Final answer:
To determine the degree of the Maclaurin polynomial required for the error in the approximation of f(x) = cos(x) at x = 0.4 to be less than 0.001, we can use Taylor's theorem with the remainder term.
Step-by-step explanation:
To determine the degree of the Maclaurin polynomial required for the error in the approximation of f(x) = cos(x) at x = 0.4 to be less than 0.001, we can use Taylor's theorem with the remainder term. The error in the Maclaurin polynomial approximation of a function can be given by the formula:
Error = |Rn(x)| = |f(x) - Pn(x)| ≤ M|x-a|^(n+1)/(n+1)!
Where Rn(x) is the remainder term, Pn(x) is the Maclaurin polynomial of degree n, M is the maximum value of the (n+1)th derivative of f(x) on the interval between x and a, and a is the center of the Maclaurin expansion.
In this case, we want the error to be less than 0.001. The maximum value of the (n+1)th derivative of cos(x) is 1 for all x. Substituting the values into the formula, we have:
|Rn(0.4)| ≤ 0.001
M(0.4 - 0)^(n+1)/(n+1)! ≤ 0.001
Simplifying and solving for n, we can find the degree of the Maclaurin polynomial required for the error to be less than 0.001.