Lagrange error bound to estimate sin4° to five decimal places( maclaurin series)

In summary, to estimate sin4° accurately to five decimal places using the Maclaurin series, we can use the Lagrange error bound. Plugging in 4° as pi/45 radians, we get an error bound of |Rn(pi/45)| < 1*(pi/45)^n+1/(n+1)! < 5*10^-6, where n must be greater than or equal to 3. This is because, when writing out the derivatives, the ones with sines disappear in the polynomial, so we can ignore them. Therefore, for a 7th order polynomial, we only need to look at the 9th derivative. The answer key applies the remainder theorem and states that
  • #1
hangainlover
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Homework Statement



Estimate sin4 accurate to five decimal places (using maclaurin series of sin)

Homework Equations





The Attempt at a Solution


Lagrange error bound to estimate sin4° to five decimal places( maclaurin series)

4°=pi/45 radians

|Rn(pi/45)<1*(pi/45)^n+1/(n+1)! < 5*10^-6

and the answer key says n should be greater than or equal to 3.



It doesn't make sense .



Because, if you write out derivatives, the ones with sines will disappear in the polynomial. So, don't we have to ignore sin (since it is maclaurin series)

So if it is 7rd order polynomial it should be x-(1/3!)x^3 + (x^5)/5!) -(x^7)/7!.

and therefore we need to look at 9th derivative.



It seems the answer key just applied the remainder theorem.
 
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  • #2
Plus why do they say that the remainder should be less than or equal to .000005
I think it is due to the fact that the questions says it should be accurate to five decimal places.
but what about .000009 or something?
they are still less than .00001
 
  • #3
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  • #4
i think in this case our error bound inequality should be
1*(pi/45)^(2n+3)/(2n+3)! greater than or equal to .000005
and i get n=1 which means i only need the first two terms in the maclurin series to have a value accurate to the fifth decimal place.
 

1. What is the Lagrange error bound?

The Lagrange error bound is a mathematical theorem that provides an upper bound for the error in approximating a function with a finite degree polynomial. It is used to estimate the accuracy of an approximation and can be applied to Maclaurin series, which are polynomial representations of functions.

2. How is the Lagrange error bound calculated?

The Lagrange error bound is calculated using the formula:
Error bound = (M * |x-a|^(n+1))/(n+1)!
where M is the maximum value of the (n+1)th derivative of the function on the interval [a,x]. In the case of the Maclaurin series for sin4°, n = 4 and a = 0, so the error bound can be calculated using the maximum value of the 5th derivative of sin(x) on the interval [0,4°].

3. Why is the Lagrange error bound important?

The Lagrange error bound is important because it allows us to estimate the accuracy of our approximations. By knowing the maximum possible error in our approximation, we can determine how many terms of the Maclaurin series are needed to achieve a desired level of accuracy. This is crucial in many applications, especially in engineering and scientific calculations.

4. How do we use the Lagrange error bound to estimate sin4° to five decimal places?

To estimate sin4° to five decimal places using the Lagrange error bound, we first need to determine the value of M, the maximum value of the 5th derivative of sin(x) on the interval [0,4°]. Then, we can plug this value into the error bound formula along with n = 4 and x = 4°. This will give us an upper bound for the error in our approximation. To achieve five decimal places of accuracy, we need to find the smallest value of n that satisfies the error bound.

5. Are there any limitations to using the Lagrange error bound?

Yes, there are some limitations to using the Lagrange error bound. It assumes that the function being approximated is continuous and has derivatives of all orders on the given interval. It also assumes that the function can be represented by a finite degree polynomial. In some cases, these assumptions may not hold and the error bound may not accurately estimate the error in the approximation.

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