a-let-fx-be-a-function-satisfying-fx-leq-x2-for-1-leq-x-leq-1-show-that-f-is-differentiable-at-x0--3

Question

a. Let $f(x)$ be a function satisfying $|f(x)| \leq x^{2}$ for $-1 \leq x \leq 1$
Show that $f$ is differentiable at $x=0$ and find $f^{\prime}(0)$
b. Show that
$$f(x)=\left\{\begin{array}{ll}{x^{2} \sin \frac{1}{x},} & {x 
eq 0} \ {0,} & {x=0}\end{array}ight.$$
is differentiable at $x=0$ and find $f^{\prime}(0)$

Karl Schaefer · Accepted Answer

in this problem in part A. We&#x27;re said we&#x27;re given what f of x b a function satisfying the absolute value that of FX is less than or equal X squared for all ex Betweennegative one and one and were asked to prove that as his defensible it zero. And then we&#x27;re asked to find f prime of zero. So it&#x27;s just right down the limit that we know. So the limit is h goes to zero of f of x plus h minus FX over age. Oh, sorry. This should be f zero plus agents were interested in X equals zero. This is F zero plus H minus zero. So again, the question is what is f primavera? So we need to kind of figure out what all these terms are. So for the F zero term, we aren&#x27;t told what death of euro is explicitly, but we can still figure it out. We know that the absolute value of F zero is less than or equal to zero squared because this is the condition that&#x27;s given to us. This is his condition here, which is just zero. So the absolute value of zero is less than or equal to zero. This tells us that F zero has to be equal to zero because there&#x27;s no numbers other than Jiro whose absolute value is less than or equal to zero. So the zero is is nice. That&#x27;s good flattery, right? This is a limited H goes zero of half of age over eight. Now get in. We&#x27;ve got a kind of understand what FF H looks like. So once again, let&#x27;s use this condition. So we know that f of H in absolute value is less from ableto age squared for negative, one less cynical age last Niccolo won. So in particular, we only care about what&#x27;s happening with age. It goes to zero, so we only care about very small values of age. So we can rewrite this as negative age where, less than or equal to f of H less than or equal to H Square. So we get rid of these absolute value signs. Now it&#x27;s divide everything by H. So we have h negative age less than or equal to half of H over h less than or equal to H. So once we have this, we can finally sort of plug in plug into our women. Well, I really plug in, but we can use the squeeze The&#x27;re, um for limits or the sandwich theorem for limits. So we know that the limited H goes to zero of H is equal to zero, and the limit is h ghost. Zero of minus age is also equal to zero and f f h over h is squeezed between or sandwich between negative agent H. So this tells us that the limit is Asia goes to zero of death of H over A. JJ must be equal to zero by the sandwich there. Great. Now, in part B, you were asked to consider specifically the function f of X equals X squared times sine of one over X x is not equal to zero and zero if X equals zero and were asked to calculate f prime of zero. This is kind of hard to just to do, but luckily we just did part a. So let&#x27;s show that that FX satisfies the assumption from part eh, namely, we just need to check if the absolute value of FX is less than or equal to x squared. If we can show this, then then we&#x27;LL be done. So. Is this just for negative one less stable excess, Michael The one. So let&#x27;s calculated the absolute value. Beth of X is equal. The absolute value of X square times sine of one over x chaps, the value of X squared times the absolute value of sign of one over X. But we know how sign works. We know that the absolute value of sign is always less or equal to one, because sign only goes between minus one and one. So we&#x27;re just left with the absolute value of X squared times one but X is always positive or seen. The X squared is always positive. So the absolute value that&#x27;s where it is just X squared. So we do see that Oh, this should have been less than or equal to once we plugged in. Our minds are one there she have to value. SFX is less than or equal to x squared. So dysfunction satisfies the hypothesis from part A. So if we apply part, eh? We know that f prime of zero exists and is equal to zero

