Power Series Estimation/Error Problem

In summary, the conversation discusses using power series to estimate the integral of cosine of x squared with an error no greater than 0.005. The attempt at a solution involves finding the power series representation of cosine of x and then using the Lagrange error formula to find the error. Alternatively, the conversation suggests integrating the series for cosine of x squared and finding the term with an absolute value less than 0.005 when x equals 1.
  • #1
chimychang
5
0

Homework Statement


Use power series to estimate [tex] \int_0^1 \cos(x^2)dx [/tex] with an error no greater than 0.005

Homework Equations



Lagrange Error Formula [tex] \frac{f^{(n+1)}}{(n+1)!}(x-a)^{(n+1)} [/tex]

The Attempt at a Solution



My original attempt was to find the series for [tex] \cos(x^2) [/tex], integrate it, and then subtract the functions to find where the error is less than or equal to .005. However since actually integrating [tex] \int\cos(x^2)dx [/tex] is very difficult and it's ridiculously slow to graph I am now thinking about using the Lagrange error formula. I'm not exactly sure how that would work though.
 
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  • #2
so the power series representation of cos(x) is
sum_(k=0 to infinity) (-1)^k *x^(2k)/(2k)!

So the first few terms are
cos(x) = 1-x^2/2!+x^4/4!-x^6/6!+x^8/x!-x^10/10!

Now for cos(x^2), put x^2 in for every x
cos(x^2) = 1-(x^2)^2/2!+(x^2)^4/4!-(x^2)^6/6!+(x^2)^8/x!-(x^2)^10/10!
= 1-x^4/2! + x^8/4! -x^12/6! +x^16/8!-x^20/10!

then integrate each term (not that hard, its a polynomial)
-x^21/76204800+x^17/685440-x^13/9360+x^9/216-x^5/10+x+constant
^copied from wolfram alphafor the error to be less than .005, that means that one of the integrated terms has to have an absolute value of less than .005 when you put x=1

basically, you guess and check. or you can do about 5 terms since its pretty accurate (9 terms is accurate to 7 decimal places)
 

1. What is a power series?

A power series is a function that is represented as an infinite sum of terms, where each term is a constant multiplied by a variable raised to a non-negative integer power. It is a useful tool in mathematics and science for approximating and representing complicated functions.

2. How is a power series used for estimation?

A power series can be used for estimation by truncating the series at a certain point and using the remaining terms to approximate the function. This is often done when the function is difficult to evaluate directly, but can be expressed as a power series with known coefficients.

3. What is the error associated with power series estimation?

The error associated with power series estimation is the difference between the actual value of the function and the estimated value calculated using a truncated power series. This error can be minimized by using more terms in the series or by choosing a better approximation point.

4. What are the limitations of power series estimation?

Power series estimation is limited by the fact that it can only approximate functions that can be expressed as a power series. If a function cannot be represented in this form, then power series estimation cannot be used. Additionally, the accuracy of the estimate depends on the number of terms used in the series and the choice of approximation point.

5. How can power series estimation be applied in real-world problems?

Power series estimation has many real-world applications, such as in physics, engineering, and economics. It can be used to approximate complex physical phenomena, model economic data, and solve differential equations. For example, it is commonly used in circuit analysis and in modeling the motion of objects under the influence of gravity.

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