Bit fiddling.
Solution to Exercise 2.2:
(define (make-segment start end)
(cons start end))
(define (start-segment s)
(car s))
(define (end-segment s)
(cdr s))
(define (make-point x y)
(cons x y))
(define (x-point p)
(car p))
(define (y-point p)
(cdr p))
(define (avg a b)
(/ (+ a b) 2))
(define (midpoint-segment s)
(let ((start (start-segment s))
(end (end-segment s)))
(make-point (avg (x-point start) (x-point end))
(avg (y-point start) (y-point end)))))
Solution to Exercise 2.1:
(define (make-rat n d)
(let ((new-n (if (negative? d) (- n) n))
(new-d (abs d)))
(let ((g (gcd new-n new-d)))
(cons (/ new-n g) (/ new-d g)))))
Solution to Exercise 1.46:
(define (iterative-improve good-enough? improve)
(define (iter initial-guess)
(if (good-enough? initial-guess)
initial-guess
(iter (improve initial-guess))))
iter)
Solution to Exercise 1.44:
(define (average-3 a b c)
(/ (+ a b c) 3))
(define (smooth f dx)
(lambda (x) (average-3 (f (- x dx))
(f x)
(f (+ x dx)))))
(define (n-fold-smooth f dx n)
(repeated (lambda (g) (smooth g dx)) n) f)
Solution to Exercise 1.43:
(define (compose f g)
(lambda (x) (f (g x))))
(define (inc i) (+ i 1))
(define (repeated f n)
(define (iter i result)
(if (= i n)
result
(iter (inc i) (compose f result))))
(iter 1 f))
Solution to Exercise 1.41:
(define (double f)
(lambda (x) (f (f x))))
Output:
> (define (inc x) (+ x 1))
> (((double (double double)) inc) 5)
21
Solution to Exercise 1.39:
(define (cont-frac n d k)
(define (iter i sub-expr)
(if (= i 0)
sub-expr
(iter (- i 1) (/ (n i) (+ (d i) sub-expr)))))
(iter (- k 1) (/ (n k) (d k))))
(define (tan-cf x k)
(let ((x-squared-negated (- (* x x))))
(cont-frac (lambda (i) (if (= i 1) x x-squared-negated))
(lambda (i) (- (* 2 i) 1))
k)))
Solution to Exercise 1.38:
(define (cont-frac n d k)
(define (iter i sub-expr)
(if (= i 0)
sub-expr
(iter (- i 1) (/ (n i) (+ (d i) sub-expr)))))
(iter (- k 1) (/ (n k) (d k))))
(define (e k)
(+ (cont-frac (lambda (i) 1.0)
(lambda (i) (if (= (remainder (+ i 1) 3) 0)
(* 2 (/ (+ i 1) 3))
1.0))
k)
2))
Solution to Exercise 1.37:
; part a (recursive process)
(define (cont-frac n d k)
(define (sub-expr i)
(if (= i k)
(/ (n i) (d i))
(/ (n i) (+ (d i) (sub-expr (+ i 1))))))
(sub-expr 1))
(define (golden-ratio k)
(cont-frac (lambda (i) 1.0)
(lambda (i) 1.0)
k))
(define (iterations-required)
(define (iter i previous)
(let ((current (golden-ratio i)))
(if (= (round (* current 10000))
(round (* previous 10000)))
(- i 1)
(iter (+ i 1) current))))
(iter 1 0.0))
; > (iterations-required)
; 12
; part b (iterative process)
(define (cont-frac-i n d k)
(define (iter i sub-expr)
(if (= i 0)
sub-expr
(iter (- i 1) (/ (n i) (+ (d i) sub-expr)))))
(iter (- k 1) (/ (n k) (d k))))
Ken Dyck 