Suppose that you are estimating cos(0.3) using the second Maclaurin polynomial for cos(x) . Use Taylor's Theorem for Remainders to bound the error.
Question
Suppose that you are estimating cos(0.3) using the second Maclaurin polynomial for cos(x) . Use Taylor's Theorem for Remainders to bound the error.
Solution
The Maclaurin series for cos(x) is given by:
cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + ...
The second Maclaurin polynomial for cos(x) is P_2(x) = 1 - x^2/2!.
Taylor's Theorem for Remainders states that the error R_n(x) in approximating a function f(x) by its nth degree Taylor polynomial P_n(x) is given by:
R_n(x) = f^(n+1)(c) * x^(n+1) / (n+1)!,
where c is some number between 0 and x.
The third derivative of cos(x) is -sin(x), so the error in approximating cos(0.3) by P_2(0.3) is given by:
R_2(0.3) = |-sin(c) * 0.3^3 / 3!|,
where c is some number between 0 and 0.3.
Since -1 <= sin(c) <= 1 for all c, we have:
R_2(0.3) <= |0.3^3 / 3!| = 0.3^3 / 6 = 0.0045.
So, the error in approximating cos(0.3) by the second Maclaurin polynomial for cos(x) is at most 0.0045.
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