Maximum Error in Taylor Polynomial for cos(x) on Interval [-.25, .25]

In summary, the maximum error in approximating cos(x) by its Taylor polynomial of order 2 on the interval [-.25, .25] can be found by using the Remainder Estimation Theorem and plugging in the maximum of the absolute value of the 3rd derivative of cos(x) on that interval.
  • #1
negatifzeo
66
0

Homework Statement


Find the maximum error in approximating cos(x) by its Taylor polynomial of order 2 on the
interval [
−.25, .25]. Justify your answer using the Remainder Estimation Theorem.



Homework Equations


|R3(x)<=M/3! |x|^3


The Attempt at a Solution


|R3(x)<=M/3! |x|^3 Plugging in the 3 is easy enough, but I don't understand where the M comes from. What is M here? I initally thought it might be the value of the 4th taylor polynomial, but that would make the remainder less than or = zero, right?
 
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  • #2
M is the maximum of the absolute value of the 3rd derivative of cos(x) on [-1/4,1/4].
 
  • #3
Thank you so much!
 

1. What is the Remainder Estimation Theorem?

The Remainder Estimation Theorem is a mathematical theorem that is used to approximate the value of a series or an integral. It is also known as the Lagrange Remainder Theorem or Taylor's Remainder Theorem.

2. How does the Remainder Estimation Theorem work?

The theorem states that the difference between the actual value and the approximation of a series or an integral is equal to the remainder term, which is the next term in the series or the next term in the Taylor polynomial of the function. This remainder term can be calculated using the Lagrange form of the remainder or the integral form of the remainder.

3. What is the significance of the Remainder Estimation Theorem?

The Remainder Estimation Theorem is an important tool in mathematics as it allows us to estimate the error in our calculations. This is particularly useful in situations where it is difficult or impossible to calculate the exact value, but an approximation is still needed. It is also used in many fields such as physics, engineering, and economics to make accurate predictions and calculations.

4. Can the Remainder Estimation Theorem be applied to any series or integral?

Yes, the Remainder Estimation Theorem can be applied to any series or integral as long as certain conditions are met. These conditions include the function being continuous and differentiable within the given interval, and the series or integral should satisfy certain convergence criteria.

5. Are there any limitations to the Remainder Estimation Theorem?

While the Remainder Estimation Theorem is a powerful tool, it has certain limitations. It can only provide an estimate of the error, and the actual error may be larger than the estimated value. It also assumes that the function is differentiable within the given interval, which may not always be the case. Additionally, the theorem may not work for all types of functions, such as non-polynomial functions or functions with discontinuities.

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