Newton's Method on Linear Approximation - Examples Part 2: Where it fails to Converge by mes

hive-128780 · @mes · Nov 18 '22 (edited)

$3.81

Newton's Method on Linear Approximation - Examples Part 2: Where it fails to Converge

https://youtu.be/mDSyBPsiv1Q

In this video I recap on Newton's Method on Linear Approximation and go through 4 very useful examples illustrating when Newton's Method either fails to converge or converges very slowly depending on the initial starting approximation you select.

---

Watch video on:

- 3Speak: []()
- Odysee: []()
- BitChute: [https://www.bitchute.com/video/LWO4fCGIJCTZ/](https://www.bitchute.com/video/LWO4fCGIJCTZ/)
- Rumble: []()
- DTube: []()
- YouTube: [https://youtu.be/mDSyBPsiv1Q](https://youtu.be/mDSyBPsiv1Q)

Download video notes: 

- PDF notes: https://1drv.ms/b/s!As32ynv0LoaIiaFkGGIFSe1q9PwLuA?e=h3PCu4
- Excel notes: https://1drv.ms/x/s!As32ynv0LoaIiaFf-oKGyfL1ZOW9OA?e=aDNrqg

---

# View Video Notes Below!

---

>Download these notes: Link is in video description.
>View these notes as an article: https://peakd.com/@mes
>Subscribe via email: http://mes.fm/subscribe
>Donate! :) https://mes.fm/donate
>Buy MES merchandise! https://mes.fm/store 
>More links: https://linktr.ee/matheasy
>**Follow my research in real-time on my MES Links Telegram:** https://t.me/meslinks
>Subscribe to MES Truth: https://mes.fm/truth
>
>Reuse of my videos:
>- Feel free to make use of / re-upload / monetize my videos as long as you provide a link to the original video.
>
>Fight back against censorship:
>- Bookmark sites/channels/accounts and check periodically
>- Remember to always archive website pages in case they get deleted/changed.
>
>Recommended Books:
>- "Where Did the Towers Go?" by Dr. Judy Wood: https://mes.fm/judywoodbook
>
>Join my forums!
>- Hive community: https://peakd.com/c/hive-128780
>- Reddit: https://reddit.com/r/AMAZINGMathStuff
>- Discord: https://mes.fm/chatroom
>
>Follow along my epic video series:
>- #MESScience: https://mes.fm/science-playlist
>- #MESExperiments: https://peakd.com/mesexperiments/@mes/list
>- #AntiGravity: https://peakd.com/antigravity/@mes/series
>-- See Part 6 for my Self Appointed PhD and #MESDuality breakthrough concept!
>- #FreeEnergy: https://mes.fm/freeenergy-playlist
>- #PG (YouTube-deleted series): https://peakd.com/pg/@mes/videos
>
>---
>
>NOTE #1: If you don't have time to watch this whole video:
>
>- Skip to the end for Summary and Conclusions (if available)
>- Play this video at a faster speed.
>-- TOP SECRET LIFE HACK: Your brain gets used to faster speed!
>-- MES tutorial: https://peakd.com/video/@mes/play-videos-at-faster-or-slower-speeds-on-any-website
>- Download and read video notes.
>- Read notes on the Hive blockchain #Hive
>- Watch the video in parts.
>-- Timestamps of all parts are in the description.
>
>Browser extension recommendations: 
>
>- Increase video speed: https://mes.fm/videospeed-extension
>- Increase video audio: https://mes.fm/volume-extension
>- Text to speech: https://mes.fm/speech-extension
>--Android app: https://mes.fm/speech-android

---

# Newton's Method on Linear Approximation - Examples Part 2: Where it fails to Converge

![Newton's Method Examples Part 2.jpeg](https://files.peakd.com/file/peakd-hive/mes/23vspBV1NicctLZ956B4N6zLFNkubANbcPkGEvRg6NVmmHjh737x3AB7NjyQvsBLSAP4S.jpeg)

## Recap on Linear Approximation

![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8CfKPvXW7EWXop2Gti83gXXpfuUM3tXeg7Au36B3ygxxmNaV2y2XHPWUSDR9K7S3G1.png)

## Example 1

Explain why Newton’s method doesn’t work for finding the root of the equation x<sup>3</sup> – 3x + 6 = 0 if the initial approximation is chosen to be x<sub>1</sub> = 1.

