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Question

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

Grace Muhihu · Accepted Answer

Okay, So for this problem, we&#x27;re gonna need to find off of x zero. So that means we&#x27;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&#x27;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&#x27;re gonna see what that looks like when it comes to finding the linear approximation. So let&#x27;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&#x27;s this X equal to one. Is this, uh, this X value here and then you can see has a height of two.

Ahmad Reda · Answer

in problem. One. We want to give them any approximation of X at X equals X node, and we have X node equals one first step we get. If there&#x27;s of X, differentiate the function. It was X toe. Get if the tropics we have root X. It&#x27;s differentiation is half X to the bar of negative off or one divided by two roads. X. The second step is to get the value. If dish off exclude, we have explored equal to one. Then we get if they should one. The substitute here by X equals one. We have one divided by two. Route one, which is half. Third step is to get the aneurysm ation hair off X, which equals it will be if it&#x27;s not, if one plus f dish off excellent, multiply it by X minus X node, which is woman we can get if one by substituting in the main function. My X equals one. We have Route one plus, if they should. One we&#x27;ve got. If there&#x27;s one from step to Uh huh, multiplied by X minus one and to simplify, we have one minus half, which is half, then we have half multiplied by X plus one. And this is the linear approximation off F of X X node equals one. Now let&#x27;s graph the function along with its linear approximation. If we have X axis here and y axis here 1234 and we have year 123 If we have root X, then we can get three points of root of X. The first point is zero on bond zero zero and zero. The first point here. Second point is one on one can point here. The third point, for example, is four on two four to here. Now we can estimate that would of X has this she This is root of X. That&#x27;s that&#x27;s a graph l affects hell of X. When we substitute by X equals zero we have they&#x27;re of X equals half. When we substitute X equals one you have And of X one, they must be the same here at X node equals one and, for example, visibility substitute X Y three. We get a lot of X equals four. Divide by two, which is two. The first point 0.5 here. The second point is here. Third point is three and two Here we can see that the linear approximation is tension. Toe the function F x at the point, exclude equals one.

Ahmad Reda · Answer

in problem two. We want to give that inner approximation of the function of X at X equals explode, which is here is zero. The first system is to get the differentiation off the function of X. When we differentiate dysfunction, we right the power third multiplied by the inner function X plus one and then we differentiate the inter function. The inner functional differentiation is one. Multiply it by own. The second step is to get if this off explode 30 equals we substitute here by X equals zero third multiplied by one equals one third. The third step is to get than in your ization it of x which equals if explode from zero plus f dash off extra note dash zero multiplied by x minus x a node X minus zero We can get f of zero by substituting in the main function by X equals zero equals zero plus one. So the bar off one third equals one plus f dash of zero. We&#x27;ve got evidence of zero from this step which is third multiplied by X and this is the notarization off all of X. That&#x27;s a graph f of X, along with aerobics Mm, It&#x27;s we have here. 1234 every year. 123 Let&#x27;s start by f of X We got here every of X on here. Alot of X We got many points to draw F of X, for example. Let&#x27;s put X equals zero we get every X equals one we what? Ever x equals one, for example. Mm old ah p one. We&#x27;ve got X equals to the ward off two thirds. It&#x27;s about f of X equals minus one. A point before exclude way we bought minus one. We got X equals zero and get out of X. When we substitute by X equals zero, we got all of X equals one. They must be the same at X node. And when we put X equals one or let&#x27;s put it three to get hell of X two and appointed before it snowed, for example, minus three toe. Get the love X equals zero. Now let&#x27;s start by every weeks. Zero on one. The second point is one on two toe about off. One third is not 11 3rd not to search. 1 30 year equals 111.26 one 0.26 Almost here. And when we have X equals minus one have the point equals zero. It&#x27;s something like that. Now let&#x27;s grab all of X with another color. We have zero on one. The same point here. Three and two. We have a point here. Um, minus three and zero is minus three and zero. Here, you can find that this delay notarization is the tangent of every X at X unload, which is you.

