solve-the-given-problems-by-newtons-method-find-the-smallest-positive-root-of-the-equation-tan-x2--2

Question

Solve the given problems.
By Newton's method, find the smallest positive root of the equation $	an x=2 x .$ Verify the result by using the zero feature of a calculator.

Helen Latting · Accepted Answer

so the equation were given is negative. X minus one equals the inverse tangent of X. Um, and we&#x27;re supposed to start at X zero, equals one and find ex of two. So the equation we&#x27;re gonna use is, um, acts of n plus one equals access end minus F of X over the derivative of F of X. And to use this equation because this is finding the roots, we set, um, one side of this equation equal to zero. So you do that by putting all the components on one side. So the inverse tangent of X plus X plus one equals zero that we plug this in for, um, off of acts and I should say, x event here. Um, so access en Plus one. Actually, we&#x27;re going to say x of one because we&#x27;re starting with X zero equals Come. And I also messed that up. That&#x27;s X zeros one. Um, certain doing X of one equals one minus the inverse tangent of X zero plus ex severe plus one over. Um, the derivative of the tangent is one of her X squared plus one. Um, plus one. Um and so you plug in only there were plug in, um, one for the other exes. Heroes. And you get except one equals, um, negative. 0.85 six. Ah, And, um, and to find, except to actually just copied and pasted the number I got here. So, um, these were not the same number of just what points that used. Um, So now, to find exit, too, we plug in x of one, um, into our equation one. So we get ex of two equals negative. 0.85 69 minus the inverse tangent. Uh, zero point negative. What should make that clear? Negatives. You. 85 69 Oh, minus zero point a bubba Bob plus one. And I should have been Not the whole doesn&#x27;t point, but in that room. Over what Over negatives. Your point. A 56 I&#x27;m sward plus one plus one. And when you plug this in, um, you will get that this part here the, um f of X over f prime of X. That&#x27;s equal to negative 0.3586 and you want this to be a smallest possible Um, this is not the best, um, number that we want. This is a little big, but it&#x27;s good enough because we were asked to find an exit, too. Um, and so if you plug all those numbers in excess of two equals negative zero point for 98 three.

Amrita Bhasin · Answer

The first step of Newton&#x27;s method is to take the derivative after Prime is gonna be three X squared, minus one over one plus X squared. Now that we have this, we know we have Newton&#x27;s method formula, which means that if we say exit zeros negative one plug in negative one. We end up with Native 0.91 for exits to night of 0.90 Ex of three negative 0.90 Again, which gives the first rule, is approximate native 0.90 Do the same for the second route. Acts of Wanda&#x27;s 0.91 bucks of Jews 0.90 exit 3.90 which gives us our two routes as negative 0.90205 25

Oswaldo Jiménez · Answer

we&#x27;re going to solve the newly near equation X plus tangent if X equals zero. So, uh, four decimal places using Newton&#x27;s method and a computer or calculator and we get your shoes any initial gas X zero that is not except truant. And here we&#x27;re going to shoot 60 in the interval, negative by half and by half. So we identify first the function f of X equals do X plus tangent of eggs. So we&#x27;re solving the knoll in immigration f of X equals zero. To use the news, Newton&#x27;s method we got to find the reality of 1/2 which is equal to one plus second of Eggs Square, which is you derivative of tangent of X sinking square. So, um, with this we can write down the Newton situation, which is X seven plus one equals X, have been minus If affects of Ben divided by the derivative of Jeff, it takes event, and that&#x27;s equal to X of Ben, minus ah, pics of end plus the tangent of except been divided by one plus sickened of except then square for and greater than or equal to zero. So we have these situation. We can use these in a calculator through the dating expression Ex Sedated Bye and expression X minus X plus tangent of eggs divided by one plus the second of eggs square. Choosing an initial gas X different from the excitement of the equation. We know that these knowing your aggression has zero&#x27;s a PSA route, and in fact, it&#x27;s the only route of the known immigration. In the end of our interval, negative by have by half. So we&#x27;re going to shoot. Sending child gets in this interval different from Syria, for example, we&#x27;re going to shoes the initial guess equal to one, and with that, we obtained the following. It reads ex one approximately two syrup on for 2 to 12 x to approximately equal to see your 0.0 to six for one x three, approximately equal to I see your point 000006 and eggs for every seaman equal to 7.7 2 to 4 time. Stand to the negative 17 which is already a very small number. So from the fourth de Mar, we have all the terms appeared to be very small. So, in fact, the Newton&#x27;s method is convergent this case to value eggs equal to Syria with fort. It&#x27;s more places. So he, uh, methods going to converge to the solution X equals zero, which was any and you shall guess you from forearm suit.

Julian Wong · Answer

from. You wanna solve this equation for answers in between zero and two pipe So 10 to X is an identity. So we can write that as to 10 X over one minus 10 squared X plus 10 X equals zero and I&#x27;m gonna multiply old terms by one minus 10 Squared X. To get rid of the denominator, we&#x27;ll have to tan x plus, uh, tan X minus 10 cubed X equal zero. Which means we have er 3 10 x minus 10. Cubed X equals to zero. We can take out a ah common factor of 10 X. We&#x27;re left with three minus 10 squared X equals zero. So tan X equals zero or tan squared X is equal to three. Or, in other words, tan X is equal to plus or minus root. Three. Those are the two cases, so tan X equals zero comes from the graph. Um, I&#x27;m not going to sketch up the entire Tang Graf again, but X would be equal to zero pi and two pies not inclusive in the domain. So those are the only two answers from the first case. 10. X equals two plus or minus root. Three. That is a special triangle soap I over six by over three, Route 31 and two so we can see that the reference angle here is opposite over adjacent, So pi over three and tan is plus or minus. So we want all four answers so X would be equal to pi. Over three second quadrant would be pine minus. The reference angles to pi over three third quadrant has pie, plus the reference angles of four pi over three and two pi minus the reference angle for the fourth quadrants of five pi over three. So we have these six answers for this problem.

Ankit Gupta · Answer

according to question. Panics decried by second equals Nutanix Right eggs belong scissor to fight. Now what we&#x27;re going to do is we&#x27;re going to see a return prison in the right inside could be left inside. So this will give us panic multiplied by second minus off Google an ex equals zero. So we&#x27;re going to take panics common from both its terms on this will give us panics multiplied by stick. It minus of cool equals zero. So either Danny equals zero or thicket Take it equals two foreign tan X equals zero. Then either X will be equal zero our fight. And when seconds equals two, then cause X equals 1.2 and when course X equals one important too the next equals fire country brain cortex is inquest Gordon, and it will be equals two by minus a fire country that it&#x27;s hype i upon three. When Cossacks is in court quadrant because inquest and fourth quarter it only politics is quality. So in total were getting four different values are picks and there zero by by upon three and finally five play upon three, which satisfies the given initially question. So this will be the final answer

Problem

Solve the given problems. Find the minimum value …

Problem 12 Easy Difficulty

Solve the given problems.
By Newton's method, find the smallest positive root of the equation $\tan x=2 x .$ Verify the result by using the zero feature of a calculator.

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

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus