-- Exercise 4.4
-- 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 t = cos t

-- exact derivative of our function of interest
dmyfunc :: R -> R
dmyfunc t = - sin t

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

-- compare the exact derivative with the numerical derivative using dt=1
abserr :: R -> R -> R
abserr dt x = abs (dmyfunc x - dOne dt x)

-- compare the differences between dt=0.1 and dt=1 for a value of x
dteffect :: R -> R
dteffect x = abs(abserr 0.1 x - abserr 1 x)

-- Solution
-- The numerical derivative is insensitive to a (dt) around multiples of pi,
-- where the slope is fairly constant. In contrast, derivative is more sensitive
-- around pi/2, where the derivative changes more quickly.