Grace Muhihu · Answer

Okay, so two panel on the approximation for this, we&#x27;re going to plug zero into this f of X function, so that&#x27;ll be this part of our formula, So f of X of zero is going to be signed of three times zero. And this will just be zero then if they take the derivative so f prime of X. So it&#x27;s gonna be the German tive of Sign the Reacts, which will then be three co sign of three X. And if we go ahead and plug zero and here, that will give us this value here. So when we do that, we get three co sign of three times zero so co sign of zero is one. So one times three will give us three. Now, if we go ahead and plug in these two values into our formula, you have l of X is approximately equal to effort vax, which is equal to so f prime of X sub zeros three and then times x minus X sub zero, which is zero which were given and then plus f of X of zero, which is up here just also zero. From here, we can just distribute so out of X is gonna be able to three X and this will be our linear approximation for the function sine three x at X equal to zero. So let&#x27;s see what that looks like in a giraffe. So in a graph this ah sine wave, which is this soon you saw it&#x27;ll wave, uh, is not perfect, um, graphing, but it&#x27;s basically just a simple wave of the function sine three x Where, um, the tangent line off the graph at X equal to zero. So the equation of this line is three X, which we found here.

Grace Muhihu · Answer

Okay, So the first thing that we need to do to solve their linear approximation is to find f, uh, x sub zero. So to get that would just plug in zero into this function. So we with that we will get to time zero plus mine, and that will give us three. So then, from there, we need to find the derivative of the function of So what? That would be news, the change, or when you use a channel, um, we will get that the derivatives gonna be won over the square of two X plus nine. Okay, so then, from there we want also plug in Xerox or when a pawn zero into the derivative. So it will be one over slurred of two time zero plus nine. And with that, we&#x27;ll get 1/3. Okay, some here. We just need a plug in. Now what we have found in toil in your approximation formula so we have l affects is approximately equal to half of axe is equal to so f prime of X subzero, which is 1/3 then times X minus X here, which is zero and then plus f of X subzero, which is three. So here, if we just simplify a little bit, this is gonna be equal to 1/3 X plus three. So this is gonna be the equation for the linear approximation of the function square toe to X plus nine at X, equal to zero. So let&#x27;s see what that looks like you grab for. Okay, so this is what it looks like. So it&#x27;s the tangent line to this curve of this function squared of two X plus nine at X, equal to zero, um, where it touches the graph. So this is where the linear approximation takes place at X equal to zero.

Grace Muhihu · Answer

Okay. So to find a linear approximation, we&#x27;re going to start by plugging pi into the function to find half of X some zero. So we do that. Get sign of pie. So sign of pie gives us zero. And then now we take the derivative of that function. So, during derivative of signed acts, which is co sign axe, then we have to do the same thing. We did hear one a plug in pie, so f prime of x of zero is equal to. So we plugged that in. It will be co sign of pie, which will then give us negative one. Okay, so now we just plug in these two things into our formula. So l a X is approximately equal to f of X, which is equal to f prime of X sub zero, which is going to be negative one and then times x minus X subzero, which is pie and then plus f of x subzero, which we found to be zero. So if we rewrite this basically ella backs is going to be equal to negative X plus pi. Okay, so this will give us our linear approximately shin for except zero equal to pi. So now we could see what that looks like. Intergraph. Okay, so we&#x27;re looking at the graph. This is the function signed accents. So leave. And then the linear approximation is the linear one that&#x27;s going through this point pie. This was X sub zero equal to pi. So this is gonna be It&#x27;s Tangela nor the linear approximation at that point.

Problem

Find the linear approximation to $f(x)$ at $x=x_{…

Problem 4 Medium Difficulty

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

Answer

$4-2 x$

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Robert T. Smith, Roland B. Minton

Calculus: Early Transcendental Functions

Anna Marie V.

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Precalculus Review - Intro

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$4-2 x$

Calculus: Early Transcendental Functions

Anna Marie V.

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Robert T. Smith, Roland B. MintonCalculus: Early Transcendental Functions

Anna Marie V.

Caleb E.

Samuel H.

Michael J.

Robert T. Smith, Roland B. Minton

Calculus: Early Transcendental Functions