Week 1 Solutions

1/1.1.rkt

@@ -0,0 +1,41 @@
+#lang racket
+
+(+ 5 3 4)
+;; 12
+
+(- 9 1)
+;; 8
+
+(/ 6 2)
+;; 3
+
+(+ (* 2 4) (- 4 6))
+;; 6
+
+(define a 3)
+(define b (+ a 1))
+
+(+ a b (* a b))
+;; 19
+
+(= a b)
+;; #f
+
+(if (and (> b a) (< b (* a b)))
+    b
+    a)
+;; 4
+
+(cond ((= a 4) 6)
+      ((= b 4) (+ 6 7 a))
+      (else 25))
+;; 16
+
+(+ 2 (if (> b a) b a))
+;; 6
+
+(* (cond ((> a b) a)
+         ((< a b) b)
+         (else -1))
+   (+ a 1))
+;; 16

1/1.2.rkt

@@ -0,0 +1,6 @@
+#lang racket
+
+(/ (+ 5 4
+      (- 2 (- 3
+              (+ 6 (/ 4 5)))))
+   (* 3 (- 6 2) (- 2 7)))

1/1.3.rkt

@@ -0,0 +1,13 @@
+#lang racket
+
+(define (f a b c)
+  (cond ((and (< a b) (< a c)) (+ (* b b) (* c c)))
+        ((and (< b a) (< b c)) (+ (* a a) (* c c)))
+        (else (+ (* a a) (* b b)))
+        ))
+
+(f 1 1 1) ; 1 + 1
+(f 1 2 2) ; 4 + 4
+(f 2 1 2) ; 4 + 4
+(f 1 2 3) ; 9 + 4
+(f 3 2 1) ; 9 + 4

1/1.4.rkt

@@ -0,0 +1,6 @@
+#lang racket
+
+(define (a-plus-abs-b a b)
+  ((if (> b 0) ; if b is {nonegative, negative} ...
+       + -)    ; choose the {+, -} operator and ...
+   a b))       ; apply to (a b)

1/1.5.rkt

@@ -0,0 +1,13 @@
+#lang racket
+
+(define (p) (p))
+(define (test x y)
+  (if (= x 0) 0 y))
+
+(test 0 (p))
+
+;; [applicative-order]
+;; Never returns; infinite loop eager evaluating (p).
+
+;; [normal-order]
+;; Returns 0; only the consequent inside test is evaluated.

1/1.6.rkt

@@ -0,0 +1,26 @@
+#lang racket
+
+(provide average square)
+
+(define (average x y)
+  (/ (+ x y) 2))
+
+(define (square x) (* x x))
+
+(define (improve guess x)
+  (average guess (/ x guess)))
+
+(define (good-enough? guess x)
+  (< (abs (- (square guess) x)) 0.001))
+
+(define (new-if predicate then-clause else-clause)
+  (cond (predicate then-clause)
+        (else else-clause)))
+
+(define (sqrt-iter guess x)
+  (new-if (good-enough? guess x)
+          guess
+          (sqrt-iter (improve guess x) x)))
+
+;; (sqrt-iter 1 4)
+;; Never returns; infinite loop eagerly evaluating the alternative.

1/1.7.rkt

@@ -0,0 +1,24 @@
+#lang racket
+
+(require "1.6.rkt")
+
+;; 1. Small numbers (less than 0.001) result in guesses that are never good enough.
+;; 2. Large numbers will blow up the squaring inside the test.
+
+(define (improve guess x)
+  (average guess (/ x guess)))
+
+;; Works better as delta scales (so for big numbers). Relative accuracy goes down the other way.
+(define (good-enough? guess last-guess)
+  (define delta (- guess last-guess))
+  (< (abs (/ delta guess)) 0.001))
+
+(define (sqrt-iter-rec guess last-guess x)
+  (if (good-enough? guess last-guess)
+      guess
+      (sqrt-iter-rec (improve guess x) guess x)))
+
+(define (sqrt-iter guess x)
+  (sqrt-iter-rec guess x x))
+
+(real->decimal-string (sqrt-iter 1 4))

1/1.8.rkt

@@ -0,0 +1,20 @@
+#lang racket
+
+(require "1.6.rkt")
+
+(define (cube x) (* x x x))
+
+(define (improve guess x)
+  (/ (+ (/ x (square guess))
+        (* 2 guess))
+     3))
+
+(define (good-enough? guess x)
+  (< (abs (- (cube guess) x)) 0.001))
+
+(define (cbrt-iter guess x)
+  (if (good-enough? guess x)
+      guess
+      (cbrt-iter (improve guess x) x)))
+
+(real->decimal-string (cbrt-iter 1 8))