. In this problem is x, squared minus 1. From this we find f 2 to be 2 squared minus 1, which is equal to 3. Now, in part c, we have negative. 2 is equal to 3, so negative 2 square plus c is equal to 3. So c is equal to negative 1 and f of x is 10 x, squared minus 1, so f of 2, just like in part straight line 1 that is equal to 3.

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