TEL AVIV UNIVERSITY
Department of computer science
0368.1105 – Extended Introduction To computer Science
Spring semester, 2012
Homework 2
Due Date: April 5th, 2012
*For all questions, you can assume that inputs are valid and correct
1.
What is the result of the evaluation of the following code:
(define (foo x1 x2 x3)
(display “Inside foo…”)
(cond (x1 x2)
(else x3)))
(define x 0)
(foo (= x 0) 0 (/ 5 x))
a. In case the interpreter uses applicative order evaluation? Explain your answer.
b. In case the interpreter uses normal order evaluation? Explain your answer.
2.
Write a procedure (deep-sum-digit x) that gets a positive integer and calculates the sum of
digits of the sum of digits up to a number smaller than 10. For example:
4526 -> 4+5+2+6=17 -> 8
> (deep-sum-digit 4526)
8
1342 -> 1+3+4+2=10 -> 1
> (deep-sum-digit 1342)
1
a) Write a recursive version of deep-sum-digit-r. Show a substitution model evaluation
for (deep-sum-digit-r 14).
b) Write an iterative version of deep-sum-digit-i. Show a substitution model evaluation
for (deep-sum-digit-i 14).
3.
What is the time complexity order of growth of func below as a function of x:
(define (func x)
(func2 (* 2 x)))
(define (func2 x)
(if (< x 1) 0 (+ (factorial x) (func2 (/ x 2)))))
(define (factorial n)
(if (= n 0) 1 (* n (factorial (- n 1)))))
4.
Consider the function defined as follows:
pow2(0) = 1
pow2(n) = 2*pow2(n-1), n > 0
a)
b)
Write a procedure that computes this function recursively: (define (pow2-r n)…)
Write a procedure that computes this function iteratively: (define (pow2-i n)…)
*Note: Use the algorithm shown above; do not use the primitive function expt.
5. Write a procedure that computes sin(x) using the following approximation (Taylor series):
N
sin x  
n 0
 1
n
(2n  1)!
 1  x2 N 1
x3 x5
 x    ... 
3! 5!
(2 N  1)!
N
x
2 n 1
For example:
N
sin(0.5)  
n 0
 1
n
(2n  1)!
 1  0.52 N 1
0.53 0.55
 0.5 

 ... 
3!
5!
(2 N  1)!
N
2 n 1
 0.5
The procedure should be defined as (define (find-sin x epsilon)…), where x is the
degree in radians, and epsilon is a parameter that defines the number of iterations N.N should
be such that the result would not increase or decrease in more than an epsilon if the number of
iterations was N+1.
For example, for (find-sin 0.5 0.0001) N is 3, and the answer is 0.4794.