5

QUESTION(8 MARKS)Find the first five terms of the Maclaurin series expansion for f(z) = marks)Hence, show the Maclaurin series expansion for g(x) e2 $ is given by22...

Question

QUESTION(8 MARKS)Find the first five terms of the Maclaurin series expansion for f(z) = marks)Hence, show the Maclaurin series expansion for g(x) e2 $ is given by22"n=umarks)From (i) , evaluate the following integrale2r' 21 1) dx_maks)

QUESTION (8 MARKS) Find the first five terms of the Maclaurin series expansion for f(z) = marks) Hence, show the Maclaurin series expansion for g(x) e2 $ is given by 22" n=u marks) From (i) , evaluate the following integral e2r' 21 1) dx_ maks)



Answers

Find the Maclaurin series for the functions in Exercises 11-22.
$$e^{-x}$$

Okay, so here. Well, we have our function, right. F racks is equal to hyperbolic sign of acts. Now this what I have everyone out here is for a while, um, hyperbolic co sign of X, which was equal to e to the X plus e to the negative X over two. Now I've brought Sign that a change too. Well, instead of each of the X plus eat the native X, this plus here is gonna become a minus. So we had ik of the x minus. Eat negative X over to now are MacLaurin series. Start saying right are MacLaurin series as well equal to the Keith derivative of our function. Evaluated at zero off the K factorial times extricate is given here. Okay, Now, if we start taking derivatives well, if you take, don't take any derivatives were just valuing the function itself. F um 00 Okay, we have to be quick. Tt zero Now instead of plus, we have a minus again. So t e 20 minus e to the negative 0/2, which is no longer equal one, but is equal to zero. Okay, And then well, if we do the first derivative. We then have e to the X, um, plus e to the negative axe over two. So evaluating that at zero is going to be right now, is there any more? But now one So and then another derivative. When you're back to the function itself, you back to e to the x minus. Oops. You back to either the X minus e to the negative x okay. And then evaluating that zero is going to be you. So again, it's the same. It's just like you're off by one instead of I mean, you start basically at the what the derivative waas of hyperbolic co sign is A instead of work you right now that have peroxide. But you still going to just switch right back and forth between your value waiting That is our Keith derivative at zero. It's going to either be 01 still right. So then well, if we look at, um, I'm MacLaurin series formula here we did the Keith derivative evaluated at zero over Cape factorial Extra que. We're gonna switch everything off by my ways. We want you to start with, um, in sort of one here. Units start with zero you start with zero and men, you see well, plus one. So plus one over. Well, it's still want factorial times X But then every numerator is going to just change from being. If it was one, they changed zero. It was What if was zero change toe one. So, again, this is starting at the previous problem he had, which was high product, um, coastline. And now we're doing hair products. Say so. Then this last numerator here is gonna be changed with zero over a perfect at all times extra fourth. Okay, so this is now going to become well, okay, So lacking. That's just a race. This Okay? You know, we can erase, um, the actual answer here. Okay, just a second. T risk this knowledge. So, um what we have no. Is the some going from Well, okay, well, this this this right here can be certified to, um, get an estimate. There was no starting with. Well, one x over one pictorial, which is just X. So it's his acts. Um, plus, now 1/3. Factorial, um, times X cube. Right. And then, um, plus well, so on, Which is the sum where n goes from zero to infinity. Oh, x to the or now we get every odd right Every on exporting is all the even Exponents are going to be zero because numerator is there. There. So now we get we get every ad one instead of every even one. So we get X to be in some extra 20 x to the two n plus one, um, divided by two n plus one victory divided by two n plus Warren Territorial. So there we have it. There is our McLaurin Siri's condition, but thank you.

