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Calculus Of One Real Variable – By Pheng Kim Ving Chapter 8: Applications Of The Derivative Part 2 – Section 8.5: Approximations Of Roots Of Functions Newton's Method |
8.5 |
Return to roots. Powered by Create your own unique website with customizable templates. Home Social Studies 8 Social Studies 8 Textbook. Which must return into it again. 16:1 – 23 One child who does God's will is a greater blessing than many sinful offspring (vv. 1 – 4 ), for history and experience show that God punishes sin (vv.
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1.Newton's Method |
A root of a function f is a point x0 in dom( f ) such that f(x0) = 0. It's a solution or root of the equation f(x) = 0, ie, a
point where the graph of f intersects the x-axis. It's also called a zero of f. Let f be a differentiable function and suppose
f has a root. Newton's method is used to find a sequence of approximations a1, a2, a3, .. to the root that approaches the
root (ie, an is closer to the root than an–1 is). The idea is as follows.
Refer to Fig. 1.1. Guess a1. The tangent line to f at a1 intersects the x-axis at a2. The tangent line to f at a2 intersects the
x-axis at a3. And so on. This process can be repeated over and over to get closer and closer to the root. The equation of
Fig. 1.1 f(x0) = 0, |
the tangentline at a1 is y = f(a1) + f'(a1)(x–a1). Since (a2, 0) is on that line, we have 0 = f(a1) + f'(a1)(a2–a1), so:
In general, since (an, 0) is on the tangent line at an–1, we have:
Thus:
Recursion Formula
The formula an = an–1 – f(an–1)/f'(an–1) is a recursion formula. In mathematics, a recursion formula is one in which a
value in a sequence is calculated in terms of one or more previous values.
Example 1.1
Let f(x) = x3 + x – 1.
a. Show that f has a root.
b. Use Newton's method to find an approximate value of that root accurate to 6 decimal places. You may use a calculator.
Solution
a.f(0) = –1 < 0 and f(1) = 1 > 0. So by the intermediate-value theorem f has a root x0, which is in (0, 1).
b.f(x) = x3 + x – 1, f'(x) = 3x2 + 1. Let's take the initial guess a1 = 0.5. Then:
EOS
As f(0) = –1 < 0 and f(1) = 1 > 0, we have f(0) < 0 < f(1), so, as f is continuous, by the intermediate-value theorem
there exists c in (0, 1) such that f(c) = 0, ie f has a root in (0, 1).
We take the midpoint of the small interval in which the root is located as the initial guess.
The required accurate number of decimal places of the approximation is 6. In each calculation of an we should keep more
than 6 decimal places, for example 7 or 8, in order to get more accuracy in subsequent calculations and to see whether
the last an should be rounded up or down to be used as the answer. In this example we keep 8. In general if the required
accurate number of decimal places is k, you should keep k or k + 1 decimal places in the calculations of the an's and
round the last one to get an approximate value of the root x0.
The approximations stabilize at 6 decimal places after 5 iterations. So we're confident that they approach a limit, which
is a root x0 of f. Thus we're confident that the root x0 of f is approximately 0.682328 rounded up and accurate to 6
decimal places.
a5 and a6 have the same value up to at least 6 decimal places. We say that the approximations stabilize at 6 decimal
places at a5. We then stop and use the rounded value of a6 as the approximate value of the root. In general the
approximations stabilize when 2 consecutive an's have the same value up to at least the required accurate number of
decimal places. We then stop and use the rounded value of the last an as the approximate value of the root. Yes we use
the rounded value of the last an, say am, not that of the previous am–1, as am and am–1 may have different digits after the
required accurate number of decimal places. Note that in our example a6 and a5 have the same value up to at least 8
decimal places.
Contactkirby: triple deluxefree download. Go To Problems & SolutionsReturn To Top Of Page
2.Solving The Equation f(x) = 0 |
The equation x5 = 20 – 7x3 can be expressed as x5 + 7x3 – 20 = 0. The latter is of the form f(x) = 0 where f(x) = x5 +
7x3 – 20. In general an equation in an unknown x can always be expressed in the form f(x) = 0 by moving everything to
the left-hand side. The solutions of for example the equation x5 = 20 – 7x3 are the same as those of the equation x5 +
7x3 – 20 = 0 and thus are the same as the roots of the function f(x) = x5 + 7x3 – 20. Mathematics is often used to
determine a quantity by finding an equation that the quantity satisfies and then solving the equation. In algebra we
learned to solve simple equations such as x2 – 2x – 3 = 0. However we didn't learn to solve equations such as x5 + 7x3 –
20 = 0, which should be regarded as quite simple too. In the example below we'll utilize Newton's method to solve the
equation x5 = 20 – 7x3.