Amit Srivastava · Answer

According to the question we need to find secondary 30 we have given is viol. Affects is equal to minus F of X upon half of white. We&#x27;re using Jane rule to differentiate though function we have the function is that off axis? It will do minus f effects upon every bite. When we use Jane rule, we have differentiation with respect Wax Why does your fax is it will do Curly by Daschle fax Gurley x Upon girly X plus Gotta levi a dash upon Gurley via bond Girl Levi a bone girly X We have the value off curly upon Carly X minus apple vax upon ever white plus minus apple vex upon half of I gurley a bond girl Levi minus aftereffects upon after white Wendy, differentiate all the functions We know that the differentiation with respect to X and with respect why, when we differentiate the using caution pool we applied caution rule here when he applied question rule. We know that re square there is apple pie hole square and V. It means minus Afro white differentiation off you It means girly affects a bone girly X minus the it means f of x and definitions. You know, you girly ever by a bone Gurley x There is a question tool when we applied next side discussions rule we have plus minus aftereffects A bone ever white Duri&#x27;s minus f off Why girly affects a bone curly by minus aftereffects Gurley half of i a bone got a Levite whole Upon after vie whole scare There is a Kocian tool applied in differentiation I really respect to X and right When the soul this Tom we have minus after of I hold square Careful x x means double duty Apple X apple vita After all X fight well upon ever by hold you There is close eval Why f x every vie X minus half off x old square f off vai vai vitamins double a rarity There is whole effort by Q Jerry&#x27;s multiplication and solving previous equation When we sold this aggression here we have minus f y hold square apple eggs X los to half off X ever white off white X minus Every ex whole square Every vai vai their whole denominator trees have off like you. This is our first part. This is a double derivative of the function then second parks is that differentiation. The function X and minus one is equal to zero. Read respect toe. Thanks we say is that x Y minus? One is equal to zero with respect to X different shit. When we differentiate with respect wax, we know that curly upon Corey X X y minus one is equal to zero When we differentiate, we know that. Why girly bongo re x x plus x curly upon Carly Jacks. Why there is used product minus curly upon Carly X one is equal to zero. We differentiate the question we have white plus X Y dash is equal to zero And get the value of my dash Your EF minus by a phone That&#x27;s dude ermine The second daddy Ready Second Daddy. Ready? Oh, function when we differential. Secondary ready. We have X Y minus. One is equal to zero With respect to Thanks When were different Shoot with respect, Lex Curly upon Corey X via dash Is it called do curly upon Corey X minus by a bone Eggs We use caution. Probably you know that the caution rule nominator in unit First off in denominator hold square and but I denominator differentiation of New minute and miners New miniter differences No denominator Gurley a boundary X Why miners? Why Gary Bonk reacts. Thanks. When we saw these differentiation, we have minus X y dash plus by a bone ex quill We substitutes the value of minus via bone x four y dash In the gration Vai double dash is a call to minus X y dish, less vital upon X squared when we substitute the value divided double dash is a cool too minus X receptacle the value of by dash U minus by a bone X blessed by a bone x squared. Then we solve this equation we have. The final conclusion is so why a bone act square in this case reduced her mind the partial derivative of the function director Mine The partial daddy Very Dave off the function. Therefore, Exco Muay is equal to x y minus one. Differentiate with respect to x and y When we differentiate x and y we differentiate x Carla Bond Carly X The function is same half affects coma. Why is it cool to Carly upon curry x for different shared the function X comma y they R is X Y minus one When we differentiate with respect to X your function we know denardo differentiation One is equal to zero because the differentiation with respect to X you&#x27;re a final dessert is white When we differentiate with respect to white half off X com A wide theory function Gurley Bunker Levi XVIII minus one Duri&#x27;s X deter mine The double activity Daddy Ready Off the function half off Exco Muay is a little x Y minus one differentiate with respect to X and white When we differentiate with respect wax we know that curly blond curly x They&#x27;re full x go My way is equal to Gurley a bone girly X dirties vihk When we differentiate with respect wax there is why we know that the differentiation is then we differentiate with respect Why girly avant garde Levi? The function is evolve Exco Muay We differentiate with respect to why girly a Bond girl Levi X We know that their differentiation is also zero because no variable by in here the differentiation really respect wax f off by function Duri&#x27;s differentiation abilities back to X if X we know that there is one Then we differentiation off why the function is X We know that girly a bunker Levi Why is equal to one The substitutes the values 14 f off XX and 14 effort vai vai and x four f of x and vie for F off right and Vito for F off VIII IX. In the equation, we have voidable ash is equal to minus f of I hold square half off. Expects close to f of X are for white F off by X minus f off X holes where f off five i half off voidable slash xv Substitute all the values we have minus half off Why hold square apple xx blows to aftereffects ever Why after all, Vie X minus half off x whole square Apple pie right Hold up on half of I hold you When we substitute all the values we have the financial clues Unease Thanks Quill zero plus two x y minus y squared zero hole upon x Q. The final dessert is we get do X y upon x Q