Okay, so let's consider the function. Um, FX is equal to x times e to the X, so wish to find the McLaurin Siris for this function. So, um, we were called right that the MacLaurin series is while piece of Interfax rate centered at zero where X equals zero. So we have f of zero plus x times f prime of zero plus x squared over two factorial times. Um, the second derivative evaluated at zero plus. Right. Plus, it's going and then we have the ETS derivative. Um oh, um, so x abandoned me in the distributive or or yes, derivative, evaluated at zero times x sub end over and or X to the end over And Victoria. Okay, um, so we don't want to find some derivatives here, right? And eventually, what we want to find is one find the insta relative. So, um, we consider the function after backs is equal to X e v X. Okay. Been to find its first derivative. What we take f crime of X. Okay, So notice, um well, is a product will here. Right. So, um, department wrote what you get for your first derivative is well, x e V X So the first derivative is X You the X, um plus either the ex and second factor on it, even the X This is equal to e to the X Times X plus one in the first derivative, then our second derivative. Um, if there were prime of X is equal to well, even the X plus x plus one times E V x, which is equal to yeah, to the X Times X plus two. Okay. And then, um well, the third derivative Oh, I s, um very waited. Well, OK, so we'll just take the derivative here, um, is well equal to eat it The X plus X plus two times even the X, which is equal to e to the axe Times X plus, um three. Okay. And then we did 1/4 derivative. Um, we would see we have eat to the X Times X plus four and then, while in general, the Enstrom votive. So the and it's derivative of F our backs is equal to well e to the X Times X plus and write whatever the end through your ends derivative happens to be, we get each of the X times X plus ed for the derivative. Okay, so, um Well, then we can evaluate, Right? These ads zero. So, at zero, what we get? Um Well, first we get f of zero, right? Just a function of the function value at zero. So f of zero is just equal. Chu what? Zero. Okay, now f prime of zero we put in jail here. You eat with zero, which is one times zero plus one times one, which is equal to one. So f primes f prime or this is Well, okay, that's room F prime right up. Zero is equal to a 11 Well, f double crimes evaluated at zero is equal. Chew. Well, um, looks just ask you this actually should be X plus. You must stay here. This should be experts to Okay, So if double Pirate zero is well, circles to is two times zero, which is one. So one time two, which is 2.0, is equal to two. How about F Tropical Joe? Well, you put in zero plus three, which is three times even zero. Once again, it's one. So one times three is three. So f no crime of zero is equal to three and that weaken general last year. And safety ends derivative, um, guy with it at their also the ends derivative right value with that zero is just equal to well, it's equal Teoh e to the zero times zero plus n So it's jail plus planets and times were each The zero is one. So one times and it just end whatever the answer group is the f derivative. If I would let you know it's just equal to men. Okay, Um so substitute these values end to our formula here for the ends. Um um polynomial here to find the, uh McLaurin. Siri's becomes right. MacLaurin series for F of X is equal to x times Each of the X is equal to what equal to the sum right and is equal to zero corner going, or it will very off and times X to the end over and factorial. Okay, so there we have it. And this is equal to well, zero plus x, um, and one. So it was X plus while x squared over two factorial, um, times two. So two factorial times two plus X cube over three factorial times three. Um plus right, um X to the n over n factorial. Ah, times. And so there we have our MacLaurin series. No.

It's question using that 1/1 minus plight equals one plus my plus y por toe on so on. So one over x four to minus one equals negative 1/1 minus exports toe which is equal negative 1/1 minus y we're four x equal access player So negative 1/1 minus X s play is equal negative one plus x a square class excess square old horse toe Ansan Pointing Verification. We can get the negative one plus exports to plus export or and so on, which is equal negative information off. And it would give you can exports to win. Oh Heinz that four X minus two over x or to minus one equals four X minus two but the play negative one or what? One look to buy one minus x a square, which is equal for X minus tool marked by negative one minus exports to minus export nine and so on. By simplification, we can get that this factor is equal to minus four x plus to export to minus for exports three and so on.

