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Linearize $f$ near $x_{0}$.$f(x)=\sqrt{1+x}$(a) $x_{0}=0$(b) $x_{0}=1$(c) what about $x_{0}=-1,$ does it present a problem?

(a) $y=1+ {x} / 2$(b) $y=\frac{\sqrt{2}}{4} x+\frac{3 \sqrt{2}}{4}$(c) yes, $f$ is not differentiable at -1

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 6

Linearization and Differentials

Derivatives

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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.

02:07

Linearize $f$ near $x_{0}$…

01:05

$f(x)=\sqrt{x}, \quad x_{0…

02:25

$f(x)=\frac{1+x}{1-x}, \qu…

02:37

$$\begin{aligned} f(x) &am…

01:22

Given $\begin{aligned} f(x…

in this question we're going to linearize to function If X is equal to one plus X. To the both mentors have mentors have. So it's the same thing as root of one class x. So first We're told that we are to linearize near the points. X. Not as good to zero. It's not it's got one and check if we're gonna have a problematic not. It's got a -1. So first we have to two to find the derivative of ethics which is half one plus X. To the post minus half. Mhm. And this gives us one over two who TF one plus X. Now At the first point it's not as good to zero. F zero is equal to one substituting zero. The X value in the ethics function. F. Prime of zero. Is he? Co two? You substitute zero here we get one over two. Now our linear arised function equation is equal to health of A plus F. Prime of a. Multiply by X minus a. So we will use this equation substituting the values that we calculated hence L. Of X. At the point X notice go to zero is equal to F. A. Which is one plus health prime of a which is half x minus zero. It gives us one plus Yeah. One plus X over two. As our linea arised expression near. Mhm. It's not easy equal to you. It it it's not Is equal to one. You calculate F zero is equality. We substitute one in the fx function we get F not easy co two We get F one is equal to Rudolph too and we get f prime of one substituting Wuhan. Here We get 1/2. Wtf two now substituting into there linearize expression equation. We get our X As equal to f. of one which is root of two plus if prime of one. Mhm. Is one over to root of two. My play by x minus one. Now we can simplify this further by multiplying um using the laws of Sirte. So um first we separate the X value from the numbers Plus one over to root of two. X -1/2 12 2. A shortcut will be applying everything by two Who told to root of 2/4. And this is just to simplify the calculations, It's the same like multiplying by 4/4 which is uh impact in practically doing nothing to the to the question. We are not changing anything which gives our final answer is who talked to over four X plus 32 over four. And this is our final answer for for them Yeah. For the linear arised uh equation near X. Notice go to one. Uh Next we're asked that Can there be a problem trying to linearize near -1. Looking at our fx, we see that fx Would beat 1 -1 years ago to zero. And we can not differentiate this, still coming back to zero. And if we differentiate it we're gonna have a defined an undefined number. So this is how we we get a problem because fx he's unverifiable at X not is equal to 21 two minus one. Mhm. Mhm. Yeah. Mhm.

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