4

# Advance Maths Problem 45Please donâ€™t answer compulsory I will givedislikeProblem 45: Lagrangeâ€™s identity (âœ“âœ“âœ“) 1987 Paper II If y = ...

## Question

###### Advance Maths Problem 45Please donâ€™t answer compulsory I will givedislikeProblem 45: Lagrangeâ€™s identity (âœ“âœ“âœ“) 1987 Paper II If y = f(x), the inverse of f is given by Lagrangeâ€™s identity: f âˆ’1 (y) = y+ âˆ‘âˆž 1 1 n! d nâˆ’1 dy nâˆ’1 [ y âˆ’ f (y) ]n , when this seriesconverges. (i) Verify Lagrangeâ€™s identity when f (x) = ax. (ii)Show that one root of the equation x âˆ’ 1 4 x 3 = 3 4 is x = âˆ‘âˆž 0 32n+1 (3n)! n!(2n + 1)! 43n+1 . (â€ ) (iii) Find a solution for x, asa series in Î

Advance Maths Problem 45 Please donâ€™t answer compulsory I will give dislike Problem 45: Lagrangeâ€™s identity (âœ“âœ“âœ“) 1987 Paper II If y = f (x), the inverse of f is given by Lagrangeâ€™s identity: f âˆ’1 (y) = y + âˆ‘âˆž 1 1 n! d nâˆ’1 dy nâˆ’1 [ y âˆ’ f (y) ]n , when this series converges. (i) Verify Lagrangeâ€™s identity when f (x) = ax. (ii) Show that one root of the equation x âˆ’ 1 4 x 3 = 3 4 is x = âˆ‘âˆž 0 3 2n+1 (3n)! n!(2n + 1)! 43n+1 . (â€ ) (iii) Find a solution for x, as a series in Î», of the equation x = eÎ»x . [You may assume that the series in part (ii) converges and that the series in parts (i) and (iii) converge for suitable a and Î».] Comments This looks pretty frightening at first, because of the complicated and unfamiliar formula. However, its bark is worse than its bite. Once you have decided what you need to find the inverse of, you just substitute it into the formula and see what happens. Do not worry about the use of the word â€˜convergenceâ€™; this can be ignored. It is just included to satisfy the legal eagles who will point out that the series might not have a finite sum. In part (ii) you can, as it happens, solve the cubic by normal means (find one root by inspection, factorise and use the usual formula to solve the resulting quadratic equation). The root found by Lagrangeâ€™s equation is the one closest to zero. Equation (â€ ) turns out to be a very obscure way of writing a familiar quantity.29 Lagrange was one of the leading mathematicians of the 18th century; Napoleon referred to him as the â€˜lofty pyramid of the mathematical sciencesâ€™. He attacked a wide range of problems, from celestial mechanics to number theory. In the course of his investigation of the roots of polynomial equations, he discovered group theory (in particular, his eponymous theorem about the order of a subgroup dividing the order of the group), though the term â€˜groupâ€™ and the systematic theory had to wait until Galois and Abel in the first part of the 19th century. Lagrangeâ€™s formula, produced before the advent of the theory of integration in the complex plane, which allows a relatively straightforward derivation, testifies to his remarkable mathematical ability. It is practically forgotten now, but in its day it had a great impact. The applications given above give an idea how important it was, in the age before computers #### Similar Solved Questions

