Really - Taylor Polynomial Approximation Error

In summary, the conversation discusses using Taylor's theorem to approximate cos(.3) and calculating the upper bound and exact value of the error. The upper bound is found to be approximately .00002, but the exact value of the error can only be calculated by subtracting the exact value of the cosine from the series value, which can be done using a calculator.
  • #1
bcjochim07
374
0

Homework Statement



Use Taylor's theorem to obtain an upper bound of the error of the approximation. Then calculate the exact value of the error.

cos(.3) is approximately equal to 1 - (.3)^2/2! + (.3)^4/4!

Homework Equations





The Attempt at a Solution



I came up with upper bound saying

Rn = (-sinz/5!)*(.3)^5 < (.3)^5/5!

so the upper bound is (.3)^5/5! which is about .00002

But for the exact error I have no idea how to calculate it. There aren't any examples in my textbook.
 
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  • #2
I think they want you to use a calculator to actually find cos(.3) then subtract the approximation to see if how well your estimate worked.
 
  • #3
The only way to calculate the exact value of the error is to take the exact value of the cosine and subtract the value of the series. I think they just mean punch cos(.3) into your calculator and then subtract the series value and see that it's less than the remainder term.
 

FAQ: Really - Taylor Polynomial Approximation Error

Q: What is a Taylor polynomial approximation error?

A: A Taylor polynomial approximation error is the difference between the value of a function at a given point and the value of its Taylor polynomial approximation at that same point.

Q: Why is it important to understand Taylor polynomial approximation error?

A: Understanding Taylor polynomial approximation error allows us to estimate how accurate our approximation of a function is at a given point, and to determine how many terms of the Taylor polynomial we need to include in order to achieve a desired level of accuracy.

Q: How is Taylor polynomial approximation error calculated?

A: Taylor polynomial approximation error is calculated using the remainder term of the Taylor polynomial, which is given by the formula R_n(x) = f^(n+1)(c)(x-a)^(n+1) / (n+1)!, where n is the degree of the Taylor polynomial, c is a value between the point of approximation a and the point of evaluation x, and f^(n+1) is the (n+1)th derivative of the function f(x).

Q: Can Taylor polynomial approximation error be reduced?

A: Yes, Taylor polynomial approximation error can be reduced by including more terms in the Taylor polynomial or by choosing a closer value for c in the remainder term formula. However, it is important to note that the Taylor polynomial is only an approximation and the error cannot be completely eliminated.

Q: In what real-life applications is understanding Taylor polynomial approximation error useful?

A: Understanding Taylor polynomial approximation error is useful in various fields of science and engineering, such as physics, chemistry, and economics. It can be used to model and analyze data, make predictions, and develop more accurate numerical methods for solving equations and differential equations.

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