using Plots
using Roots
using ForwardDiff
D(f, k=1) = k > 1 ? D(D(f),k-1) : x -> ForwardDiff.derivative(f, float(x))
using QuadGK
using SymPy

function taylor(f, n=2; c::Real=0)
	 x -> f(c) + sum([D(f, k)(c)/factorial(k)*(x-c)^k for k in 1:n])
end

f(x) = cos(x)
plot([f, taylor(f, 2, c=0), taylor(f, 2, c=pi/4)], -pi/2, pi/2)

f(x) = exp(x)
fs = [f, [taylor(f,n) for n in 1:6]...]
plot(fs, 0, 4) # 6 polynomial approximations









f(x) = exp(-x^2)
quadgk(f, 0, 0.5)





k1(v) = v^2 / 2
k2(v; c=10) = c^2 * (1/sqrt(1 - (v/c)^2) - 1)



@vars a0 a1 a2 a3 a4 a5 b
f(x) = exp(x)
g(x) = a0 + a1*x + a2*x^(2//1) + a3*x^(3//1) + a4*x^(4//1) + a5*x^(5//1) - f(x)
di = solve(Sym[g(i*b) for i in 0:5], [a0, a1,a2,a3,a4,a5])

limit(di[a5], b, 0)