##### I. A subset Y of Fn is closed (that is, V(J(Y)) = Y) if and only if Y is an afline K-variety determined by some subset S of K[xt Xn]:
I. A subset Y of Fn is closed (that is, V(J(Y)) = Y) if and only if Y is an afline K-variety determined by some subset S of K[xt Xn]:...
##### (4 pts) Use any numerical integralion technique to approximate sin(0 2 deto al loast six significant digits Lef n be the number of subintervals Explain which numerical procedure YOu are using _ which value of n YOU are Using nd why you Ihink YOU have Usod large enough value of value of n. Make sure your calculalor In radians_4, | ' ; -Ie ' .'0 ( ^:tw4} P ` -ddine %-l4^ | u -'"0Fe
(4 pts) Use any numerical integralion technique to approximate sin(0 2 de to al loast six significant digits Lef n be the number of subintervals Explain which numerical procedure YOu are using _ which value of n YOU are Using nd why you Ihink YOU have Usod large enough value of value of n. Make sure...
##### T/2 tan x dx .5. Evaluate
T/2 tan x dx . 5. Evaluate...
##### 3) What is magnitude of the total gravitational force on the fifth sphere?Submit
3) What is magnitude of the total gravitational force on the fifth sphere? Submit...
##### Identify the expected product for the following reactionAICl;AICI}AICl~AICl}~AIClzDD
Identify the expected product for the following reaction AICl; AICI} AICl ~AICl} ~AIClz DD...
##### AACUGAAUGGGGCCUACAUUGUAAAUCGUCGUAUUAUGADecode Ihe RNA sequence above into an amino acid sequence I you aller the Cylosine (with the slar above W) to an adenine; what arino acid would it code Ior and would you expect the mutation lo be conservalive Or radical?
AACUGAAUGGGGCCUACAUUGUAAAUCGUCGUAUUAUGA Decode Ihe RNA sequence above into an amino acid sequence I you aller the Cylosine (with the slar above W) to an adenine; what arino acid would it code Ior and would you expect the mutation lo be conservalive Or radical?...
##### Centa"Thaore TAccording Wrigicy;ucre cact the probability thatthe proporonEacoutsealou conbin 2570 cach calom slarhurst tha bag los5 than 22567prngc Ycllow Jnd pink]Eeiatics Raungani WanITnstni chim suppostrndomn 3JMPLn KatburrCnn MnnaConceNAnANcOWhatW"e 4ncurSianurat candysmped Mche ansierceamalWnat We wwan;Kor0.22 /80.2240.22}Nong onthg aboyeClick - selbctVol answerls
Centa" Thaore T According Wrigicy;ucre cact the probability thatthe proporon Eacoutsealou conbin 2570 cach calom slarhurst tha bag los5 than 22567 prngc Ycllow Jnd pink] Eeiatics Raungani WanI Tnstni chim suppost rndomn 3JMP Ln Katburr Cnn Mnna ConceNAnANcO WhatW"e 4ncur Sianurat candy sm...
##### Homework: Section 5.2 ath Homework Sizore: 0 of 1 pt 9 of 20 (3 complete)SaveHW Score: 10.63_Bus Econ 5.1.87Question HelpMonthly sales of a particular computer are expected to decline at the following rate of S'(t) computers per month_ where lime in months and S() is the number of computers sold each month:S' (t) = 30tThe company plans to stop manufacluring Ihis compuler when monthly sales reach 900 computers monthly sales now ( 0) are 2,160 compulers_ find S(t) . Use graphing calculat
Homework: Section 5.2 ath Homework Sizore: 0 of 1 pt 9 of 20 (3 complete) Save HW Score: 10.63_ Bus Econ 5.1.87 Question Help Monthly sales of a particular computer are expected to decline at the following rate of S'(t) computers per month_ where lime in months and S() is the number of computer...
##### Determine whether the lines with the given slopes are parallel, perpendicular, or neither.$$m_{1}=3 ; m_{2}=- rac{1}{3}$$
Determine whether the lines with the given slopes are parallel, perpendicular, or neither. $$m_{1}=3 ; m_{2}=-\frac{1}{3}$$...
##### Usc this tablc about diffcrcntiablc functions and to answcr thc following:17 . If h(z) = ezf(a), first find formula for W' (c) and then find h' (0)18. If j(2) = -Af(e)g(e), first find formula for j' (2) and then find j' (1)rf(c) 19. If k(z) first find formula for k' (2) and then find k' (-2) 9(1)20. IF l(z) = -ig(z) and "'(2) = -48 first fird formula for "' (#) and then find g (2)In(z) =t2 f()g(r), [irst firuc] forla for "' (*) atd] the
Usc this tablc about diffcrcntiablc functions and to answcr thc following: 17 . If h(z) = ezf(a), first find formula for W' (c) and then find h' (0) 18. If j(2) = -Af(e)g(e), first find formula for j' (2) and then find j' (1) rf(c) 19. If k(z) first find formula for k' (2) a...
##### A thin plate is bounded by the graphs of $y=e^{-x}, y=-e^{-x}, x=0,$ and $x=L .$ Find its center of mass. How does the center of mass change as $L \rightarrow \infty ?$
A thin plate is bounded by the graphs of $y=e^{-x}, y=-e^{-x}, x=0,$ and $x=L .$ Find its center of mass. How does the center of mass change as $L \rightarrow \infty ?$...
##### (NOTE: For this problem; hand-drawn visualizations will be accepted for full credit:)Consider a sphere of radius 9 centered at the origin Use Cartesian coordinates only to set up a triple integral that represents the volume of the spnere_ b) Use Cartesian coordinates only to set up a double integral that represents the volume of the sphere. Jse 'spherical coordinates to set up a triple integral that represents the volume of the sphere: Evaluate the integrals from parts b) and c) Which was e
(NOTE: For this problem; hand-drawn visualizations will be accepted for full credit:) Consider a sphere of radius 9 centered at the origin Use Cartesian coordinates only to set up a triple integral that represents the volume of the spnere_ b) Use Cartesian coordinates only to set up a double integra...
##### The gene PRNP produces the prion protein. Inhumans there are two alleles for the gene, knownas Met and Val. Only people whoare homozygous for Met cancontract variant CJD, the fatal human form ofmad cow disease associated with the consumption of infected beef.However not everyone who is homozygousfor Metactually contracts the disease, even if theyhave consumed infected beef; the disease haslow penetrance, meaning it does not developin everyone who has the associated genotype.In Denmark, a survey
The gene PRNP produces the prion protein. In humans there are two alleles for the gene, known as Met and Val. Only people who are homozygous for Met can contract variant CJD, the fatal human form of mad cow disease associated with the consumption of infected beef. However not everyone who is homozyg...
##### Henry is interested in the amount of time it takes to completean online order. He samples 100 orders and finds the mean to be 4minutes with a standard deviation of 30 seconds.Construct a 99% confidence interval mean amount of time.-not sure if you have to convert SD or mean to seconds orminutes a little confused!
Henry is interested in the amount of time it takes to complete an online order. He samples 100 orders and finds the mean to be 4 minutes with a standard deviation of 30 seconds. Construct a 99% confidence interval mean amount of time. -not sure if you have to convert SD or mean to seconds or minutes...
##### It is claimed that sports-car owners drive on the average 20,000 miles per year. A consumer firm believes that the mean annualmileage is actually lower. To check, the consumer firm decided totest this hypothesis. The modeling assumptions are that the annualmileage is normally distributed with an unknown mean Âµ and with thestandard deviation of 1200. The consumer firm obtained informationfrom 36 randomly selected sports-car owners that resulted in asample average of 19, 530 miles. What is the de
It is claimed that sports-car owners drive on the average 20, 000 miles per year. A consumer firm believes that the mean annual mileage is actually lower. To check, the consumer firm decided to test this hypothesis. The modeling assumptions are that the annual mileage is normally distributed with an...