Robert Daugherty · Answer

Okay, section 3.6, and we&#x27;re looking at problem number 97% is were there is a piece wise function. And so in looking at this function, we&#x27;re asked to determine a few things about it. So in part a, let&#x27;s show that the function is continuous at X equals zero. So if this function is continuous at X equals zero, then I should be able to look at the limit as I approach from the left. And as I approach from the right and it should be the same So if I look at this function, then the limit as X approaches zero from the left of F of X. So for negative values of X, that has to be zero just by definition. So what I need to find out now is I need to see that the limit is zero. Um, as I approach from the right. Okay, So if I look at this as I approach from the right, I&#x27;m looking at X times a sign of one over X. So first of all, let&#x27;s just look at the sign of one over X for those positive values of X. It&#x27;s gonna be a number between zero and one, just by nature of the range of the sine function. So I know that&#x27;s a number between zero and one. Okay, so what would happen if I look at X sign of one over X? Okay. What happens now? As X approaches zero. This approaches zero as X approaches zero. I approach zero on the right. And so I know that the limit of Exxon one over X has to be a number of this between zero and zero. This has to approach zero. So I get that f of X must be continuous because I have proven that the limit as X approaches zero of f of X equals zero. Okay, so I have continuity at the point X equals zero because the limit from the left and the right is the same. Part two tells me to find the derivative for ex, not equal to zero. Okay, so f prime of X. So to find for ex not equal to zero, you&#x27;re going to have f prime of X and we&#x27;ll ride our final answer here. So when X is not equal to zero for the negative values, I&#x27;m gonna get the derivatives equal to zero. So that&#x27;s for X values less than zero for the X values that are greater than zero. How do I differentiate ex? Find one over X So if when x is positive, I know that f prime of X is going to be equal to so the derivative of X is just simply one times a sign of one over X plus x The derivative of the sine function is gonna be co sign one over X times the derivative of one over X minus one over X squared. So this turns into you. Get what sign one over X minus one over X co sign one over X for X values that are positive. So that is the derivative for XVIII is not equal to zero Now show that F is not defensible at X equal to zero. So what I would need to look at is the derivative so f prime of X. So when X is positive, we had just determined that that was the sign a one over X minus one over x co sign one over X. Okay, So what would happen as X approaches? Zero. Okay, as X approaches zero. You know that this is a number that is bounded. Um, this is a number that is bounded between negative one and one. This coastline is a number that is bound to betweennegative one and one. But as I approach zero, this is going to go toward negative infinity. So what I&#x27;m gonna have is, as I approach if I approach zero from the left, Um, I know that the derivative f prime of X is going approach zero as I come from the left. As I approach from the right, it&#x27;s approaching negative infinity. So what I have is as I approached zero the derivative from the left and the right are not matching. I&#x27;m approaching zero from the left. Negative. Punitive, right? So I&#x27;m not defensible at that point. X equal zero.

Robert Daugherty · Answer

section 3.6 problem 1 11 So they give us a piece wise to find function Half of X is X signed one of her ex for positive values of X f of x zero for bodies of X less than or equal to zero show. This function is continuous at X equal to zero. So the easy cases for um for negative values leading up to zero f of X is equal to zero. So we know that f of zero is going to be equal to zero. So I have fo zero is equal to zero. If I can show that the limit his ex approaches zero of f of X is equal to zero, then I will have continuity if I can show that. So what I&#x27;m after I need to figure out what is the limit. I know as I approach from the left, the lemon is going to be zero. So I need to see what is the limit as X approaches zero from the right of X sign one over x it This could be shown to be zero then I&#x27;ve got continuity of this function. So it&#x27;s just a matter of showing this particular limit. Um, So what I know is that the sign value so I know that the sign of one of her ex is a number between one and negative one. Now, let&#x27;s multiply both sides of this equation by X So negative X is less than X Sign one over X, which has to be less than X and sorry about this is was less than or equal to now. What would happen if you took the limit as X approaches zero of this entire expression? Well, what happens on the two extremes? You&#x27;re going to get zeros. We&#x27;re gonna have to have zero less than or equal to X sign one over X, which has to be less than or equal to zero. So if the upper bound and the lower bound are approaching zero than Exxon over one Iraq&#x27;s this has to approach zero. So this function must be continuous. I&#x27;ve got F zero is equal zero, and I have the limit. As I approached zero from the right and trivially, the limit is a purchase here from the left all approaches zero. So this function has to be continuous at X equal to zero now The next thing wants to do is to find the derivative when X is not equal to zero. So for negative values, if f of x zero that you can see that F prime of X is also going to be zero now for positive I use. Now we just need to find the derivative of X times, a sign of one of Rex, and this would require use of the chain role. So all I need to do here is Look at this from a product standpoint. So the derivative of X is one time to sign a one over X plus X, the derivative of sign of one of Rex. That&#x27;s a co sign of one of our own Iraq&#x27;s times a derivative one of Rex&#x27;s minus one over X squared. And so this turns into F Prime of X is equal to sign one of her ex and then minus, um, one over X co signed one of her ex, and this is for X greater than zero. So I know that in this particular case, F prime of X is equal to sign one of her X minus one over x co sign one of our X. So here is my derivative for negative values. There is my derivative for positive values of X. And then in the third part of this were asked to look and see to show that F is not different trouble at X equals zero. Okay, so if I approach from the left, So in this particular case, right here, f of X zero, that&#x27;s a horizontal line. No problem. The derivative zero. So I have no problem at zero with that particular case. But what I need to look at is what happens from the from the right side. So what would be the limit would have to exist, So I would have to get the limit as X approaches zero of X, sign one of her X minus f of zero over X minus zero. This limit would have to exist in order for this function to be differential, um, at, um, at zero. And so, in this case, I&#x27;m approaching from the right, so the limit has to exist from the left or right, But there&#x27;s gonna be a problem with this limit as I approach from the right. So what is F of zero? We know that F zero is equal to zero. So this reduces down to the limit as X approaches zero from the right of X sign one over X over X, which is the limit X approaches zero from the right of the sign of one of her ex. So that number is bounded, his bounded between negative one and one. But what happens is it starts oscillating as you get to zero. So this limit does not exist. Okay, so since this limit does not exist, I&#x27;m not defensible at X equals zero. So there is, um this f of X is not differential at X equals zero because the limit definition of their derivative does not exist at zero. So we&#x27;ve looked at it from the right. We did not have a problem from the left because from the left you&#x27;ve got this horizontal line at zero. So that is defensible zero. But what you have on the right is you have these extreme oscillations with the sine function, so the limit does not exist from the right. If the limit does, the limit definition of the derivative doesn&#x27;t exist. This is not defensible at that particular point

