1. (20 points) Excercise 1.32 (page 61) from the textbook.

hw3_1.scm:

;recursive procedure

(define (accumulate combiner null-value term a next b)
  (if (> a b)
      null-value
      (combiner (term a) (accumulate combiner null-value term (next a) next b))))

(define (sum term a next b)
  (accumulate + 0 term a next b))

(define (product term a next b)
  (accumulate * 1 term a next b))

;iterative procedure

(define (accumulate-iter combiner null-value term a next b)
  (define (iter a result)
    (if (> a b)
        result
        (inter (next a) (combiner (term a) result))))
  (iter a null-value))

---

2. (20 points) Exercise 1.34 (page 66) from the textbook.

hw1_2.txt:

Trying to evaluate (f f) will get error "object 2 is not applicable".  The
process of evaluating (f f)
is as follows:

(f f)     ---> apply prodedure f with parameter f
(f 2)     ---> apply procedure f with parameter 2
(2 2)     ---> at this point, scheme procedure tries to find a procedure named
               "2", which does not exist, thus gives back the error message.

So in this case, the parameter of procedure "f" should always be a procedure.    

---

3. (20 points)Excercise 1.40 (page 77) from the textbook.

hw3_3.scm:

(define tolerance 0.00001)
      
(define dx 0.00001)

(define (fixed-point f first-guess)
  (define (close-enough? v1 v2)
    (< (abs (- v1 v2)) tolerance))
  (define (try guess)
    (let ((next (f guess)))
      (if (close-enough? guess next)
          next
          (try next))))
  (try first-guess))
    
(define (deriv g)
  (lambda (x)
    (/ (- (g (+ x dx)) (g x)) dx)))

(define (newton-transform g)
  (lambda (x)
    (- x (/ (g x) ((deriv g) x)))))

(define (newtons-method g guess)
  (fixed-point (newton-transform g) guess))
  
(define (cubic a b c)
  (lambda (x) (+ (* x x x) (* a (* x x)) (* b x) c)))
               

hw3_3.txt:

;No value

1 ]=> (load "hw3_3.scm")

;Loading "hw3_3.scm" -- done
;Value: cubic

1 ]=> (newtons-method (cubic 10 5 0) 1)

;Value: 7.942367246941882e-11

1 ]=> (newtons-method (cubic 5 15 10) 1)

;Value: -.8788890222278306

1 ]=> (transcript-off)

---

4. (20 points) Excercise 2.1 (page 87) from the textbook.

hw1_4.scm:

(define (make-rat n d)
  (let ((g (gcd n d)))
    (cond ((< d 0) (cons (- (/ n g)) (- (/ d g))))
          ((> d 0) (cons (/ n g) (/ d g)))
          ((= d 0) (cons n "Illegal Denom!")))))

---

5. (20 points) Excercise 2.4 (page 92) from the textbook.

hw5_5.scm:

(define (cons x y)
  (lambda (m) (m x y)))
  
(define (car z)
  (z (lambda (p q) p)))
  
(define (cdr z)
  (z (lambda (p q) q)))

(define (gcd a b)
  (if (= b 0)
      a
      (gcd b (remainder a b))))

(define (numer x)
  (car x))

(define (denom x)
  (cdr x))

(define (print-rat x)
  (newline)
  (display (numer x))
  (display "/")
  (display (denom x)))

(define (make-rat n d)
  (let ((g (gcd n d)))
    (cond ((< d 0) (cons (- (/ n g)) (- (/ d g))))
          ((> d 0) (cons (/ n g) (/ d g)))
          ((= d 0) (cons n "Illegal Denom!")))))

hw5_5.txt:

;No value

1 ]=> (load "hw3_5.scm")

;Loading "hw3_5.scm" -- done
;Value: cdr

1 ]=> (define a (make-rat 2 3))

;Value: a

1 ]=> (print-rat a)

2/3
;No value

1 ]=> (numer a)

;Value: 2

1 ]=> (denom a)

;Value: 3

1 ]=> (transcript-off)