6.090 IAP '05 - Lecture 2 Solutions

Problem 1:

#[compound-procedure 10]

17

42

Error: division by zero

17

Problem 2:

a)

(define cube
  (lambda (x)
    (* x x x)))

(cube 2)
;Value: 8
(cube 3)
;Value: 27

b)

(define the-answer?
  (lambda (answer)
    (= answer 42)))
or
(define the-answer?
  (lambda (answer)
    (if (= answer 42)
        #t
        #f)))

(the-answer? 3)
;Value: #f
(the-answer? 42)
;Value: #t

c)

(define sign
  (lambda (x)
    (if (> x 0)
        1
        (if (= x 0)
            0
            -1))))
or
(define sign
  (lambda (x)
    (if (= x 0)
        0
        (/ x (abs x)))))

(sign 5)
;Value: 1
(sign 0)
;Value: 0
(sign -10)
;Value: -1

d)

(define usable-page
  (lambda (m)
    (* (- 8.5 m m) (- 11 m m))))

e)

(define beautiful-rectangle
  (lambda (width)
    (* width (/ (+ (sqrt 5) 1) 2))))

f)

(define positive-root
  (lambda (a b c)
    (if (< (- (* b b) (* 4 a c)) 0)
        "complex roots"
        (/ (+ (- b) (sqrt (- (* b b) (* 4 a c))))
           (* 2 a)))))
or
(define det
  (lambda (a b c)
    (- (* b b) (* 4 a c))))
(define positive-root
  (lambda (a b c)
    (if (< (det a b c) 0)
        "complex roots"
	(/ (+ (- b) (sqrt (det a b c)))
	   (* 2 a)))))


Problem 3:

a)

(define biggie-size
  (lambda (combo)
    (+ combo 4)))

b)

(define unbiggie-size
  (lambda (combo)
    (- combo 4)))

c)

(define biggie-size?
  (lambda (combo)
    (> combo 4)))

d)

(define combo-price
  (lambda (combo)
    (if (biggie-size? combo)
        (+ .5 (combo-price (unbiggie-size combo)))
	(* 1.17 combo))))

Problem 4:

Plan:

Base case:      x^0 = 1
Recursive case: x^n = x^(n-1) * x

(define expt
  (lambda (x n)
    (if (= n 0)
        1
        (* x (expt x (- n 1))))))

(expt 2 2)
;Value: 4
(expt 2 3)
;Value: 8

Problem 5:

Plan:

Base Case:      x < y => remainder x y = x
Recursive Case: remainder x y = remainder (x-y) y

(define remainder
  (lambda (x y)
    (if (< x y)
        x
        (remainder (- x y) y))))

Problem 6:

Plan:

Base Cases:     fib 0 = 1, fib 1 = 1
Recursive Case: fib n = fib (n-1) + fib (n-2)

(define fib
  (lambda (n)
    (if (< n 2)
        1
        (+ (fib (- n 1)) (fib (- n 2))))))