Taylor Polynomial with Remainder Question

• JustinLiang
In summary, the minimal degree Taylor polynomial you need to calculate sin(1) to 3 decimal places is 6 decimal places. However, an easier way is to start computing the terms of sin(1) one-by-one, and noting that you have an alternating series. What do you know about the "truncation" (remainder) error in an alternating series?f

Homework Statement

What is the minimal degree Taylor polynomial about x=0 that you need to calculate sin(1) to 3 decimal places? 6 decimal places?

Homework Equations

R_nx = f^(n+1)(c)(x-a)^(n+1)/(n+1)(factorial)

The Attempt at a Solution

I have attached my attempt. I am stuck on the last step, how do I solve for n? Did I even do it right up until now?

Attachments

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Homework Statement

What is the minimal degree Taylor polynomial about x=0 that you need to calculate sin(1) to 3 decimal places? 6 decimal places?

Homework Equations

R_nx = f^(n+1)(c)(x-a)^(n+1)/(n+1)(factorial)

The Attempt at a Solution

I have attached my attempt. I am stuck on the last step, how do I solve for n? Did I even do it right up until now?

A MUCH easier way is to start computing the terms of sin(1) one-by-one, and noting that you have an alternating series. What do you know about the "truncation" (remainder) error in an alternating series?

RGV

A MUCH easier way is to start computing the terms of sin(1) one-by-one, and noting that you have an alternating series. What do you know about the "truncation" (remainder) error in an alternating series?

RGV

But don't you need a calculator for that? You would have to calculate sin1 and compare your approximations to see the difference (remainder).

But don't you need a calculator for that? You would have to calculate sin1 and compare your approximations to see the difference (remainder).

No, you don't need to know the value of sin(1)---remember, sin(1) is the thing that you are trying to compute!

RGV

No, you don't need to know the value of sin(1)---remember, sin(1) is the thing that you are trying to compute!

RGV

Wow, I totally misread the question... Ok so now I have

Pn(x) = x - x^3/3! + x^5/5! - x^7/7! + x^9/9! - x^11/11! + ...
Pn(1) = 1 - 1/6 + 1/120 - 1/5040 + 1/362880

How do I know which is 3 decimal and 6 decimal places without a calculator?

Wow, I totally misread the question... Ok so now I have

Pn(x) = x - x^3/3! + x^5/5! - x^7/7! + x^9/9! - x^11/11! + ...
Pn(1) = 1 - 1/6 + 1/120 - 1/5040 + 1/362880

How do I know which is 3 decimal and 6 decimal places without a calculator?

Are you not allowed to use a calculator to do simple addition, subtraction and division? if not, then welcome to the world of manual computation from 50 years ago: this CAN be done by hand, but it is unpleasant.

RGV

Are you not allowed to use a calculator to do simple addition, subtraction and division? if not, then welcome to the world of manual computation from 50 years ago: this CAN be done by hand, but it is unpleasant.

RGV

Haha, our prof said we don't need a calculator for his course. But it seems like we do for the assignments.

Back to the question... I am still somewhat clueless. First off, when they said 3 decimals places, would that mean <10^-2? It seems like the 7th derivative at 1/5040 would be a plausible answer but the answer key says 6... What do I do :S

Haha, our prof said we don't need a calculator for his course. But it seems like we do for the assignments.

Back to the question... I am still somewhat clueless. First off, when they said 3 decimals places, would that mean <10^-2? It seems like the 7th derivative at 1/5040 would be a plausible answer but the answer key says 6... What do I do :S

No. Three-decimal places of accuracy require an |error| < 0.5*10^-4 = 1/2000, so stopping at the term -1/5040 will do (but be sure to INCLUDE that term). Six decimals of accuracy need an |error| < 0.5x10^-7 = 1/20,000,000, so you can figure out where you have to stop the series.

RGV

Haha, our prof said we don't need a calculator for his course. But it seems like we do for the assignments.

Back to the question... I am still somewhat clueless. First off, when they said 3 decimals places, would that mean <10^-2? It seems like the 7th derivative at 1/5040 would be a plausible answer but the answer key says 6... What do I do :S

No. Three-decimal places of accuracy require an |error| < 0.5*10^-4 = 1/20,000, so stopping at the term -1/362,880 will do (but be sure to INCLUDE that term). Six decimals of accuracy need an |error| < 0.5x10^-7 = 1/20,000,000, so you can figure out where you have to stop the series.

RGV