## 2.5-2:
##
## Plot the fourth derivative of f(x)
##
xs = seq(0, pi/2, length.out=100)
f_prime_4 = -8 * xs^3 * cos(xs^2) - 8 * xs * sin(xs^2) - 4 * xs * sin(xs^2)

print(min(f_prime_4))

plot(xs, f_prime_4, type='l', xlab='x', ylab='f^4(x)', ylim=c(-16.5, max(f_prime_4)))
abline(h=-16.5, col='red', lty=1)
grid()

derivative_bound = 16.5
print((derivative_bound/24)*4*0.1^4)


## 2.5-4:
##
xs = c(1, 4/3, 5/3, 2)
xbar = 3/2
P = prod(xs - xbar)
C = P/factorial(4)
print(sprintf('C= %f; error_lower_bound= %f; error_upper_bound= %f', C, C/2^4, C))