the-linearization-is-the-best-linear-approximation-suppose-that-yfx-is-differentiable-at-xa-and-that

Question

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 conditions
1. $E(a)=0$ The approximation crror is zero at $x=a$
2. $\lim _{x \rightarrow a} \frac{E(x)}{x-a}=0$ The error is negligible when compared
with $x-a$ .
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 $x-a$ .

Fasiha Binat Zafar · Accepted Answer

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&#x27;s prove that first of all, let&#x27;s consider the first condition that is this Your fix is equal to ffx. Let&#x27;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&#x27;s use the second condition and see what we find from the second condition. So let&#x27;s make use of the second condition. Second condition is Lim X goes to mhm. Let&#x27;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&#x27;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&#x27;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&#x27;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&#x27;ll see. Plus, if prime affects was M X minus A, which is exactly same as geo fix hence proved.

Matt Just · Answer

Okay, so we have a function on like himself attacks and we know that have prime of exists. In other words, F is differential today, and we have that she of x is a linear approximation. Two. Yes. Okay, so it&#x27;s just in times X minus a plus thing. Yes, they c c and em and see your constant. So it&#x27;s just it&#x27;s just some line that&#x27;s supposed to be approximating f OK. And so we&#x27;re going to define the error, which is Ethel X minus G of X. And now, But there is small near a we can think of actually about using the linear approximations of F instead of the one you&#x27;re ization. Okay, so what we actually want to show is that if GI is a good enough approximation, it essentially is the legalization of that at that point. So we&#x27;re going to impose the condition that, um well, first of all, the air at hey is zero. And then secondly, the limit, his ex goes to a, uh v of X over X minus A Is your, uh okay, So what we want to show is that Jia Vex is actually equal to roll in here approximation of slips that&#x27;s actually do it. All right, so let&#x27;s look it. What is the same he of a zero We ve is just today minus jia. But this tells us that FAA equals g ave. But if we look at G g ave is m times a minus eight zero So see, this is actually telling us that seem is equal to as a bell. It&#x27;s coming from the first condition in the second condition is telling us Is that okay? The limit sexy. Purchase A of Candia Becks is death of X minus J of X over X minus A. Well, this is the limit is experts say f of X, uh, times your plot start minus jean, which is in times X minus a of course, Death of a right plus Eve. What we just found that she was f obey. Next my essay. Okay, And then you just take this limit. So see, that&#x27;s the limit. Experience is a of and write this as FX mama&#x27;s f Obey. It&#x27;s over X minus a and then minus the limit. His exit Purchase A in times X minus a over X minus a Remember, this is all supposed to be zero. Well, this guy is the derivative of that today. And then this guy that cancels this is just em. That means that in is that today? But now look at what that means about jean ends of prime of a so f prime of a times X minus A and C is up today. So imposing these two conditions allowed us to conclude that G really is the linear ization of s at a single day.