Example 2.1
Utilize Newton's method to solve the equation x5 = 20 – 7x3 with the solution or solutions accurate to 8 decimal places.
You many need a calculator.
Solution
EOS
To solve an equation we must find all of its solutions if they exist. If there's a unique solution then we must show this
uniqueness
As f(1) = –12 < 0 and f(2) = 68 > 0, we have f(1) < 0 < f(2), so, as f is continuous, by the intermediate-value theorem
there exists c in (1, 2) such that f(c) = 0, ie f has a root in (1, 2).
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3.Approximations Of Roots Of Numbers |
discussed in Section 8.3. Newton's method can also be used to approximate such roots.
Example 3.1
Solution
EOS
Remarks 3.1
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4.Newton's Method Doesn't Always Work |
Newton's method doesn't always work. For example, as seen in Fig. 4.1, the choice of a1 as shown causes the sequence
{an} to move farther away from x0 rather than closer to it. In this case the sequence {an} diverges to infinity. For another
example, as seen in Fig. 4.2, the choice of a1 as shown causes the sequence {an} to oscillate between a1 and a2 without
converging to x0.
Fig. 4.1 This choice of a1 causes |
Fig. 4.2 This choice of a1 causes |
Example 4.1 Monkey quest free download.
Let f(x) = x4 + 4x3 + 4x2 – x – 1.
a. Show that the equation f(x) = 0 has a solution in (–1, 0).
b. Attempt to use Newton's method to find a solution of f(x) = 0 starting with a1 = 0. What happens?
Solution
a.f(–1) = 1 > 0, f(0) =–1 < 0, so by the intermediate-value theorem there exists x0 in (–1, 0) such that f(x0) = 0, ie the
equation f(x) = 0 has a solution x0 in (–1, 0).
b.f(x) = x4 + 4x3 + 4x2 – x – 1, f'(x) = 4x3 + 12x2 + 8x – 1, a1 = 0,
Thus the an's oscillate between –1 and 0 and consequently don't converge to x0.
EOS
Note that the x0 in this example can't be either –1 or 0. However even if it were either –1 or 0, we would still be correct
to say that the an's oscillate between –1 and 0 and don't converge to x0, because they don't converge to either –1 or 0.
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Chapter 16 Return To Roots
5.Various Approximations |
Section 8.3 discusses the approximation of values of functions using tangent-line approximation. Section 8.4 deals
with the approximation of errors in measurement. This section, Section 5.3.3, handles the approximation of roots of
functions utilizing Newton's method.
Problems & Solutions |
Return To My Roots
1. Let f(x) = x – e–x.
a. Show that the function f has a root.
b. Use Newton's method to find an approximation of that root accurate to 6 decimal places. You may use a calculator.
Solution
a.f(0) = –1 < 0, f(1) = 1 – e–1 > 0, and f is continuous. So by the intermediate-value theorem f has root x0, which is in
(0, 1).
b. We have f'(x) = 1 + e–x. Let the initial guess be a1 = 0.5. Then:
2. Utilize Newton's method to solve the equation 3t5 = 10t3 – 15t + 15 accurate to 10 decimal places. You may use a
calculator.
Solution
Solution
4. Let f(x) = 2x3 – 3x2 – x + 1.
a. Show that the equation f(x) = 0 has a solution in (0, 1).
b. Attempt to use Newton's method to find a solution of f(x) = 0 starting with a1 = 1. What happens?
Solution
a.f(0) = 1 > 0, f(1) = –1 < 0, so by the intermediate-value theorem there exists x0 in (0, 1) such that f(x0) = 0, ie the
equation f(x) = 0 has a solution x0 in (0, 1).
b.f(x) = 2x3 – 3x2 – x + 1, f'(x) = 6x2 – 6x – 1, a1 = 1,
Thus the an's oscillate between 0 and 1 and consequently don't converge to x0.
