Final answer:
To approximate cos(1.7) using the Taylor series expansion, we find the smallest degree of the polynomial that meets the 10⁻´ accuracy by adding terms of the series centered at 0 until the absolute value of the term falls below that threshold. Each term is calculated with x=1.7 and we continue until sufficient accuracy is achieved.
Step-by-step explanation:
To approximate cos(1.7) to an accuracy of 10⁻´ using the Taylor series expansion for cos(x), we consider the Taylor series centered at 0:
cos(x) = 1 - x²/2! + x⁴/4! - x⁶/6! + ...
We are looking for the smallest degree of the Taylor polynomial that would give us the desired accuracy. With each term, the accuracy improves. For the term of the Taylor series, cos(x) at 0, with the factorial in the denominator and the power term being even, it's obvious that the term's absolute value is pretty small after a few terms even for values of x slightly larger than 1. By checking the size of each term, we can find when the term's value falls below 10⁻´ to meet our accuracy requirement.
If we substitute x = 1.7 into the Taylor series and add up the terms until the absolute value of a term is less than 10⁻´, we get an approximation of cos(1.7). You might need to use a calculator to find when the terms are sufficiently small. Once we have that term, we add up all the terms prior to it to get the approximate value of cos(1.7).