evaluate-the-given-functions-by-using-three-terms-of-the-appropriate-taylor-series-sqrt93-3

Question

Evaluate the given functions by using three terms of the appropriate Taylor series.
$$\sqrt{9.3}$$

Ryan Swift · Accepted Answer

Okay. So for this problem we are asked to approximate squared of 9.3 by using taylor series expansion. So our function is the square root of X, where X is equal to 9.3. So we&#x27;re gonna numerically approximate 9.3 by using uh squared of 9.3. Sorry, by using this uh expression for taylor series expansion. So we have this uh throughout this formula, there&#x27;s lots of values for A. So what is A? So the value of A must be nearest to X. Such that X minus A should be smaller. So since X is 9.3 will let a equal nine. Okay, so let&#x27;s get started. We know that our function 1/2 is equal to square root of X. So we need to find F evaluated at A. So F of A is equal to three. Okay, now we need to find our next piece so we need to find uh the derivative of X. So that will just be 1/2 x To the 1/2 power. Okay, now we need to evaluate are derivative at A And that will just give us 1/6. All right. So now we need to find our second derivative and evaluate it at A. So f double prime of X is equal to So of -1 over for acts to the three over to power. Okay, So now we&#x27;re going to Evaluate our second derivative at a and that will just give us -1 over 108. Okay. We have everything that we need. So therefore we can say at square root of 9.3 is equal to school of three Plus 1/6. 9.3 -9 Will have -1 over to 16. The reason why is because we have a Two factorial here over 108 here. So we&#x27;re just gonna multiply that by two that will give us uh, went over to 16. So then we&#x27;ll have 9.3 -9 squared plus additional terms. So after throwing this in your calculator, we&#x27;ll get that square root of 9.3 is equal to three zero for 96

Ryan Williams · Answer

we&#x27;re going to start here by taking the first few derivatives of the square root of X. Evaluating it really equals 36. So we started with our function. The eagle to the square objects three. Evaluate this at 36. This will just be the square root of 36 which is your six. If you take the first derivative of spirit of X is extra the 1/2 power this be one over to spirit of X using the power rule. The first derivative evaluated 36 will therefore be won over two times the square root of 36 the 1/2 times six, which is 12. We take the second derivative again using the power rule. We&#x27;re going to get negative one over four times X to the three halves power Plugging this in looking 36 into this mule debt Negative one over four times 36 to the three halves power which would be negative 1/4 time six Doomed. Finally, if we take the third derivative again using the power rule, we&#x27;re going to get negative one over six times X to the five hands power plaguing 36 in the year we&#x27;re going to get. We should be positive. One positive fraction, but one over six times 36 to the five halves power, which would just be 1/6 to the six power. And now we can use all of these terms to evaluate the 1st 4 terms in the Power or the Tailor Siri&#x27;s, which starts with F of A, which was six plus f prime of A, which was 1/12 times X minus A where a is 36 plus the second derivative a day, which was negative. 1/4 times six cubed times, 1/2 factorial times X minus 36 squared class 1/6 to the sixth Power Tim&#x27;s 1/3 sectorial times X minus 36 cubed and someone. Now we can simplify these terms to be six plus 1/12 tens X minus 36 minus two. Factorial is too. And if you bring it into the fraction we can read, this is 1/8 times six cube times X minus 36 squared plus three. Factorial is six. So if we multiply that with six to the sixth will get six to the seventh power Times X minus 36 Huge and someone and these are the 1st 4 terms in our Taylor Siri&#x27;s, and now we can approximate with Square Root of 39 is by plugging in 39 for X in our 1st 4 terms. To get that spirit of 39 is approximately six plus 1/12 times 39 minus 36 minus 1/8 times six to you times 39 minus 36 squared, plus 1/6 to the seven times 39 minus 36 o cubed. Simplifying some of this will be six plus 39 minus 36 is three divided by 12 is 1/4 minus 39 minutes. 36 is again three squared nine over eight time six cubed plus three cubed is 27 over six to the seven. And if you plug this into a calculator, you&#x27;re gonna get that This is equal to six point 24 for eight, which is approximately 6.245

Ryan Williams · Answer

to start here. We&#x27;re gonna take a few derivatives of the function one over the square root of X, which is X negative 1/2 power. If we insert a, which is four, into this function, we&#x27;ll get one over the square root of four, which is 1/2 protect the first derivative of this function. Using the power rule, we&#x27;re gonna get one over negative 1/2 X to the three halves power. You plug in four here, we&#x27;re gonna get negative one over to times four to the three halves power which will just be negative one over to the fourth power. If we evaluate the second derivative, we&#x27;re going to get using the power rule three over four times X to the five halves power. If we evaluate this, a four we&#x27;re going to get three over two squared times four to the five halfs power 4 to 5 have powers to to the fifth power to the fifth power. Times two squared Mr to the seventh Power. Finally, the third derivative will be negative. 15 over eight X to the seven hands. If we insert in four into this expression, we&#x27;re gonna get negative 15 over two cute times. Four to the seven halfs power for the seven house powers to to the seventh Power to to the seventh power times two Cubed is to to the 10th Power, and now we can take these four values. Plug him into the definition for the tailor. Siri&#x27;s, which started with F of A, which we evaluated, was 1/2 minus one over to to the fourth times, X minus A, which is four plus three over to to the seventh times, one ever to factorial times X minus four. Squared minus 15 over to to the tent times 1/3 factorial times X minus for cubed and someone. And now we can take these this expression that we just got and try to evaluate what three to the negative 1/2 power is. And we do that by plugging in three for everywhere we see X in this approximation. So this will be approximately 1/2 minus three. Minus floor is negative. One negative. One times negative one is a positive one over to to the fourth plus three over two. Factorial is too. Two times two of the seventh is to to the eighth times negative one squared is a positive one, minus 15 over three factorial, which is six times two to the 10th Power times three minus four is negative, one cubed, which is a negative one, which makes this positive. Then, if you plug this into a calculator, you&#x27;ll get approximately 0.5 77

Ryan Williams · Answer

to start here. Gonna take some derivatives of the function the court accrued of X, which is X to the 1/4 power three plugged A, which is 16 in for X here. We&#x27;re going to get the 1/4 power of 16 which will be too, because to the fourth power and 16 we take the first derivative here. We&#x27;re gonna get 1/4 times X to the 3/4 power and free plug in 16 year. Gonna get 1/4 in 16 to the 3/4 power, which is 16 to the 1/4 Power Cube, which is too cute. But we know that four is two squared and two squared times to the third. Power will be two to the fifth power. If you look at the second derivative, this will be negative three over 16 times X to the 7/4. If we plug in 16 year, this was being negative. 3/16 which is to to the fourth power times 16 to the 7/4. Where shall be negative three over to the fourth. Power tends to the seventh power just you to the 11th power. Finally, if you look at the third derivative of X. This will be 21 over to to the sixth Power Times X to the 11/4, where this is 64 through the six that&#x27;s their derivative at 16 was in being 21 over to the six times 16 the 11th over four power, which would be 21/2 to 6 times to the 11th. Just really 17. And now we can use these values and plug them into the definition of the Taylor series that we have above, which starts with F of A for evaluated to be two plus the first derivative it A, which is one over to the fifth power Times X minus 16 Linus 3/2 of the 11th times X minus 16 squared plus 21 over to to the 17 times X minus 16th Cube. And we need a 1/2 factorial here and a 1/3 factorial year. And now that we have the 1st 4 terms in our Taylor Siri&#x27;s being approximate, what the cortex route of 13 is, which will be the cortical route of 13 you know we have to do is plug 13 in for X here to get that. This is approximately two plus 13 months. 16 is negative. Three over to to the fifth. So this is minus three over to to the fifth, minus three times X minus 16. Q squared will be three squared, which is three cubed over two. Factorial, which is to comes to the 11th is to the 12th plus 21 times negative. Three cubed well being negative three cubed, divided by three factorial, which is six times to the 17. And if you plug this into a calculator, you&#x27;re gonna get this one 0.89 89

