find-the-linear-approximation-of-the-function-fxsqrt1-x-at-a0-and-use-it-to-approximate-the-number-4

Question

Find the linear approximation of the function
$f(x)=\sqrt{1-x}$ at $a=0$ and use it to approximate the
numbers $\sqrt{0.9}$ and $\sqrt{0.99}$ . Illustrate by graphing $f$ and
the tangent line.

Linda Hand · Accepted Answer

find the linear approximation for the function. F of X equals the square root of one minus X at X equals zero. Use it to approximate the square 2.9 and the square 2.99 So even though x zero were ableto calculate the square roots near one because we&#x27;re subtracting, um, numbers from one. Okay, but if you get very far away from one, then it&#x27;s not gonna work. Okay, So, ffx, that&#x27;s this red graph right here, the square root of one minus X. And now we&#x27;re going to write the linear approximation, which is the blue line. So first, we need to find a point on the line. Okay, Can the point they want us to use is where x zero and you plug zero in and you get why is one cool and then we need the slope and find the slope. We take the derivative of prime of X. It&#x27;s one half one minus X to the minus one half times minus one. That minus one, came from the derivative of the inside. So it&#x27;s minus 1/2 times the square toe, one minus x. So that&#x27;s the slope of any tangent to that curve, and we want the specific one where x zero. So we plug in X equals zero. And now it&#x27;s the slope. Am equals f prime of zero, which is minus 1/2. And now I&#x27;m going to write the equation of the line. Yeah, slow equals. Why? Minus y over X minus X cross. Multiply solve for why, um two y equals minus X plus two. Or why equals minus one half X plus one. Okay, so let&#x27;s see if that&#x27;s the line I have drawn. Hopefully it is Y intercept is one. A slope is negative. One half. All right, so we want to calculate squared of 0.9. So we started equal to the square root of one minus X so we can figure out what X is so square. Both sides 0.9 equals one minus x, so X must be equal 2.1. Oh, snap 0.9 point one. So then here&#x27;s our approximation Then the square root of 0.9 is approximately equal to minus one half times 10.1, which is 1/10 plus one. This is minus 1/20 plus one. So 19 20th, which is 0.95 So I&#x27;m saying the square root of 0.9 is approximately 0.95 So I got my calculator out. I see what it is. 0.9 square root, right? Actual value 0.9 0.94868 blah, blah blah. Okay, here&#x27;s the beauty of this look. I never had to take the square root. I didn&#x27;t have to get out my calculator. I could get a good approximation to the square 2.9 without even doing hardly any calculations. That&#x27;s the beauty of the linear approximation. Now let&#x27;s use it to 5.99 square root a 0.99 So this time X is equal 2.1 So then 0.99 is approximately equal to minus one half times point No. One, which is one over 100 plus one so minus one over 200 plus one. That&#x27;s 199 over 200. Okay, yeah, so that&#x27;s 99.5 over 100. So that&#x27;s 0.995 So I&#x27;m saying the square root of 0.99 is 0.995 approximately. Let&#x27;s see what calculator says. 0.99 square roots, according to my calculator is 0.994987 more stuff. Okay, It&#x27;s a very good approximation to the square root of this. Okay, now, if you look back at the picture here as long as you&#x27;re taking the square root of numbers that fall within here, then, um, this line is a good approximation. But if, for example, you wanted to use X equals one, it&#x27;s not a good approximation. Or over here, it&#x27;s not a good approximation. You need to stay where the line and the curve or close to each other.

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.

Amrita Bhasin · Answer

Okay, Jim zero is the cube root of one plus zero, which is one therefore, G prime of acts is 1/3 times. Now we&#x27;re playing in zero. So one plus zero to the negative to over three simplifies to 1/3 plug into the blender or ization formula with sequels one plus 1/3 axe. So now we have one plus X equals 0.95 they&#x27;re for ox equals negative 0.5 Therefore, we have one plus 1/3 times negative 0.5 0.9833 You know The cube root of 1.11 plus X is 1.1 an ax 0.11 plus 1/3 time&#x27;s Your 0.1 is one point 1.3 three.

Doruk Isik · Answer

in this problem. We&#x27;re given a function G anywhere. Esther. The year in the right spot. Joanna used at airports. Mate, you know. So let&#x27;s start with writing function. G s one plus extra probable heard. And let&#x27;s find a day with Tochigi Perm affects would be 1/3 well plus eggs. Negative two firms. So now, unless you let the functions of every threat given point Sarge easier. Zero text Joe would be one and g primal zero b. We&#x27;ll burn so that the musicians function, uh, is G zero plus g prime zero explain. It&#x27;s given point. So zero on from this. This would be one plus. We&#x27;ll third. Oh, yes. So the first what we&#x27;re given is going by. So this is a cook, too. One was negative. Orange. You&#x27;re high. So if you plug nectar is your part in X, we find this to be one plus 1/3 of negatives. You&#x27;re five. Then this is 50.983 2nd 1 is won&#x27;th boy form, and that is ICO to one plus one. So we&#x27;re gonna put point fun. One, uh, warm was 1/3 0.1 and we find it to be one point zero