Doruk Isik · Answer

in this problem. We&#x27;re given that a function after the first X were also given that bunch into blues and one for easy one, and we&#x27;re going out there with hope. Dysfunction is less than zero in this interval and it is a positive in this given trouble. And where&#x27;s to show whether fom fice is great? Really work for all? Let&#x27;s assume this to be true. So let&#x27;s assume that this is true. Less Look at what exes betweennegative immunity and one assist is defensible and continues. We can apply mean value there. So let&#x27;s do that. You have a crime that is equal to, uh, one minus some effort Banks. This is where the, uh, this interval that one will be greater. We have won my stance, so we don&#x27;t have one is one. So we have one minus x. You want my one line in space now? Since exes betweennegative unity and war, we know that one minus sites will be positive And in this interval right here were given that the narrative that time is less than zero. So it is negative. So these two means that one minus eggs should then me created and I&#x27;m sort of less than zero. It means that that ffx should be great to doing too well. Let&#x27;s look at the utter interval. The 2nd 1 assist your ex this time is between one infinity against is it is defensible in this ranch as well. We can apply me, Tara, and let&#x27;s try to find this time Best X minus one. You want my X minus one here. We know that, um, we know that X minus one is positive and given in this interval that crime is also positive. So these dude implies that f X minus one should be positive. So it means that perfect should be greater than or even to one hear from board intervals by a shooting. This to be true. We proved actually that this is the case. So we can say for bull X ffx is greater than or equal to what now? In party. We want to show that Taiwan is zero were given that f is differential before or less. It means that they&#x27;re it off. X has some has continued for all the ex volumes. So the critical point with us will be cut off for So let&#x27;s look at X What X is equal to one? We know that when X is less than one. So what X is less than one? That crime is less than zero. His negative one x is greater. Don&#x27;t want f time is positive. So in order for this damage to have continued throught a ll the real numbers This is it is changing this sign when X is one from this region say that when X is one f crime then should be zero because otherwise it wouldn&#x27;t have continuity, so it wouldn&#x27;t be continuous and it wouldn&#x27;t be differential.

Sajin Shajee · Answer

first we have. We&#x27;re trying to show that this is half of X is greater than one for all the values for party F of X minus one X squared, zero x rated, then one. Okay, Greater than or equal one excess less than run one When his effort x way CSOs big toe. So then for part B have a limit. So this is a test to be for this one part based on this way. Have that effort vaccine X ray nous ranges one divide by X minus one, which is greater than zero. And, uh, and one means X, which is provided by one minus X ray. Listen equals zero, which implies that as X approaches as as X approaches one, uh then if one well, coaches little purchase sear zero since F of X is is defensible Incan Here&#x27;s yeah

Problem

Graph $y=1 /(2 \sqrt{x})$ in a window that has $0…

Problem 60 Hard Difficulty

Answer

(a) $f^{\prime}(0)$ exists and equals $0 .$ See inside for the detailed explanation.

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(a) $f^{\prime}(0)$ exists and equals $0 .$ See inside for the detailed explanation.

Thomas Calculus

Heather Z.

Kayleah T.

Kristen K.

Samuel H.

Precalculus Review - Intro

Functions on the Real Line - Overview

George B. Thomas Jr.Thomas Calculus

Heather Z.

Kayleah T.

Kristen K.

Samuel H.

George B. Thomas Jr.

Thomas Calculus