Solution
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Under the figure of an unfaithful wife, God upbraids Jerusalem with her ingratitude and manifold disloyalties: but promiseth mercy by a new covenant.
[1] And the word of the Lord came to me, saying: [2] Son of man, make known to Jerusalem her abominations. [3] And thou shalt say: Thus saith the Lord God to Jerusalem: Thy root, and thy nativity is of the land of Chanaan, thy father was an Amorrhite, and thy mother a Cethite. [4] And when thou wast born, in the day of thy nativity thy navel was not cut, neither wast thou washed with water for thy health, nor salted with salt, nor swaddled with clouts. [5] No eye had pity on thee to do any of these things for thee, out of compassion to thee: but thou wast cast out upon the face of the earth in the abjection of thy soul, in the day that thou wast born.
[2] 'Make known to Jerusalem': That is, by letters, for the prophet was then in Babylon.
[6] And passing by thee, I saw that thou wast trodden under foot in thy own blood. and I said to thee when thou wast in thy blood: Live: I have said to thee: Live in thy blood. [7] I caused thee to multiply as the bud of the field: and thou didst increase and grow great, and advancedst, and camest to woman's ornament: thy breasts were fashioned, and thy hair grew: and thou wast naked, and full of confusion. [8] And I passed by thee, and saw thee: and behold thy time was the time of lovers: and I spread my garment over thee, and covered thy ignominy. And I swore to thee, and I entered into a covenant with thee, saith the Lord God: and thou becamest mine. [9] And I washed thee with water, and cleansed away thy blood from thee: and I anointed thee with oil. [10] And I clothed thee with embroidery, and shod thee with violet coloured shoes: and I girded thee about with fine linen, and clothed thee with fine garments.
[11]I decked thee also with ornaments, and put bracelets on thy hands, and a chain about thy neck. [12] And I put a jewel upon thy forehead and earrings in thy ears, and a beautiful crown upon thy head. [13] And thou wast adorned with gold, and silver, and wast clothed with fine linen, and embroidered work, and many colours: thou didst eat fine flour, and honey, and oil, and wast made exceeding beautiful: and wast advanced to be a queen. [14] And thy renown went forth among the nations for thy beauty: for thou wast perfect through my beauty, which I had put upon thee, saith the Lord God. [15] But trusting in thy beauty, thou playedst the harlot because of thy renown, and thou hast prostituted thyself to every passenger, to be his.
[11] 'I decked thee also with ornaments': That is, with spiritual benefits, giving you a law with sacrifices, sacraments, and other holy rites.
[16] And taking of thy garments thou hast made thee high places sewed together on each side: and hast played the harlot upon them, as hath not been done before, nor shall be hereafter. [17] And thou tookest thy beautiful vessels, of my gold, and my silver, which I gave thee, and thou madest thee images of men, and hast committed fornication with them. [18] And thou tookest thy garments of divers colours, and coveredst them: and settest my oil and my sweet incense before them. [19] And my bread which I gave thee, the fine flour, and oil, and honey, wherewith I fed thee, thou hast set before them for a sweet odour; and it was done, saith the Lord God. [20] And thou hast taken thy sons, and thy daughters, whom thou hast borne to me: and hast sacrificed the same to them to be devoured. Is thy fornication small?
[21]Thou hast sacrificed and given my children to them, consecrating them by fire. [22] And after all thy abominations, and fornications, thou hast not remembered the days of thy youth, when thou wast naked, and full of confusion, trodden under foot in thy own blood. [23] And it came to pass after all thy wickedness (woe, woe to thee, saith the Lord God) [24] That thou didst also build thee a common stew, and madest thee a brothel house in every street. [25] At every head of the way thou hast set up a sign of thy prostitution: and hast made thy beauty to be abominable: and hast prostituted thyself to every one that passed by, and hast multiplied thy fornications.
[21] 'Thou hast sacrificed': As there is nothing more base and abominable than the crimes mentioned throughout this chapter; so the infidelities of the Israelites in forsaking God, and sacrificing even their children to idols, are strongly figured by these allegories.
