Continuing from yesterday, we made our first function that takes a function and returns a function. Go to 43:00 in this video to see why (you can stop at 54:00):



In this example, we take the first-order functions idea further and define a derivative function (that takes and returns functions) and a newton function (which takes a function and uses fixed-point). Here it is in clojure (you can find fixed-point and other functions in yesterday's post):

    (defn square [x] (* x x))

    (defn deriv [f]
        (let [dx 0.000001]
            (fn [x] (/ (- (f (+ x dx)) (f x)) dx))))

    (defn newton [f guess]
        (let [df (deriv f)]
            (fixed-point (fn [x] (- x (/ (f x) (df x)))) guess)))

    (defn sqrt [x] (newton (fn [y] (- x (square y))) 1))

(clojure day (March 20) in Northampton, MA is coming soon)