So we have ever vex. It's equal to sign of X over two. And our objective here is to find the MacLaurin series while at X equals zero right off this function. So, um, recall right that the MacLaurin series when we have the tale of Siri's I'm sending that. Well, um, after secret, you a or or A is equal to zero. So we have, Well, f of axe is equal to the sum K goes from zero to infinity of the case, derivative of F evaluated at a or B that zero over k factorial times X minus a rex minus zero to the case power. So, um, do this. We're going to find the case derivative right above. So we gotta find while some derivatives and then find a general derivative. So, um, given that f of X is equal to sign up excellent, too. Well, they don't take any derivatives. Zero derivative. We just evaluate the function at zero. So we have f of zero is just equal to well, sign of zero, which is equal to zero. Okay, so everything is equal to zero. Now we differentiate. Um, well respected ex. So we have our first derivative or F crime of X is equal to, well, 1/2 times co sign of, um X over two. Okay, this is a chain rule there. So then we evaluate at X equals zero. So we get f prime of zero or the first derivative evaluated at zero is equal to, well, 1/2 times co sign of zero cosign. A zero is one so f prime of zero is equal to one house. Um, then we continue to find, um, while the second derivative. So we differentiate f prime of X, right? The derivative off the derivative would be the second derivative, So f double prime of X is equal to 1/2 times. Um, negative one halftime sign of X over two. So right, trend will again. And we get, um, second derivative is equal to negative 1/4 times sign of axe over to. Okay, so then, um, to find while the second derivative evaluated at zero. So f prime evaluated at zero. Well, gonna have sign of zero, which is just zero. Right. So native, 1/4 times zero is still zero. The second derivative evaluated. Add zero is equal to zero. Okay, then to find, um, the third derivative. So the third derivative here is going to be off the derivative all the second derivative. So if third do every of F little crime looks people, time of X is equal to what we have. Native, 1/8 times negative. One halftime sign of actual for two, which is equal to a positive 1/16 times sign of, um, time sign and sign of X over two UK and, um, the fourth derivative. Well, this is a velvet at zero, but those at zero we get, um Well, I actually wait a minute. Wait a minute. Wait a minute. This is actually mistake here because this should actually be right. Derivative off sign is not science co sign. Right. So this is going to be, um well, negative. OK, so actually and jumped ahead here, so I jumped ahead. So this is the third derivative off off back. The axe is going to actually be equal to, um, negative 1/8 times co Sign of X over to right, derivative of sign of actually a sign of black. It's called stand up lap times driven off the block. So we get native 1/8 co sign affects over two and therefore at X equals zero or F. Um, the third derivative of F evaluated at X equals zero is equal to well, just native one is right. This is becomes one coastline of zero is one becomes negative one. OK? And if we were to do, um, or the fourth derivative the floor through it, we differentiating the third derivative. So the fourth derivative of with respect of axe right is equal to, um Well, now we get 1/16 times. Sign X older, too. And the fourth derivative evaluated at zero is just equal to zero. Okay, um, and if we did one more Well, you get the fifth derivative right? Is 1 32nd times co sign of X over two and evaluating the fifth derivative at zero would give us 1/32. Okay, so we see a pattern here, right? We can basically get that the, um and do I MacLaurin series. We get that f of axe as equal to well zero plus let's see 1/2 times, while one factorial times x plus zero times 2/2 factorial times x squared. But this term is gonna be gone. Right? Because you just zero let me get a minus 1/8 times three factorial times x cube, and then a plus. We'll just zero. But you have zero over 0/4 factorial times X to the fourth. And, um um right. So the general what we have here, Well, this is gonna become, um, this is equal Teoh X over two. Because this first was just zero. Every second term is gonna be gone, so we get zero, um, minus X cube over two cubes, times three, factorial plus x to the fifth over, uh, to to the fifth times five factorial. Okay. And continues. So, what this becomes this becomes the some. This is equal to the some of well, n goes from zero all the way to infinity of negative one to the end Times x to the, um, to end plus one. He gives us the odd values right to in plus one. Just the idea, um, and native one to the end, while the being that becomes positive. Negative, positive, negative. Right. And we're dividing by to to the end. Plus one times two and plus one factorial. All right, so there we have our McLaurin. Siri's. Okay,


