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The linearization is the best linear approximation Suppose that $y=f(x)$ is differentiable at $x=a$ and that $g(x)=$ $m(x-a)+c$ is a linear function in which $m$ and $c$ are constants. If the error $E(x)=f(x)-g(x)$ were small enough near $x=a$ , we might think of using $g$ as a linear approximation of $f$ instead of the linearization $L(x)=f(a)+f^{\prime}(a)(x-a) .$ Show that if we impose on $g$ the conditions1. $E(a)=0$ The approximation crror is zero at $x=a$2. $\lim _{x \rightarrow a} \frac{E(x)}{x-a}=0$ The error is negligible when comparedwith $x-a$ .then $g(x)=f(a)+f^{\prime}(a)(x-a) .$ Thus, the linearization $L(x)$ gives the only linear approximation whose error is both zero at $x=a$ and negligible in comparison with $x-a$ .

$g(x)=f^{\prime}(a)(x-a)+f(a)$

Calculus 1 / AB

Chapter 3

Differentiation

Section 9

Linearization and Differentials

Derivatives

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Hello, everyone. The question is, uh, if some function is, um by equal to why, equal to f of X and the other function given us do you off x equal to mm x minus eight plus C And the edit function is, uh, defined as every fix minus geo fix letter between these two functions. And we know that the linear approximation, for example, for this one linear approximation that is linear approximation is to find this, uh, at e will be fl fi f prime offi X minus a. Now what we have to do is if this this um added function, if it follows fulfills Do these two conditions that your faith is equal to zero and Lim X goes to a e off X, divided by X minus A is equal to zero then the linear approximation is equal to this. Then these two are seen. So let's prove that first of all, let's consider the first condition that is this Your fix is equal to ffx. Let's plug in the value off geo affects. That will be m X minus a plus c no sorry. Because of the negative sign, it will be negative. C now you off. They will be f off a minus m a minus a minus c So this is zero and this should be zero according to this condition. So this will be zero equal to F off a minus C, which means f off e equal to see. This is what we found from the first condition. Now let's use the second condition and see what we find from the second condition. So let's make use of the second condition. Second condition is Lim X goes to mhm. Let's plug in the value off you off eggs from here that will be fo fix minus m. Explain this a minus. C divided by X minus e. This should be equal to zero. Let's do that. No, um Lim X goes to a This is F affects now as C is equal to wear for faith. And let's plug in F or fe for this and X minus a negative M X minus a over X minus savior separating the denominator. These two will cancel out. Let's settle Now you can see here that this is the definition of deliberative at a so this will be a prime off a equal to mm. So now let us plug in this and what We found this in this linear ization formula. So this linear ization formula will be alot fix equal to f off A. We'll see. Plus, if prime affects was M X minus A, which is exactly same as geo fix hence proved.

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