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Find the linear approximation to $f(x)$ at $x=x_{0}$ Graph the function and its linear approximation.$$f(x)=2 / x, x_{0}=1$$

$4-2 x$

Calculus 1 / AB

Chapter 3

Applications of Differentiation

Section 1

Linear Approximations and Newton's Method

Derivatives

Differentiation

Applications of the Derivative

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Okay, So for this problem, we're gonna need to find off of x zero. So that means we're gonna plug one into this function, so that will give us to over one which is just able to and from here, we need to also find the derivatives of Prime of X. So derivative of two over acts is negative. Two over X squared. Then from here, we want to plug one into the derivative. So this will give us negative to uber warms squared, So this will just be negative too. So now we can plug what we know into our fill of ex formula. So l A X is approximately equal to f of X, which is equal to so f prime of X here, which you found Ah, here to be negative too. And then times x minus except here, which is one and then plus f of x subzero, which is true. It's not from here. Goodness, rewrite that. So this will be negative two x plus two by distribute. And then another plus to someone from here was gonna add that choose together to get negative two acts plus for so this is gonna be our L A X. So now when we graph this and the original function two over acts. So we're gonna see what that looks like when it comes to finding the linear approximation. So let's see what that looks like him in a graph. Okay, so here we have a function to over X, and then we have So this is the tangent line or the linear approximation of the function, Uh, two over X at axe Evil toe want. So that's this X equal to one. Is this, uh, this X value here and then you can see has a height of two.

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