Similar Solved Questions

5 answers
0 3 4'0 a434M'ZAq + P uLLo} 311 Jo suaqunu [pa4 1ID J0 135 341 uopejiidpinw Jo uoippv Jo uoqiuijap Iensn a41 01 Jjadsaj 4WM piai} JO ulewop '/ej831U! ue S! Jas BuImollo} 341 Jay1aym apijaa
0 3 4'0 a434M'ZAq + P uLLo} 311 Jo suaqunu [pa4 1ID J0 135 341 uopejiidpinw Jo uoippv Jo uoqiuijap Iensn a41 01 Jjadsaj 4WM piai} JO ulewop '/ej831U! ue S! Jas BuImollo} 341 Jay1aym apijaa...
5 answers
(Simpllly 1 2 40x - 1 Wunluixew 1 3 3 Round to Ihe 1 11 number 0i unins Mm 1 proquced must be produced ano sold eold in ordet 1 Maxun pfol2 and cost; 8
(Simpllly 1 2 40x - 1 Wunluixew 1 3 3 Round to Ihe 1 11 number 0i unins Mm 1 proquced must be produced ano sold eold in ordet 1 Maxun pfol 2 and cost; 8...
2 answers
Suppose that A and B are square symmetric matrices in Rnxn Show that ifA and B have the same eigenvectors, then they commute; i.e , AB = BA: (6) Show that ifAB = BA and A has no repeated eigenvalues, then Aand B have the same eigen vectors_ Clearly show where you use the assumption that there are not repeated eigenvalues:
Suppose that A and B are square symmetric matrices in Rnxn Show that ifA and B have the same eigenvectors, then they commute; i.e , AB = BA: (6) Show that ifAB = BA and A has no repeated eigenvalues, then Aand B have the same eigen vectors_ Clearly show where you use the assumption that there are no...
5 answers
3 N 81. [8 pts] Find the QR factorization of the matrix A
3 N 8 1. [8 pts] Find the QR factorization of the matrix A...
5 answers
Find thu relative exlreme polnts tho function, itttoy exist Then skolch gmph of Iho function G(r)=x' Idontily all the ralalivo minMun points Sulect the correct cholce bolow and, nocossary; (ha ans48complete your choiceTha telallvo minimum poini(e) ilaro (Simplity Your anauUr Use intogers Tfaclons (OrAnv numbers Tharo Oro rolntwo minlmum pointe axprexecan Typeodeted pair Use _ commabeparate ants wensneadad )
Find thu relative exlreme polnts tho function, itttoy exist Then skolch gmph of Iho function G(r)=x' Idontily all the ralalivo minMun points Sulect the correct cholce bolow and, nocossary; (ha ans48 complete your choice Tha telallvo minimum poini(e) ilaro (Simplity Your anauUr Use intogers Tfac...
5 answers
Tio Kaniedl capacinr C and C = conkoclod n paralit , ar antutatea Bouica t DC voltage Vo. Each acquttos cherge Q0 . Tha vohaou souico Osconaecad and thon an insulabng Ma uih delectro constant (K insoried fll tte Cpa0t between tha Plales ol 5PartDolermine chigo capacto dieloctrc Wneemem Expreet Your anewi valng tro aIgnhicent nquretAEdsubmmAcaueatananaPart BEEDraanawot UlingduqniticontIla4noiSubmliAcquobpanic
Tio Kaniedl capacinr C and C = conkoclod n paralit , ar antutatea Bouica t DC voltage Vo. Each acquttos cherge Q0 . Tha vohaou souico Osconaecad and thon an insulabng Ma uih delectro constant (K insoried fll tte Cpa0t between tha Plales ol 5 Part Dolermine chigo capacto dieloctrc Wneemem Expreet You...
5 answers
91,2 Find Isgn(a) and &-L where a=( } ; & ; & 3 9)
91,2 Find Isgn(a) and &-L where a=( } ; & ; & 3 9)...
5 answers
Maxwell started a home theater business in 2012. The revenue of his company for that year was $240,000. The revenue grew by 20% in 2013 and 30% in 2014. Maxwell projected that the revenue growth for his company in the next 6 years will be at least 25%/year: If that is true; how much would Maxwell expect his minimum revenue to be for 2017?