Jack Chen · Answer

Okay, So for this program, we start away that a function Joe fix. So first we need to compute its curative, since we want to fight the memorization that you prime mix, which is 1/3 1 Prospects to the, um minus 2/3 set of Familiarization is given by she off. Hey, plus G prime off eight times X minus A. So, in this case, a course zero. So we immediately have one plus 1/3 picks. So this is the remilitarization and the we can usually visit glamorization to approximate the following values. So first is, um, 0.95 to the cubic roots. Um, in fact, this is the value off G off miners, their 0.5 So by arleen Ooh, approximation we have We can use ello off miners syrup going 05 to approximate this value, which is, um, 0.983 30 meets the wrong this and for the other one, we just simply we do the same thing because this equals two g off their 20.1. And we&#x27;re just project and put 0.1 into ella. So we have this result 1.33 toe. Approximate Miss given value

Joseph Lentino · Answer

so in This problem we&#x27;re looking at is were given a function. F of X is equal to one plus specs and the value that&#x27;s close to the values that we&#x27;re gonna be working with is going to be zero. Okay, ultimately, we want to evaluate f of the square. Sorry, we want to evaluate Is the square of 0.9. We want to evaluate that. Want the square root of 1.1. So the first thing I wanted to is find the slope. So now, in order to find the slope, I need to take the derivative. So in this case, I&#x27;m going to rewrite my function as one plus x two. The one half power taking the derivative f prime of X, which will be my slope, is going to be one half one plus x two the negative one half times the truth of the inside, which is just one substituting f prime plus subsidies zero and F prime. We&#x27;re going to have one half times one plus zero to the negative one half power times one which is one half times one to the negative one half power. This is still going to be one. So my slope is one half. So the equation that I&#x27;m gonna be using will have a slope of one half the coordinate that we need to use. Zero. What? Now I gotta plug that back in, and that&#x27;s gonna be the square root of one plus zero, which is going to be the square root of one, which is zero. So the equation that I&#x27;m going to be working with is why equals one half X minus zero plus one. And so what I&#x27;m gonna be doing here is I&#x27;m going to be substituting come the value, in this case of the value here, right into the equation in order to approximate it. So we&#x27;re going to be playing in point us. We have why equals one half times 9/10 plus one, which is gonna give us one half Sorry, which is going to give us 9/20 plus one. So our answer for this guy&#x27;s 29. 20 it&#x27;s or one and 9 20 It&#x27;s perform the same operation here. We&#x27;ve got why is equal to and again, what I want to do here is I want to work in terms of fractions. That&#x27;ll be the easiest to work with, So 1.1 is 11. So one half is 11 tense, plus one which is 11 20. It&#x27;s plus 1 20/20 gives me 31 20th or one and 11 20th, and these would be good approximations to

Linda Hand · Answer

in this question, we are given the function if and they want to use a fee to approximate root off 0.9 and the route off zero point. And I know. Okay, So first we need to use We didn&#x27;t find out of the memorization. I&#x27;m so using euros Ages is given by the tension line. Why minus stay for Faye? Because to have prime of a time six minus eight in this question, a close to zero. So first, if off a Because the way for zero which is just one and the for the for the curative for this curative which is a several popular attention line going to figure out what the purity of first certain pilot. Um, General, we have the duty because toe one have times one minus x two B minus 1/2. And don&#x27;t forget, we had another minus one, which comes from this How one minus x by the general, you know we have minus one have one minus X to the minus 1/2. So the slope off attention line at X equals toe A is given by F prime zero uh, which is minus 1/2. So the tension line or the demilitarization because the Y minus one, because to minus one have tons X cellini are there is given by it kept alof X equals two minus one over to 1/2 X plus one. So the notarization means that we can approximate if, by this linear function minus one have expressed one. Okay, So in order to figure out what&#x27;s, um Groot off 0.9 so far, so were we realized. If he calls to root off one minus X so we want approximate on route off their point that so if he&#x27;s Ruto one minus X, that means route off 0.9 equals toe f off their 0.1, and by our militarization or the knee approximation we have, we can approximate if off 0.1 by Project X equals 0.1 into this steamy approximation. So we have minus one have tanks. Your 10.1, class one, um so result B zero point, um 95 So we can use their 0.95 to approximate grew toff 0.9 um, similarly, if we want to approximate our route top 0.99 and we realized route off the open 99 equals toe. If off their 990.1 by the in your approximation become vehicle approximate by minus one. Have time. Surprise. You&#x27;re one plus one, which is 0.0. So is there going nine time five, um, geometrically this muse, Um, suppose we have. Suppose a graph the f on the coordinates. So we have something, like so something like this. Okay. So that in the approximation mused that we have the tension line had Tex equals 20 And for the small neighborhood off zero, we can use the value off this tension line. Um, toe approximate the value, the actual value off F.

Problem

Find the linear approximation of the function $g…

Problem 5 Medium Difficulty

Find the linear approximation of the function
$f(x)=\sqrt{1-x}$ at $a=0$ and use it to approximate the
numbers $\sqrt{0.9}$ and $\sqrt{0.99}$ . Illustrate by graphing $f$ and
the tangent line.

Answer

$\sqrt{1-x} \approx 1-\frac{1}{2} x ; \sqrt{0.9} \approx 0.95 ; \sqrt{0.99} \approx 0.995$

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$\sqrt{1-x} \approx 1-\frac{1}{2} x ; \sqrt{0.9} \approx 0.95 ; \sqrt{0.99} \approx 0.995$

Essential Calculus Early Transcendentals

Grace H.

Anna Marie V.

Kayleah T.

Samuel H.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartEssential Calculus Early Transcendentals

Grace H.

Anna Marie V.

Kayleah T.

Samuel H.

James Stewart

Essential Calculus Early Transcendentals