Oswaldo Jiménez · Answer

in these problem, we have a differential function f of X at X equals e and she of excellent. Your function equals two m times X minus a plus C where M and C are constants. We also defined the era function E of X equals to gif of X minus your Vicks. So if this era function were small enough near excitable, eh? When I think of using these function is leaner French in G A. Selena approximation off f instead of the leaner ization off F at X equals C. We want to show that if we impose thes two conditions on G, that is the era function at a Is he going to zero a limit when X approaches? E off the euro fashion divided by a X minus a easy with zero then the function. Jeez, exactly the same US generalization of f X equals a. So for that, we&#x27;re going to see first that the condition one we&#x27;ve both g that he&#x27;s hearer at a secret zero. It would see definition of your function here it is secret to f off, eh minus g of a and this implies immediately the F A A is equal to G off, eh? But we know the G off a. We see the definition of she here is equal to 10 times a minus a sea. There is a good to see. So we conclude that C is equal to if of a you&#x27;re now going to study. He&#x27;s second condition. So the second condition condition too. That is limits. When X approaches a off the era function to by the by x minus e sequence zero You see that limits when eggs a brush is a off the era fashion. He&#x27;s e of a and that by condition one Is he going to Syria? So we have that. The limit of the numerator of the suspicion here is Ciro and the limit. When eggs goes to a off the denominator of this expression that his X minus a is Syria because he&#x27;s a minus a, which is serious. That means that the limit we have here he&#x27;s one of the forum Sierra divided by serious so we can apply love Tito&#x27;s rule here and say that the limit when x goes to a off e of ex about it by X minus a secret to the limit when X goes to pay off the river tive of the year, a function divided by the derivative off X minus A and that is equal to the limit when exposed to gay too derivative off their function is, if derivative minus derivative of Jean because if of ecstasy into the function ever vex minus to your base in that divided by one so is equal to the limit when X approaches a off the video off. If. Why no sir. Derivative of and that is he going to They even have a off death at a class derivative factory minus minus. They&#x27;re riveted g at a but this limit. When eggs purchase a of a year, extroverted checks minus day Is sirrah the condition two. So we can say that so the river native of Pay Highness, derivative of G. Gay is zero. So immediately we get a derivative of if a day is equal to purity of G A. But remember, remember, Dad G of X is n times takes my nest. People see, but we have then that derivative of G X is equal to him. But any eggs so it implies immediately that embassy too stare even achieve off death on A. That is because the relative of she at A is going to be good to him. And the relative of Year A is derivative off if at a so we have C equals. So it here if a see his secretive off Hey, in m is derivative of f a day. He implies that she s a secret too M times that is the relief of F a times X minus day plus c That is, if you pay and that is just we see here if a favor blast derivative of F A bluff and times X minus a just the leader is Asian off the function Death at X equals a So, uh, the militarization give us the only linear approximation whose hero is both zero at X equals a and negligible in conversation with X minus a

Doruk Isik · Answer

this problem? We are Lena Rising Function Natural law one plus extra for zero. Now let&#x27;s in port They performed this immunization being ordered the musician of a function musical function of lit that porn misstated zero plus debuting a lit that same point times distance to that point. Now our function F FX is nation log off plus X So then f 00 as natural lawyer horn which is zero using effects we can find there is a form of access one over one Plus that&#x27;s anything like that function at a given 10.0 We find it ever to be born now plugging dose and you don&#x27;t find of the immunization for zero to the zero plus one times X minus zero, which is thanks or in port beat. Now we Well, they were told to take the spacing as points one. So we are trying to find how much difference this spacing going from one point to another, uh, would make Bonaparte baby No, not decorative office function. I will let that given point is born. So, um, this means that going or moving this much would result in a change of a function. 04 point minus F off zero. This is our initial point, or in part, won&#x27;t be calculated. This to be dear, right? Yeah. So we need to park red F port one, which is natural law off one plus 41. And that is nature lover Horn for one. So using this, we can do for you to be and your law off 1.4 minus warm. This we end up with this because there it is. One, uh, using this week for a change. To be born hero, you&#x27;re or 69 Important. See, we&#x27;re on street for the function, error and the approximations off realization. Uh, this is our accesses. Why? Accesses are ex senses. We are reporting this between zero in point by zero. Uh, this is our leader explosions bags or Joe puncher. Well, looks like there&#x27;s careful backs is natural. And this is how the air order difference between those defections became. As you can see, a CZ, we go from zero point fire air increases so four smaller value. The exit. The immunization gives colter results to the actual function. Then for live your values race and or does the x o point fine we can just like in party. We could find error to be 0.0 during, uh three. Fine. And actually, we could use this as our upper Oh.

