7

(define (sqrt-iter guess x) (if (good-enough? guess x)
guess
(sqrt-iter (improve guess x)
x)))

(define (improve guess x)
       (average guess (/ x guess)))

(define (average x y)
  (/ (+ x y) 2))

(define (good-enough? guess x)
(< (abs (- (square guess) x)) 0.001))

(define (square x) (* x x))

(sqrt-iter 1 0.0001)

(sqrt 0.0001)

$| \Delta |$ =

(- 0.03230844833048122 (sqrt 0.0001))

$| \Delta% |$ =

(/ (- 0.03230844833048122 (sqrt 0.0001)) 0.0001)

Design and Implementation of better sqrt-iter:

(define (sqrt-iter lastGuess guess x)
    (if (good-enough? lastGuess guess)
        guess
        (sqrt-iter guess (improve guess x) x)
    )
)

(define (good-enough? lastGuess guess)
    (< (abs (- lastGuess guess)) 0.001)
)

Compare to the previous example:

(sqrt-iter 1 100 0.0001)

(sqrt 0.0001)

$| \Delta |$ =

(- 0.010028540197249 (sqrt 0.0001))

$| \Delta% |$ =

(/ (- 0.010028540197249 (sqrt 0.0001)) 0.0001)