Taylor Polynomials and Numerical Analysis

In summary, the conversation discusses using a Taylor Polynomial about pi/4 to approximate cos(42) degrees to an accuracy of 10^-6. The error term is used to determine the nth Taylor Polynomial to use, with x = 45 or pi/4 and x0 = 42 or 7pi/30. The polynomial term is represented by Pn(x) = ∑f^(k)(x-x0)^k/(k)! and the error term is Rn(x) = f^(n+1)(ζ(x))(x-x0)^(n+1)/(n+1)!. The goal is to solve for n, with the constraint that the error must not exceed 10^-6.
  • #1
mynorka
1
0

Homework Statement


Use a Taylor Polynomial about pi/4 to approximate cos(42){degrees} to an accuracy of 10^-6.

*To get an accuracy of 10^-6, use the error term to determine an nth Taylor Polynomial to use.

Homework Equations


x = 45 or pi/4, x0 = 42 or 7pi/30

cos(x) = Pn(x) + Rn(x)

Polynomial Term - Pn(x) = ∑f^(k)(x-x0)^k/(k)!

Error Term - Rn(x) = f^(n+1)(ζ(x))(x-x0)^(n+1)/(n+1)!

The Attempt at a Solution


(pi/60)^n/(n+1)! < ((-60/pi)*10^-6)/pi

^I get stuck at this part. I'm supposed to solve for n, but the left-hand side of this inequality confuses me.
 
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  • #2
mynorka said:

Homework Statement


Use a Taylor Polynomial about pi/4 to approximate cos(42){degrees} to an accuracy of 10^-6.

*To get an accuracy of 10^-6, use the error term to determine an nth Taylor Polynomial to use.

Homework Equations


x = 45 or pi/4, x0 = 42 or 7pi/30

cos(x) = Pn(x) + Rn(x)

Polynomial Term - Pn(x) = ∑f^(k)(x-x0)^k/(k)!

Error Term - Rn(x) = f^(n+1)(ζ(x))(x-x0)^(n+1)/(n+1)!

The Attempt at a Solution


(pi/60)^n/(n+1)! < ((-60/pi)*10^-6)/pi

^I get stuck at this part. I'm supposed to solve for n, but the left-hand side of this inequality confuses me.

What are you doing? We have ##|\text{error}| \leq (\pi/60)^{n+1}/(n+1)!##, and this must not exceed ##10^{-6}##.
 
  • #3
this is pretty tough to read. perhaps put it in latex and then we can check it out
 

1. What is a Taylor polynomial?

A Taylor polynomial is a mathematical function that approximates a more complex function near a specific point. It is made up of a finite number of terms, each representing a different order of the derivative of the original function at that point.

2. How is a Taylor polynomial different from a regular polynomial?

A Taylor polynomial is specifically designed to approximate a function at a specific point, while a regular polynomial can represent a function over a larger interval. Additionally, a Taylor polynomial includes terms for derivatives, while a regular polynomial does not.

3. Why is numerical analysis important in the study of Taylor polynomials?

Numerical analysis is important in the study of Taylor polynomials because it allows us to approximate the value of a function at a specific point using a finite number of terms. This is especially useful when dealing with complex functions that are difficult to evaluate analytically.

4. How do we determine the accuracy of a Taylor polynomial?

The accuracy of a Taylor polynomial depends on the number of terms included in the polynomial. The more terms included, the closer the approximation will be to the actual value of the function at the specified point.

5. Can Taylor polynomials be used to approximate any function?

No, Taylor polynomials can only be used to approximate functions that are differentiable at the specified point. Additionally, the accuracy of the approximation may decrease as the distance from the specified point increases.

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