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Question Suppose you are estimating V11 using the second Taylor polynomial for 1 about € = 9. Use Taylor's theorem to bound the error. Round your answer ...

Question

Question Suppose you are estimating V11 using the second Taylor polynomial for 1 about € = 9. Use Taylor's theorem to bound the error. Round your answer to 4 decimal placesSorry; that'$ incorrect. Try again?FEEDBACKVIEW ANSWERSUBMIT

Question Suppose you are estimating V11 using the second Taylor polynomial for 1 about € = 9. Use Taylor's theorem to bound the error. Round your answer to 4 decimal places Sorry; that'$ incorrect. Try again? FEEDBACK VIEW ANSWER SUBMIT



Answers

Use the second Taylor polynomial of $f(x)=\ln x$ at $x=1$ to approximate $\ln 0.9,$ and find a bound for the error in the approximation.

From exercise 20 for we know that the second container party nominal for this function vehicles. That's my nurse. one miners. 1/2 X -1 sq. So its value at 1.1 because .1 miners are half 0.1 squared, Which he calls zero lifeline. Next we want to find a bound for the arrow of this approximation. First we need To find a capital M. Such that it is greater than or Echo two. The absolute value of a triple primex, which is uh, 3rd order derivative of the function where she goes to over Arabs killed. And when X equals one, it reaches its maximum, So capital um equals two. Okay. And the procedure at 1.1 he calls Captain William Over three factorial times porn warrant cute. Which yes, approximately 3.33 times tend to empower miners for

In this exercise a call zero and function is X plus one to the one cuff. So it's derivative of the second order derivative. Half tax +12 miners. Aha. And minus one falls. Yes. Plus one to the minus three hugs. And their function values of the aurora one a half and miners one force. So the 2nd Taylor Party nominal. Yes one plus a half X miners 18 X squared. And its value at .1. Because 1.0 4875. Next we want to find a bound for the arrow of this approximation first we need to find a capital M. Such that it is greater than or equal to the absolute value of F. Triple prime acts and the third other derivative as three over eight x. Just one choose them Power of five House. And when X equals zero this several auto derogative red transits maximum in the hands of vehicles which is 3/8. hence the residual and .1 Because Captain Ryan or worse, three factorial Times .1 Cubed It tickles 6.25 times 10 to empower miners. Far

Yeah.

Okay, we're asked to solve to obtain the upper bound for the error of the approximation, we want to use the tailor serum the tailor. Sterile states rebounded equal absolute value. The upper bound is equal to you. Put you or less than the article value of to the end Post one Z over and plus one factorial times X minus C to the end, plus one. Okay, Now we know our ffx will be able to Kiev X. You know, our excess one c is equal to zero or end. Those 25 could see them in the last in the last term of our approximation. And then you could say, Well, our Siri's or our terms consists of six terms are boxing consists of six terms of those six mm to the X. Okay. Oh, see, You know, that effort would be than less than or equal to b to the X. So I'm not into the X just will be able to eat and see was for you of value that is within there's zero or what? Okay, now, if we go through and plug those values in into our equation from the tailor, stare up so we know what we have f to the end plus one which we f to the six Z He said that ISS equal Thio eat well, let's just let's write it all out for So we're saying we have absolutely after seen and then over n plus one factorial which would be 5% loss of six factorial the x minus c So one minus zero one to the end, plus and plus once of six gonna be equal to our final value. We've already determined After the sixth, Zou said less unequal e So then we could says e or six factorial and then one to any power is just one. Okay. And if we plugged that into the calculator, we're going to get an approximate value of zero 0.0, you know, 3775 Okay. And that will be our final answer upper bound. Now we need to also find the exact on the exact hour would just be taking e and you just subtract your your approximation that you have the original Siri's, which is one plus one plus one squared over two factorial was one you over three sectorial plus one to the fourth for Factorial was one to the fifth by factory. Okay? And that comes out to 0.1 61 Bye. Okay. And we can see that our well, you hear the upper bound is larger than our exact areas we know other than that satisfies eso. We're kind of, in a way approved our answer that And so, yeah, these will be your final answers. Our upper bound here is 0.375 and the exact er is 0.165 Thank you for watching.


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