**Solution:**

![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8DBnQEtNKs13eoSJ1A8rWLjTHMYvADXL4grsL9xjXhJMmqFogSfYYvq3cFjYoaC4ix.png)

## Example 2

Use Newton’s Method to find the root of the equation x<sup>3</sup> – x = 1 for the following initial approximations:

>a) x<sub>1</sub> = 1
b) x<sub>1</sub> = 0.6
c) x<sub>1</sub> = 0.57

Graph f(x) = x<sup>3</sup> – x – 1 and it’s tangent lines at x<sub>1</sub> = 1, 0.6, and 0.57 to explain why Newton’s Method is so sensitive to the value of the initial approximation.

**Solution:**

![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8CSPFBxuJYQWHeoynRu3GoSjAiJxDXd8dDWqbweepB97eXwFA2exLz8M2pvpwinRBg.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/Eo2BpKHNZbth42UeRJsbCJkv4JevbCvCnUhrhm5NDtKLwP1ZdtvTohsok491wbUq5JB.png)

## Example 3

Explain why Newton’s method fails when applied to the equation ∛x = 0 with any initial approximation x<sub>1</sub> ≠ 0. 

Illustrate your explanation with a sketch.

**Solution:**

![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8D6MYf9SWog7DV4FL5DXDD91JGAyuttiofsr2eWe1nLNazZ7NqjnExMtPhDom7odGq.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8EjqhGAA8YzbqA6fKtvetbxpM9HduGC9UwK7z4jL81ujqBWbW6qJih2bSn169fM4qm.png)

## Example 4

If:

![image.png](https://files.peakd.com/file/peakd-hive/mes/23wr9i58hzJCXfzMiCdWugwu3h7t8vjj5fesNEmgSEcoEfJP5L7U9YSo8hoqWpE1KJ94D.png)

Then the root of the equation f(x) = 0 is x = 0. 

Explain why Newton’s method fails to find the root no matter which initial approximation x<sub>1</sub> ≠ 0 is used. Illustrate your explanation with a sketch.

![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8DBnQF2rBQBSoSaRkWVgZssHrx9GvPZ2TaHGgQc5ESo27J8T1irmNv8arm2mNrnpkZ.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/48HhKJZ3D9hrTJtQ35a7wHYDMuMhymrvPNjhkAJdHXaRpRox6yWXEKFVs1vCPEbYfX.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8DBnQFDCAT1GsZLbuULrSJMsbHcA2in5B7kumWxiwFfrnRkija4G9azsWGCHbcEArG.png)

👍 mahdiyari, minnowbooster, steemstem, mes, jerrybanfield, arcange, fw206, howo, postpromoter, lemouth, quinnertronics, intrepidphotos, theouterlight, mcsvi, rocky1, curie, stemsocial, firstamendment, nbs.gmbh, sagarkothari88, dhimmel, smartvote, roelandp, gadrian, hijosdelhombre, cyprianj, ibt-survival, acont, valued-customer, stem-espanol, goblinknackers, fengchao, techslut, operahoser, iamphysical, titusfrost, smartsteem, stayoutoftherz, steemitboard, tomastonyperez, pboulet, bennettitalia, realblockchain, meanbees, therising, takowi, lavista, prosocialise, aleestra, elvigia, utopis, gunthertopp, steemean, aboutcoolscience, laruche, netaterra, devann, meritocracy, doctor-cog-diss, hivebuzz, lordvader, ew-and-patterns, reinaseq, artgrafiken, and 164 others

properties (23)

`author`	mes
`permlink`	newtons-method-on-linear-approximation-examples-part-2
`category`	hive-128780
`json_metadata`	{"app":"peakd/2022.11.1","format":"markdown","tags":["mathematics","calculus","science","stemgeeks","palnet","proofofbrain","education","math","stem","school"],"users":["mes"],"image":["https://files.peakd.com/file/peakd-hive/mes/23vspBV1NicctLZ956B4N6zLFNkubANbcPkGEvRg6NVmmHjh737x3AB7NjyQvsBLSAP4S.jpeg","https://files.peakd.com/file/peakd-hive/mes/23t8CfKPvXW7EWXop2Gti83gXXpfuUM3tXeg7Au36B3ygxxmNaV2y2XHPWUSDR9K7S3G1.png","https://files.peakd.com/file/peakd-hive/mes/23t8DBnQEtNKs13eoSJ1A8rWLjTHMYvADXL4grsL9xjXhJMmqFogSfYYvq3cFjYoaC4ix.png","https://files.peakd.com/file/peakd-hive/mes/23t8CSPFBxuJYQWHeoynRu3GoSjAiJxDXd8dDWqbweepB97eXwFA2exLz8M2pvpwinRBg.png","https://files.peakd.com/file/peakd-hive/mes/Eo2BpKHNZbth42UeRJsbCJkv4JevbCvCnUhrhm5NDtKLwP1ZdtvTohsok491wbUq5JB.png","https://files.peakd.com/file/peakd-hive/mes/23t8D6MYf9SWog7DV4FL5DXDD91JGAyuttiofsr2eWe1nLNazZ7NqjnExMtPhDom7odGq.png","https://files.peakd.com/file/peakd-hive/mes/23t8EjqhGAA8YzbqA6fKtvetbxpM9HduGC9UwK7z4jL81ujqBWbW6qJih2bSn169fM4qm.png","https://files.peakd.com/file/peakd-hive/mes/23wr9i58hzJCXfzMiCdWugwu3h7t8vjj5fesNEmgSEcoEfJP5L7U9YSo8hoqWpE1KJ94D.png","https://files.peakd.com/file/peakd-hive/mes/23t8DBnQF2rBQBSoSaRkWVgZssHrx9GvPZ2TaHGgQc5ESo27J8T1irmNv8arm2mNrnpkZ.png","https://files.peakd.com/file/peakd-hive/mes/48HhKJZ3D9hrTJtQ35a7wHYDMuMhymrvPNjhkAJdHXaRpRox6yWXEKFVs1vCPEbYfX.png","https://files.peakd.com/file/peakd-hive/mes/23t8DBnQFDCAT1GsZLbuULrSJMsbHcA2in5B7kumWxiwFfrnRkija4G9azsWGCHbcEArG.png"]}
`created`	2022-11-18 17:49:33
`last_update`	2022-11-18 18:00:45
`depth`	0
`children`	2
`last_payout`	2022-11-25 17:49:33
`cashout_time`	1969-12-31 23:59:59
`total_payout_value`	1.924 HBD
`curator_payout_value`	1.886 HBD
`pending_payout_value`	0.000 HBD
`promoted`	0.000 HBD
`body_length`	5,505
`author_reputation`	210,178,283,269,445
`root_title`	"Newton's Method on Linear Approximation - Examples Part 2: Where it fails to Converge"
`beneficiaries`	`[]`
`max_accepted_payout`	1,000,000.000 HBD
`percent_hbd`	10,000
`post_id`	118,435,951
`net_rshares`	8,147,462,990,848
`author_curate_reward`	""

vote details (228)

voter	rshares	pct
eric-boucher	1,947,065,566	0.4%
thecryptodrive	6,170,421,521	0.16%
mammasitta	1,275,824,009	0.4%
roelandp	46,941,275,317	5%
cloh76	533,014,795	0.4%
arcange	354,192,187,452	2%
djennyfloro	484,565,782	10%
lordvader	7,804,311,905	0.8%
lemouth	266,563,269,283	10%
netaterra	9,081,422,743	0.4%
lamouthe	736,792,821	10%
titusfrost	19,433,310,911	100%
tfeldman	742,421,441	0.4%
mcsvi	125,450,572,366	50%
cnfund	1,456,968,537	0.8%
justyy	3,874,237,542	0.8%
curie	97,421,396,859	0.8%
techslut	24,190,688,865	4%
roguewriter	3,509,990,610	5%
steemstem	583,510,546,290	10%
walterjay	798,509,474	0.1%
valth	1,616,688,629	5%
dna-replication	348,573,631	10%
steemitboard	18,124,778,519	2%
dhimmel	54,416,621,078	2.5%
oluwatobiloba	835,273,820	10%
elevator09	516,727,273	0.4%
detlev	3,175,132,458	0.24%
lizanomadsoul	1,680,747,592	2%
mes	524,481,830,790	100%
federacion45	1,196,958,316	0.4%
gamersclassified	595,541,353	0.4%
iansart	660,815,411	0.4%
forykw	4,906,191,655	0.4%
mobbs	2,639,622,705	5%
valued-customer	32,671,801,122	25%
jerrybanfield	384,647,060,582	100%
rt395	1,582,729,093	1.5%
bitrocker2020	1,387,520,352	0.12%
sustainablyyours	4,202,530,377	5%
yehey	1,042,034,113	0.8%
helo	2,164,337,729	5%
arunava	1,218,584,575	0.08%
juancar347	2,368,412,541	0.52%
samminator	1,824,681,223	0.5%
mahdiyari	1,640,418,410,982	80%
lorenzor	1,343,467,179	50%
firstamendment	82,074,560,203	50%
alexander.alexis	5,389,398,287	10%
jayna	855,990,877	0.16%
ew-and-patterns	7,762,637,197	3%
princessmewmew	1,061,997,721	0.4%
joeyarnoldvn	490,982,765	1.47%
theouterlight	155,582,459,027	100%
gunthertopp	10,391,462,487	0.2%
flatman	662,301,484	0.8%
analealsuarez	1,661,555,971	50%
minnowbooster	1,139,613,530,782	20%
felt.buzz	938,566,133	0.1%
howo	308,183,921,461	10%
tsoldovieri	1,002,360,228	5%
steemwizards	523,403,450	0.8%
neumannsalva	603,063,131	0.4%
stayoutoftherz	18,377,060,120	0.2%
abigail-dantes	3,658,797,150	10%
zonguin	610,296,815	2.5%
joseg	1,182,331,232	80%
investingpennies	1,947,085,484	0.8%
khalil319	541,303,645	10%
iamphysical	21,413,708,781	90%
zyx066	628,510,926	0.24%
revo	1,455,061,379	0.8%
azulear	523,432,000	100%
djlethalskillz	1,358,267,838	5%
psicoluigi	856,704,498	50%
rocky1	115,151,795,029	0.12%
thelordsharvest	662,553,267	0.8%
aidefr	1,052,427,021	5%
therealwolf	4,571,880,239	0.4%
meno	3,118,403,789	0.4%
bhoa	706,580,988	5%
enzor	633,686,437	10%
dreamm	2,902,168,831	50%
dandays	1,300,251,328	0.16%
sunsea	582,920,262	0.4%
postpromoter	269,914,518,747	10%
steveconnor	618,772,470	0.4%
nicole-st	1,435,764,739	0.4%
smartsteem	19,222,549,750	0.4%
aboutcoolscience	9,958,839,916	10%
kenadis	2,701,695,132	10%
robotics101	3,089,901,887	10%
punchline	2,027,325,591	0.8%
sco	2,749,019,206	10%
anikekirsten	529,936,183	5%
ennyta	984,122,654	50%
brotherhood	5,966,280,141	0.8%
juecoree	3,245,176,645	7%
ydavgonzalez	1,784,999,597	10%
intrepidphotos	175,408,792,719	7.5%
fineartnow	504,117,772	0.4%
hijosdelhombre	42,405,254,409	40%
oscarina	908,676,034	10%
fragmentarion	2,559,719,169	10%
bennettitalia	15,762,355,702	25%
utube	619,616,242	0.8%
neneandy	881,589,466	0.8%
sportscontest	764,990,155	0.8%
pandasquad	1,251,702,239	0.8%
miguelangel2801	788,854,424	50%
fantasycrypto	610,100,076	0.8%
operahoser	23,507,106,580	100%
geopolis	625,308,427	10%
robertbira	1,040,493,863	2.5%
alexdory	1,553,734,844	10%
takowi	14,256,513,888	0.8%
irgendwo	2,005,344,446	0.8%
charitybot	4,824,301,500	100%
cyprianj	40,615,951,017	100%
melvin7	3,531,793,408	5%
francostem	1,348,544,093	10%
endopediatria	693,226,904	20%
croctopus	1,538,846,658	100%
superlotto	2,196,796,742	0.8%
tomastonyperez	16,956,048,996	50%
elvigia	11,090,218,374	50%
sanderjansenart	656,129,276	0.4%
gadrian	42,848,870,442	7.5%
therising	14,326,276,714	0.8%
gifty-e	628,690,277	80%
de-stem	5,472,564,293	9.9%
imcore	902,473,922	10%
josedelacruz	4,615,270,530	50%
achimmertens	1,940,266,425	1%
charitymemes	523,124,666	100%
realblockchain	14,690,696,982	50%
erickyoussif	751,628,095	100%
meanbees	14,373,936,028	10%
deholt	541,609,017	8.5%
fw206	309,265,855,326	6.6%
diabonua	709,116,019	0.4%
temitayo-pelumi	844,410,586	10%
andrick	857,691,558	50%
motherofalegend	1,622,154,723	5%
doctor-cog-diss	8,089,333,984	10%
acont	33,393,358,881	50%
uche-nna	729,771,625	0.64%
cheese4ead	562,103,865	0.4%
talentclub	1,190,177,109	0.52%
bflanagin	896,910,646	0.4%
armandosodano	1,724,780,946	0.4%
goblinknackers	31,144,232,663	7%
smartvote	53,415,832,159	2.5%
reinaseq	7,019,323,033	100%
kylealex	4,538,842,228	10%
fran.frey	4,174,142,495	50%
thelittlebank	2,510,359,004	0.4%
pboulet	16,172,445,146	8%
stem-espanol	32,083,163,818	100%
aleestra	12,964,588,386	80%
agrovision	1,844,592,242	5%
brianoflondon	4,375,968,292	0.12%
giulyfarci52	1,706,179,602	50%
esthersanchez	3,960,455,466	60%
steemcryptosicko	1,608,142,972	0.16%
cakemonster	665,497,447	0.8%
stem.witness	553,189,496	10%
devann	8,636,870,870	3%
shainemata	2,980,391,796	2%
wilmer14molina	1,019,551,077	50%
steemstorage	925,836,666	0.8%
aqua.nano	1,053,936,844	100%
crowdwitness	5,491,515,867	5%
apokruphos	3,554,276,756	1%
limka	174,071,985	79%
steemean	10,039,742,501	5%
deriyon	491,043,827	100%
epicdice	480,957,678	0.24%
monsterjamgold	2,754,259,332	10%
qwerrie	496,031,414	0.07%
aicu	1,792,874,318	0.8%
stem.alfa	2,657,285,132	100%
steemstem-trig	780,344,769	10%
atheistrepublic	775,496,457	0.4%
ibt-survival	33,857,628,091	10%
darthsauron	844,665,291	10%
roamingsparrow	1,010,264,624	0.4%
chapmain	0	100%
fengchao	28,270,743,460	2%
hivebuzz	7,813,567,606	2%
laruche	9,338,067,767	0.2%
ykretz	2,400,315,186	5%
stemsocial	83,534,364,458	10%
veeart	752,631,667	50%
artgrafiken	6,405,515,938	75%
noelyss	3,075,932,924	5%
greengalletti	2,967,842,342	5%
quinnertronics	232,882,963,917	100%
aabcent	1,658,038,841	0.64%
nbs.gmbh	60,251,792,465	0.8%
stemcur	765,096,577	100%
meritocracy	8,629,472,496	0.08%
dcrops	5,875,146,371	0.4%
krishu.stem	450,943,349	100%
dronegirl	283,833,152	100%
utopis	10,806,810,822	1.58%
moraviareosa	0	100%
drricksanchez	1,865,990,698	0.4%
nfttunz	1,216,702,709	0.08%
holovision.stem	1,766,945,840	50%
lavista	14,059,925,365	100%
mayberlys	2,122,852,682	50%
irivers	3,408,232,968	100%
aries90	6,159,945,534	0.8%
migka	4,368,491,078	90%
gejzep	752,592,021	0.4%
waivio.curator	3,081,533,827	5.69%
raph1	3,475,865,306	100%
sagarkothari88	55,474,040,578	5%
dondido	3,333,310,563	0.8%
prosocialise	13,465,867,226	5%
med-hive	499,033,793	5%
wasabian	442,015,449	100%
minas-glory	744,024,849	0.4%
the-grandmaster	638,369,209	0.4%
the-burn	818,402,176	0.4%
ambicrypto	2,216,271,833	0.8%
superuser3	1,980,109,872	5%