Maxwell started a home theater business in 2012. The revenue of his company for that year was $240,000. The revenue grew by 20% in 2013 and 30% in 2014. Maxwell projected that the revenue growth for his company in the next 6 years will be at least 25%/year: If that is true; how much would Maxwell ex...
5 answers
7070.CoCOuertonOperations Fall 2020.docx boratory 2020 doc" AJ kdI boratory Operatians Ftom jru= dola; which tyR of glusuwc je Posf iccurrte (6r hxs tke bowest naecnt erut)?thisk thoull hac Excn Ikk most accuraka?WEich .ahchly Rt7i 5 reasurciGEA docs D4 47ny8 IE4 a bizhlyHulnFouldemntutdyou Rtchlcn0 FLock'PrtstBurridlet
7070.CoC Ouerton Operations Fall 2020.docx boratory 2020 doc" AJ kdI boratory Operatians Ftom jru= dola; which tyR of glusuwc je Posf iccurrte (6r hxs tke bowest naecnt erut)? thisk thoull hac Excn Ikk most accuraka? WEich . ahchly Rt7i 5 reasurciGEA docs D4 47ny8 IE4 a bizhly Huln Fouldemnt ut...
5 answers
Suppose that two continuous random variables Xand Y have a joint probability density function (PDF): f(z,y) Ax? (y - 4), ~1 <r <2,1 < y < 3and f(z,y) 0 elsewhere: What is the value of A? b) What is P(0 < X <1,2 <Y < 3)2 Construct the marginal probability density functions fx (1) and fy (y) d) Are the random variables X and Y independent? Explanation! If Y = 3, what is the conditional probability density function of X?
Suppose that two continuous random variables Xand Y have a joint probability density function (PDF): f(z,y) Ax? (y - 4), ~1 <r <2,1 < y < 3and f(z,y) 0 elsewhere: What is the value of A? b) What is P(0 < X <1,2 <Y < 3)2 Construct the marginal probability density functions fx ...
5 answers
Complete each Maclaurin Series by showing at least first 5 nonzero terms w/ the correct use of +. Orcosx1-X
Complete each Maclaurin Series by showing at least first 5 nonzero terms w/ the correct use of +. Or cosx 1-X...
5 answers
At most three projects are to be selected.Ox1+ X2 + X3 + X4 = 3Ox1 + Xz + X3 + X4 > 3Ox1+ X2 + Xa + X4 2 3Ox1+ Xz + Xa + Xa < 3Ox1 + X2 + X3 + X4 $ 3
At most three projects are to be selected. Ox1+ X2 + X3 + X4 = 3 Ox1 + Xz + X3 + X4 > 3 Ox1+ X2 + Xa + X4 2 3 Ox1+ Xz + Xa + Xa < 3 Ox1 + X2 + X3 + X4 $ 3...
4 answers
Selected values of f () are shown in the table below. What is the trapezoidal Sum approximation for f(c) dx using 3 subintervals as indicated by the table?f(z)Answer:Subin]t Answer
Selected values of f () are shown in the table below. What is the trapezoidal Sum approximation for f(c) dx using 3 subintervals as indicated by the table? f(z) Answer: Subin]t Answer...
5 answers
Find a possible formula for the graph.f(r)sin(€ v)+2
Find a possible formula for the graph. f(r) sin(€ v)+2...
1 answers
Evaluate the partial integral. $$ \int_{1}^{e^{y}} \frac{y \ln x}{x} d x $$
Evaluate the partial integral. $$ \int_{1}^{e^{y}} \frac{y \ln x}{x} d x $$...
5 answers
The equilibrium constant, Kp, for the followingreactionis 2.01 at 500 K:PCl3(g)+ Cl2(g) PCl5(g)Calculate the equilibrium partial pressures of all specieswhen PCl3 and Cl2,each at an intitial partial pressureof 1.27 atm, are introduced into anevacuated vessel at 500 K.PPCl3=atmPCl2=atmPPCl5=atm
The equilibrium constant, Kp, for the following reaction is 2.01 at 500 K: PCl3(g) + Cl2(g) PCl5(g) Calculate the equilibrium partial pressures of all species when PCl3 and Cl2, each at an intitial partial pressure of 1.27 atm, are introduced into an evacuated vessel at 500 K. PPCl3 = atm PCl2 = at...

-- 0.021058--