1. Construct a Taylor polynomial approximation for f (x) = cos x, where x ∈ [−π/2 , π/2 ], that is accurate to within 10−6 using x0 = 0. Use the fact that p2n+1(x) = p2n(x) and find an upper bound on |R2n+1(x)| for x ∈ [−π/2 , π/2 ] that is independent of x and ξ.

2. Construct a Taylor polynomial approximation for f (x) = ln(1 + x), where x ∈ [−1/4 , 1/4 ], that is accurate to within 10−6 using x0 = 0.

Please show all work.