Oswaldo Jiménez · Answer

given a function f with the properties that f at zero is equal to one and it&#x27;s derivative is equal to co sign of X square at any number. Eggs, you got to find a literacy ation. Fifth X equals zero in bar, eh? You Barbie, we&#x27;re going to estimate the value of F X equal syrup. I&#x27;m on any party. We&#x27;re going to talk about the relation between the actual value of the function at eggs, equal syrup we won and the estimate found in party. So first importantly, we&#x27;re going to calculate the scenery ization of F at X equals zero. And for that, beneath the value of the function of X equals zero, which we have, you see, but one. And now we want to calculate in the rivet eve of F at zero and this secret to the co sign of zero square that he&#x27;s to go sign of zero, which is equal to one. With that, we can calculate Vienna rhe ization of eggs of F X equals zero, and that is a love X equal to F zero that he&#x27;s won, plus derivative at zero, which is one times X minus zero. That&#x27;s equal to one plus X. So the militarization of the function f x equal zero he&#x27;s elevates equals to one plus X. With that, we can calculate now Barbie the estimate off the value of the function it x equals here when one that is, if of 0.1, is a pertinently, go to the legalization at syrup. When long and that is one plus syrup we want that is 111 So the estimate of the value of the function F and syrup I want is 111 and non party. We&#x27;re got to compare the function at syrup in one. The actual value and the estimation we have found in parties that he&#x27;s one by one. And for that we&#x27;re going to see the following. You&#x27;re going to see that their relative of his function is co sign of X Square, and we have calculated in leaner a cessation of function of X equal cereal. We have estimated the value of the French in a 0.1 so we can study function f at the interval zero. Cyril, we want to see what the properties of the function are in this interval, and here we can see that as the graph of the function goes, I kn in smaller, less this way we&#x27;re here. We have P half. We can see that the derivative of the function f ees ah positive for all eggs in the interval. Zero seriously wanted. That&#x27;s because it co sign of X Square is going to be positive for all eggs from Ciro 201 Remember that series square in serious 0.1 and that&#x27;s much less than be off. So for all the points in thes interval, the derivative, it&#x27;s positive. So the function. It&#x27;s increasing in the interval 200.1. But there is another thing we can say. And that&#x27;s we seen these graft that if prime of X derivative of F is decreasing because we see that the graph of the function co sign see creasing from Syria to see your allies Syria one. So if these function it&#x27;s increasing these interval, they were considered that it&#x27;s derivative is negative in disinter bodies to the relative off the relative of F that is the second derivative five. He&#x27;s negative remind you ready this way negative for all eggs in the interval, Ciro syrup in LA And that means that the function have he&#x27;s gone cave down in this interval. So if we draw the function f in the interval series Europe in one in at the same time a draw. He, uh, generous station at X Equal cereal. We must have something like we see in this rough. Is he here that the function he&#x27;s gonna get down increasing and God can give them from 0 to 0 ping one on DDE. In this case, it was ready. Ah, dungeon line at zero, which is a generalization. We have calculated our A This dungeon line is always above the graph of the function f and with that, uh, with that, we can say that the function at Syria everyone is always less than the unionization and syrup one, which is the approximation we have used to the value of the function as at syrupy one. So with this graph, we can say now that the actual value of the function that syrup in one it&#x27;s less than one by one, which is your approximation. We have found a party

Elham Kordzadeh · Answer

we have the shape here. It&#x27;s kitsch. It is in X equals two X that&#x27;s here. If peaks. It was 21 divided boy, too. One first extra look year. It is geeks. It wants to be doing this line x from the figure. Be understand that F cakes has the more accurate approximation at X equal toe one on for the B. The error in the liner approximation toe t near a point off a is the proportional to magnetism off the first order for supreme a frame? A. Because then, ah care nature decreases and becomes similar to the tension at points close toe A. If the value off after the crime off X decrease.

Problem

Quadratic approximations a. Let $Q(x)=b_{0}+b_{1…

Problem 53 Hard Difficulty

Answer

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

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

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Precalculus Review - Intro

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George B. Thomas, Jr. Maurice D. Weir, Joel HassThomas Calculus

Kristen K.

Samuel H.

Michael J.

Joseph L.

George B. Thomas, Jr. Maurice D. Weir, Joel Hass

Thomas Calculus