-- Exercise 4.1
-- George C. Privon
-- 2023-12-22
type R = Double
type Derivative = (R -> R) -> R -> R

-- a function to take the numerical derivative, approximated for some dt
derivative :: R -> Derivative
derivative dt x t = (x (t + dt/2) - x (t - dt/2)) / dt

-- the function we want to take the derivative of
myfunc :: R -> R
myfunc x = 1/2 * x**2

-- take the derivative with dt=10
dOne :: R -> R
dOne x = derivative 10 myfunc x

-- take the derivative with dt=1
dTwo :: R -> R
dTwo x = derivative 1 myfunc x

-- take the derivative with dt=0.1
dThree :: R -> R
dThree x = derivative 0.1 myfunc x

-- Solution discussion
-- dThree does not give an exact solution because the floating point value
-- `0.1` does not have an exact finite binary representation and so is
-- truncated. That truncation manifests as an inexact answer for the evaluation
-- of the derivative.