[26] And thou hast committed fornication with the Egyptians thy neighbours, men of large bodies, and hast multiplied thy fornications to provoke me. [27] Behold, I will stretch out my hand upon thee, and will take away thy justification: and I will deliver thee up to the will of the daughters of the Philistines that hate thee, that are ashamed of thy wicked way. [28] Thou hast also committed fornication with the Assyrians, because thou wast not yet satisfied: and after thou hadst played the harlot with them, even so thou wast not contented. [29] Thou hast also multiplied thy fornications in the land of Chanaan with the Chaldeans: and neither so wast thou satisfied. [30] Wherein shall I cleanse thy heart, saith the Lord God: seeing thou dost all these the works of a shameless prostitute?
[31] Because thou hast built thy brothel house at the head of every way, and thou hast made thy high place in every street: and wast not as a harlot that by disdain enhanceth her price, [32] But as an adulteress, that bringeth in strangers over her husband. [33] Gifts are given to all harlots: but thou hast given hire to all thy lovers, and thou hast given them gifts to come to thee from every side, to commit fornication with thee. [34] And it hath happened in thee contrary to the custom of women in thy fornications, and after thee there shall be no such fornication: for in that thou gavest rewards, and didst not take rewards, the contrary hath been done in thee. [35] Therefore, O harlot, hear the word of the Lord.
Chapter 16: Return To Roots Root
5.Various Approximations |
Section 8.3 discusses the approximation of values of functions using tangent-line approximation. Section 8.4 deals
with the approximation of errors in measurement. This section, Section 5.3.3, handles the approximation of roots of
functions utilizing Newton's method.
Problems & Solutions |
Return To My Roots
1. Let f(x) = x – e–x.
a. Show that the function f has a root.
b. Use Newton's method to find an approximation of that root accurate to 6 decimal places. You may use a calculator.
Solution
a.f(0) = –1 < 0, f(1) = 1 – e–1 > 0, and f is continuous. So by the intermediate-value theorem f has root x0, which is in
(0, 1).
b. We have f'(x) = 1 + e–x. Let the initial guess be a1 = 0.5. Then:
2. Utilize Newton's method to solve the equation 3t5 = 10t3 – 15t + 15 accurate to 10 decimal places. You may use a
calculator.
Solution
Solution
4. Let f(x) = 2x3 – 3x2 – x + 1.
a. Show that the equation f(x) = 0 has a solution in (0, 1).
b. Attempt to use Newton's method to find a solution of f(x) = 0 starting with a1 = 1. What happens?
Solution
a.f(0) = 1 > 0, f(1) = –1 < 0, so by the intermediate-value theorem there exists x0 in (0, 1) such that f(x0) = 0, ie the
equation f(x) = 0 has a solution x0 in (0, 1).
b.f(x) = 2x3 – 3x2 – x + 1, f'(x) = 6x2 – 6x – 1, a1 = 1,
Thus the an's oscillate between 0 and 1 and consequently don't converge to x0.
Solution
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Under the figure of an unfaithful wife, God upbraids Jerusalem with her ingratitude and manifold disloyalties: but promiseth mercy by a new covenant.
[1] And the word of the Lord came to me, saying: [2] Son of man, make known to Jerusalem her abominations. [3] And thou shalt say: Thus saith the Lord God to Jerusalem: Thy root, and thy nativity is of the land of Chanaan, thy father was an Amorrhite, and thy mother a Cethite. [4] And when thou wast born, in the day of thy nativity thy navel was not cut, neither wast thou washed with water for thy health, nor salted with salt, nor swaddled with clouts. [5] No eye had pity on thee to do any of these things for thee, out of compassion to thee: but thou wast cast out upon the face of the earth in the abjection of thy soul, in the day that thou wast born.
[2] 'Make known to Jerusalem': That is, by letters, for the prophet was then in Babylon.
[6] And passing by thee, I saw that thou wast trodden under foot in thy own blood. and I said to thee when thou wast in thy blood: Live: I have said to thee: Live in thy blood. [7] I caused thee to multiply as the bud of the field: and thou didst increase and grow great, and advancedst, and camest to woman's ornament: thy breasts were fashioned, and thy hair grew: and thou wast naked, and full of confusion. [8] And I passed by thee, and saw thee: and behold thy time was the time of lovers: and I spread my garment over thee, and covered thy ignominy. And I swore to thee, and I entered into a covenant with thee, saith the Lord God: and thou becamest mine. [9] And I washed thee with water, and cleansed away thy blood from thee: and I anointed thee with oil. [10] And I clothed thee with embroidery, and shod thee with violet coloured shoes: and I girded thee about with fine linen, and clothed thee with fine garments.
[11]I decked thee also with ornaments, and put bracelets on thy hands, and a chain about thy neck. [12] And I put a jewel upon thy forehead and earrings in thy ears, and a beautiful crown upon thy head. [13] And thou wast adorned with gold, and silver, and wast clothed with fine linen, and embroidered work, and many colours: thou didst eat fine flour, and honey, and oil, and wast made exceeding beautiful: and wast advanced to be a queen. [14] And thy renown went forth among the nations for thy beauty: for thou wast perfect through my beauty, which I had put upon thee, saith the Lord God. [15] But trusting in thy beauty, thou playedst the harlot because of thy renown, and thou hast prostituted thyself to every passenger, to be his.
[11] 'I decked thee also with ornaments': That is, with spiritual benefits, giving you a law with sacrifices, sacraments, and other holy rites.
[16] And taking of thy garments thou hast made thee high places sewed together on each side: and hast played the harlot upon them, as hath not been done before, nor shall be hereafter. [17] And thou tookest thy beautiful vessels, of my gold, and my silver, which I gave thee, and thou madest thee images of men, and hast committed fornication with them. [18] And thou tookest thy garments of divers colours, and coveredst them: and settest my oil and my sweet incense before them. [19] And my bread which I gave thee, the fine flour, and oil, and honey, wherewith I fed thee, thou hast set before them for a sweet odour; and it was done, saith the Lord God. [20] And thou hast taken thy sons, and thy daughters, whom thou hast borne to me: and hast sacrificed the same to them to be devoured. Is thy fornication small?
[21]Thou hast sacrificed and given my children to them, consecrating them by fire. [22] And after all thy abominations, and fornications, thou hast not remembered the days of thy youth, when thou wast naked, and full of confusion, trodden under foot in thy own blood. [23] And it came to pass after all thy wickedness (woe, woe to thee, saith the Lord God) [24] That thou didst also build thee a common stew, and madest thee a brothel house in every street. [25] At every head of the way thou hast set up a sign of thy prostitution: and hast made thy beauty to be abominable: and hast prostituted thyself to every one that passed by, and hast multiplied thy fornications.
[21] 'Thou hast sacrificed': As there is nothing more base and abominable than the crimes mentioned throughout this chapter; so the infidelities of the Israelites in forsaking God, and sacrificing even their children to idols, are strongly figured by these allegories.
[26] And thou hast committed fornication with the Egyptians thy neighbours, men of large bodies, and hast multiplied thy fornications to provoke me. [27] Behold, I will stretch out my hand upon thee, and will take away thy justification: and I will deliver thee up to the will of the daughters of the Philistines that hate thee, that are ashamed of thy wicked way. [28] Thou hast also committed fornication with the Assyrians, because thou wast not yet satisfied: and after thou hadst played the harlot with them, even so thou wast not contented. [29] Thou hast also multiplied thy fornications in the land of Chanaan with the Chaldeans: and neither so wast thou satisfied. [30] Wherein shall I cleanse thy heart, saith the Lord God: seeing thou dost all these the works of a shameless prostitute?
[31] Because thou hast built thy brothel house at the head of every way, and thou hast made thy high place in every street: and wast not as a harlot that by disdain enhanceth her price, [32] But as an adulteress, that bringeth in strangers over her husband. [33] Gifts are given to all harlots: but thou hast given hire to all thy lovers, and thou hast given them gifts to come to thee from every side, to commit fornication with thee. [34] And it hath happened in thee contrary to the custom of women in thy fornications, and after thee there shall be no such fornication: for in that thou gavest rewards, and didst not take rewards, the contrary hath been done in thee. [35] Therefore, O harlot, hear the word of the Lord.
Chapter 16: Return To Roots Root
[36] Thus saith the Lord God: Because thy money hath been poured out, and thy shame discovered through thy fornications with thy lovers, and with the idols of thy abominations, by the blood of thy children whom thou gavest them: [37] Behold, I will gather together all thy lovers with whom thou hast taken pleasure, and all whom thou hast loved, with all whom thou hast hated: and I will gather them together against thee on every side, and will discover thy shame in their sight, and they shall see all thy nakedness. [38] And I will judge thee as adulteresses, and they that shed blood are judged: and I will give thee blood in fury and jealousy. [39] And I will deliver thee into their hands, and they shall destroy thy brothel house, and throw down thy stews: and they shall strip thee of thy garments, and shall take away the vessels of thy beauty: and leave thee naked, and full of disgrace. [40] And they shall bring upon thee a multitude, and they shall stone thee with stones, and shall slay thee with their swords.
