10일차 - 1.3.4 연습문제 1.42~45

ex 1.42

(define (compose f g)
  (lambda (x) (f (g x))))

((compose square inc) 6) ;; => 49

ex 1.43

(define (repeated f i)
  (define (iter g n)
    (if (= n 1)
      g
      (iter (compose g f) (- n 1))))
  (iter f i))
((repeated inc 10) 0) ;; => 10
((repeated square 2) 5) ;; => (square (square 5)) => 625

ex 1.44

(define (smooth f)
  (lambda (x) (/ (+ (f (- x dx)) (f x) (f (+ x dx))) 3)))

(define (smooth-n f n)
  ((repeated smooth n) f))

ex 1.45

(define (cube-root x)
  (fixed-point (average-damp (lambda (y) (/ x (square y)))) 1.0))
;; (cube-root 27)

(define (4th-root x)
  (fixed-point (average-damp (lambda (y) (/ x (* y y y)))) 1.0))
;; (4th-root 16) => infinite loop

;; y -> x/y^(n-1) 은 몇번 average-damp 해야 하는지 실험하기
(define (nth-root x n)
  (fixed-point ((repeated average-damp n) (lambda (y) (/ x (expt y (- n 1)))))
               1.0))

;; 실험 결과.. 패턴이 안보인다
;; n = 4 > 2
;; n = 5 > 2
;; n = 13 > 3
;; n = 22 > 2