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Question

Find the first three nonzero terms of the Maclaurin expansion of the given functions.
$$f(x)=(2 x-1)^{2}$$

Tyler Moulton · Accepted Answer

Oh we want to find the first three non zero terms of the MacLaurin series expansion for the function after taxes? Two x minus one square. This question is child your understanding of how to utilize series expansion? Who functions as such? We appreciate we have MacLaurin series is to solve this problem, MacLaurin series states that F of X is equal to zero plus F prime zero X plus F double time zero, which is about to expire and so on. Where the term of F MacLaurin series is the derivative of F zero. By the way, N factorial times X to the app. Thus we have since our derivative rather since our function F x zero is negative, one squared equals one because at least two derivatives. So F one is two times two, X minus one times two equals eight, X minus four, F two is eight. All and after that are zero. So are MacLaurin series ends at the second derivative. Thus we have series one minus four X plus four X squared with no continuation afterwards.

Jaida L · Answer

so another MacLaurin series for Eat Our X is equal to Sigma from in equal zero to infinity exit. Part end about by in factorial, which is equal to one plus x plus x squared about a by two factorial plus X cubed whereby three factorial plus dot, dot dot the Siri&#x27;s converges on negative infinity to infinity and let&#x27;s see, let&#x27;s generate the Siri&#x27;s. But with bex, which is one minus x plus X squared times, eat part X. So we&#x27;re just gonna replace you the part x with this Read it right up here. And so that is going to equal one minus X plus X squared time Sigma for men equals zero to infinity Ex apart ended when Ryan Factorial. And so if we, um expend, we get one plus 1/2 ex squid plus 2/3 x cubed, plus dot, dot dot This Siri&#x27;s right here converges on negative infinity to infinity, just like the above Siri&#x27;s. And so the first terms the 1st 3 terms are going to be this right up here, and it&#x27;s going to converge on a negative infinity too. Positive. Infinity

Jaida L · Answer

So the model in Syria&#x27;s generated by sign of eggs is equal to and the Sigma from an equal zero to infinity Negative 12 part end about a bite to impossible on factorial times x the power of to in plus one This serious converges on negative infinity to infinity he series generated by, um X time Senate X squared is going to be so let me just read it. Also f X equals x times, um, sort of X and in squared. So you&#x27;re going to replace son of ex with what we found right up here. And so it is gonna look like X times. So we&#x27;re just refining the, uh, serious. So an equal zero to infinity Negative wants the party end by the bite to and plus one factorial times exit, part two in plus one. Remember, there&#x27;s a squared right here, so we&#x27;re going to add a squared. And so once we simplify, we get X cubed minus 1/3 x tooth there plus two divided by 45 times x, the seventh minus one divided by 3 15 Excellent. Nine plus dot dot Doc, This convergence on negative infinity to infinity and so here are the first few terms. The first few terms we&#x27;re going to be X cubed minus one. That expects, plus to to buy the 45 access seventh. So right up here, These are the 1st 3 terms, and the interval of convergence is negative. Infinity to infinity.

Jaida L · Answer

So we need to figure out the MacLaurin series for both for both coastline of X and one divided by one minus X. So we&#x27;ll start with consent of X. So once we find out the derivatives of G of X and the value of those derivatives at X equals zero weaken generate the MacLaurin series, so it&#x27;s gonna look like one plus zero times x minus X squared to barter by two factorial, plus zero times X cubed plus X. The fourth developed for factorial plus dot, dot, dot plus and then a general negative one to the part of end times, actually part to end, divided by two in factorial plus doctor dark. And so this is equal to the Siri&#x27;s form and equals zero or sigma form to infinity. They could have wants to part in intense X, the part you into but by two in factorial and that is just equal to co sign of X or G of X. Other those to the same for one divided by one minus X. So let&#x27;s that h of X equal to one&#x27;s about about one minus X and then figure out the derivatives again at X equals zero and let&#x27;s generate the MacLaurin series. So that is equal to one plus X plus X squared, plus X cubed plus doctor and then in Sigma form. That&#x27;s equal to some from zero to infinity in the next part in and so that is equal to again. Want to ride by one minus X or H backs? And so, since our original function is given as co son of Ex minus two, divided by one minus X, well, that means we can just rewrite this, which is the given in terms of our new calculated Siri&#x27;s So X is equal to remember Coast and an ex. We found that it was Sigma permanent and from n equals zero to infinity, the negative one that wants a part of end times X apart to end bye bye to in factorial Last to remember. It&#x27;s too, because there is a numerator of to appear so last two times the signal that we just found in equal zero to Infinity Ex apartment. And so this sequel, Teoh Negative one must to x slash five hives X squared. You might have started thought once you expand, so this converges on the intersection of negative infinity to positive affinity and negative 1 to 1. And so the interval convergence is negative one one, whereas the 1st 3 terms are up here. Negative one minus two x minus five halves X squared.

Jaida L · Answer

so the Mekele in Syria&#x27;s generated by side of exes Sigma from n equals zero to infinitely negative went to the party end, but by two and plus one factorial times x the part of two in post one. This converges are negative. Infinity to infinity and the Siri&#x27;s generated by, um, the natural log of one plus X is sigma for men equals one to infinity Negative once a part of n minus one. But my Aunt Times X apartment, which converges on native 1 to 1. So again this Siri&#x27;s we&#x27;re here is equal son X and this Siri&#x27;s right here is natural log of one plus X. And so the Siri&#x27;s generated by F of X, which is just these two functions multiplied together, is going to equal Sigma an equal zero to infinity. Negative wants the part and divided by two in plus one factorial times X to the power to and plus one and then this multiplied by Sigma n equals zero n equals one Teoh infinity the negative wants the part of and minus 12 by two by end times X apartment. And so once reasonable fire we should get This is equal to x squared minus 1/2 x cubed +16 x Afford this convergence on the intersection of hold on X squared. So this is X squared. This Siri&#x27;s right here converges on the intersection of negative infinity to infinity and negative one toe one. So the actual interval convergence is negative ones toe one. Remember, this is going to be minus Doctor dog. And so the first terms are going to be represented right here. So X squared minus what happened? X cubed +16 x to the fourth and then the interval. Convergence is negative 1 to 1.

Ahmed Kamel · Answer

in this question. 40. We&#x27;re expert mention or negative X. Where will sign eggs? The teachers is a Taylor series off function if off X and E is defined as if Frank&#x27;s equal and before e plus the over the X your coffee over Cody and one and so on. So by finding the directives off, give off X, which is, as we said before, E board negative excess square, we&#x27;ll sign X and let equals zero. So we will have one plus the year over factory in one X plus negative city, over factory to Access Square and so on. If we simply fight, we can get one minus city over to export toe plus 25/24 export four and so on. Similarly, in fourth B, we will get that Len one plus X over one minus. That&#x27;s thank you.

Problem

Find the first two nonzero terms of the Maclaurin…

Problem 20 Easy Difficulty

Find the first three nonzero terms of the Maclaurin expansion of the given functions.
$$f(x)=(2 x-1)^{2}$$

Answer

$\sin ^{2}(-1)+\frac{2 \sin (-2)}{1 !} x+\frac{8 \cos (-2)}{2 !} x^{2}$

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$\sin ^{2}(-1)+\frac{2 \sin (-2)}{1 !} x+\frac{8 \cos (-2)}{2 !} x^{2}$

Basic Technical Mathematics with Calculus

Anna Marie V.

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Joseph L.

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

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