[41] And they shall burn thy houses with fire, and shall execute judgments upon thee in the sight of many women: and thou shalt cease from fornication, and shalt give no hire any more. [42] And my indignation shall rest in thee: and my jealousy shall depart from thee, and I will cease and be angry no more. [43] Because thou hast not remembered the days of thy youth, but hast provoked me in all these things: wherefore I also have turned thy ways upon thy head, saith the Lord God, and I have not done according to thy wicked deeds in all thy abominations. [44] Behold every one that useth a common proverb, shall use this against thee, saying: As the mother was, so also is her daughter. [45] Thou art thy mother's daughter, that cast off her husband, and her children: and thou art the sister of thy sisters, who cast off their husbands, and their children: your mother was a Cethite, and your father an Amorrhite.
[46] And thy elder sister is Samaria, she and her daughters that dwell at thy left hand: and thy younger sister that dwelleth at thy right hand is Sodom, and her daughters. [47] But neither hast thou walked in their ways, nor hast thou done a little less than they according to their wickednesses: thou hast done almost more wicked things than they in all thy ways. [48] As I live, saith the Lord God, thy sister Sodom herself, and her daughters, have not done as thou hast done, and thy daughters. [49] Behold this was the iniquity of Sodom thy sister, pride, fulness of bread, and abundance, and the idleness of her, and of her daughters: and they did not put forth their hand to the needy, and to the poor. [50] And they were lifted up, and committed abominations before me: and I took them away as thou hast seen.
[49] 'This was the iniquity of Sodom': That is, these were the steps by which the Sodomites came to fall into those abominations for which they were destroyed. For pride, gluttony, and idleness are the highroad to all kinds of lust; especially when they are accompanied with a neglect of the works of mercy.
[51] And Samaria committed not half thy sins: but thou hast surpassed them with thy crimes, and hast justified thy sisters by all thy abominations which thou hast done. [52] Therefore do thou also bear thy confusion, thou that hast surpassed thy sisters with thy sins, doing more wickedly than they: for they are justified above thee, therefore be thou also confounded, and bear thy shame, thou that hast justified thy sisters. [53] And I will bring back and restore them by bringing back Sodom, with her daughters, and by bringing back Samaria, and her daughters: and I will bring those that return of thee in the midst of them. [54] That thou mayest bear thy shame, and mayest be confounded in all that thou hast done, comforting them. [55] And thy sister Sodom and her daughters shall return to their ancient state: and Samaria and her daughters shall return to their ancient state: and thou and thy daughters shall return to your ancient state.
Other documents viirejected scriptures. [53] 'I will bring back': This relates to the conversion of the Gentiles out of all nations, and of many of the Jews, to the church of Christ.
Chapter 16: Return To Roots Book
[55] 'Ancient state': That is, to their former state of liberty, and their ancient possessions. In the spiritual sense, to the true liberty, and the happy inheritance of the children of God, through faith in Christ.
[56] And Sodom thy sister was not heard of in thy mouth, in the day of thy pride, [57] Before thy malice was laid open: as it is at this time, making thee a reproach of the daughters of Syria, and of all the daughters of Palestine round about thee, that encompass thee on all sides. [58] Thou hast borne thy wickedness, and thy disgrace, saith the Lord God. [59] For thus saith the Lord God: I will deal with thee, as thou hast despised the oath, in breaking the covenant: [60] And I will remember my covenant with thee in the days of thy youth: and I will establish with thee an everlasting covenant.
Roots Return Policy
[61] And thou shalt remember thy ways, and be ashamed: when thou shalt receive thy sisters, thy elder and thy younger: and I will give them to thee for daughters, but not by thy covenant. [62] And I will establish my covenant with thee: and thou shalt know that I am the Lord, [63] That thou mayest remember, and be confounded, and mayest no more open thy mouth because of thy confusion, when I shall be pacified toward thee for all that thou hast done, saith the Lord God.