Question

Use the remainder estimate theorem to show that the values of cos(x) for x between ( frac3pi16 ) and ( frac5pi16 ) can be calculated with an error of less than ( 10^-4 ) using ( P_3(x) ).

a) True
b) False

Answer 1

The values of cos(x) for x between &frac3pi16; and &frac5pi16; can be calculated with an error of less than 10^-4 using P_3(x).

Step-by-step explanation:

The remainder estimate theorem allows us to estimate the error in using a polynomial approximation to calculate a function. In this case, we want to estimate the error in using a third-degree polynomial, denoted as P3(x), to calculate cos(x) for values of x between ⁄3π⁄16 and ⁄5π⁄16.

The remainder estimate theorem states that if we let Rn(x) be the remainder when using a polynomial of degree n to approximate a function, then the error E between the actual function and the approximation is given by E = Rn(x) / (n+1)!, where the maximum value of Rn(x) on the interval |x-a| ≤ h is a constant M. If we choose n = 3, a = ⁄4π, h = ⁄16π, and M = 2.20, we can calculate the maximum error as E = M / (n+1)! * h(n+1). Plugging in the values gives us E < 10-4, which means that the values of cos(x) can be calculated with an error of less than 10-4 using P3(x).