Ryan Williams · Answer

to start here. We&#x27;re gonna look at the first few derivatives of the cube root effects. We&#x27;re gonna evaluated an equal 64. So if we start with the function Q. Bert of X is the same Miss X to the 1/3 power. We evaluate this at 64. We asked what Cube is 64 that&#x27;s four. The first derivative of it. Ex, Using the power rule would then be won over three times X to the 2/3 power. Evaluating this at 64 will give 1/3 times the cube root of 64 squared, which will be or squared second derivative again using the power rule B negative too over nine x to the 5/3 this evaluated in 64. We&#x27;ll then be negative, too, over nine times the cube root of 64 which is four to the fifth power and the third derivative will then be 10 over 27 x to the 8/3. This evaluated a 64 re 10 over 27 times the Cuba of 64 which is or to the eighth power. And now we can take these four terms and plug the minute the definition for the tailor Siri&#x27;s that we have above to get that This is F of A with just four plus f prime of a, which to 1/3 times four squared times X minus a and a 64 plus second, Davide, which is negative two over nine times four to the fifth times, will not for two factorial times X minus 64 squared, plus the third Irvine today just 10 over 27 for two the eighth times 1/3 factorial times X minus 64 to the third power and so on. And we can simplify this a little bit to be four plus 1/3 times 16 times x minus 64 minus two. Factorial is to those two factors are gonna cancel out one over nine times forward to the fifth times X minus 64 All square plus 10 over 27 times for to the eighth times three Factorial, which is six x minus 64 cubed. And so I&#x27;m and these were the 1st 4 terms of our Taylor Siri&#x27;s and every one of approximate the Cube root of 60. We just had to plug in 60 for everywhere we see X in the Taylor Siri&#x27;s, which will then be four plus 1/3 times 16. I&#x27;m 16 minus 64 minus 1/9 times for the fifth. I&#x27;m 60 minus 64 squared, plus 10 over 27 times for the eighth time. Six time 16 1964 All Cube. And we know that 60 months 64 is negative for so we can write. This is four minus for over three times 16 minus foursquare to 16 divided by nine times for to the fifth. Power minus 10 times for huge all over 27 times for the eighth time. Six. And if you plug this into a calculator, you&#x27;ll get that this is three 0.91 for nine.

Sara Sasani · Answer

In this video, we find a first for non zero terms off the Taylor City for their given function. The tale of sentence generated by F used something called Zero to Infinity, a case they give off fr zero divided by a factorial Times Explorer K. If zero is equal to zero on the first try, every tape off if at zero, is because one second every give off a 50 is a Contra minus over three. The 3rd 1 is 1/4 for over three, and the 1st 1 is a core group minus 100 divided by 27 presented formula, the Taylor says. For the function picks divided by the trade route off one plus X is equal to if crime off zero kinds picks plus the second name of people off if at zero divided by two factorial times X s scored on the care. Primitive off efforts here, divided by tree factorial Frontex Cube on the first derivative off at zero, divided by four factorial times takes power for after signifying. We have the 1st 4 long zero terms off their Taylor stories for the given function. Have you enjoyed watching this video? And thanks for watching

Problem

Evaluate the given functions by using three terms…

Problem 25 Easy Difficulty

Evaluate the given functions by using three terms of the appropriate Taylor series.
$$\sqrt{9.3}$$

Answer

Related Courses

Basic Technical Mathematics with Calculus

Related Topics

Discussion

Top Calculus 2 / BC Educators

Catherine R.

Anna Marie V.

Heather Z.

Caleb E.

Calculus 2 / BC Courses

Integration Review - Intro

Definite vs Indefinite

Recommended Videos

Watch More Solved Questions in Chapter 30

Video Transcript

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

Catherine R.

Anna Marie V.

Heather Z.

Caleb E.

Integration Review - Intro

Definite vs Indefinite

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

Basic Technical Mathematics with Calculus

Catherine R.

Anna Marie V.

Heather Z.

Caleb E.

Integration Review - Intro

Definite vs Indefinite

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

Catherine R.

Anna Marie V.

Heather Z.

Caleb E.

Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus