Answer:
Step-by-step explanation:
According to Tylor theorem,
f(x)=f(a)+f′(a)/1!×(x−a)+f′′(a)/2!×(x−a)2+......+fn(a)/n!×(x−a)n
Tylor series for cosx is
1 - x^2/2! + x^4/4! + x^6/6! + ……
cos (0.3) = 1 - 〖(0.3)〗^2/2! + 〖(0.3)〗^4/4!
The R4 valur is fourth derivative of cosx
f’(x) = -sinx
f’’ (x) = -cosx
f’’’ (x) = sinx
f’’’’ (x) = cosx
Here, a = 0, n = 3, x = 0.3
max l f^4l = cos (0.3)
= 0.9999
lR_4 (x)l ≤ |0.3-0|^(4+1)/4! × 0.9999
≤ 0.00015186
= 1.51786 × 10^(-4)
R_n (x) = cos (0.3) – (1 - 〖(0.3)〗^2/2!-(0.3)^4/4!)
= 0.0446
The error of approximation R_4 (x) ≤ 1.2 × 10^(-2) and exact value of error R_4 = 0.0446