Category: Sicp Solutions

Exercise 2.56.  Show how to extend the basic differentiator to handle more kinds of expressions. For instance, implement the differentiation rule by adding a new clause to the deriv program and defining appropriate procedures exponentiation?, base, exponent, and make-exponentiation. (You may use the symbol ** to denote exponentiation.) Build in the rules that anything raised to…

Exercise 2.41.  Write a procedure to find all ordered triples of distinct positive integers i, j, and k less than or equal to a given integer n that sum to a given integer s. This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the…

Exercise 2.40.  Define a procedure unique-pairs that, given an integer n, generates the sequence of pairs (i,j) with 1< j< i< n. Use unique-pairs to simplify the definition of prime-sum-pairs given above. This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file…

Exercise 2.39.   Complete the following definitions of reverse (exercise 2.18) in terms of fold-right and fold-left from exercise 2.38: (define (reverse sequence) (fold-right (lambda (x y) <??>) nil sequence)) (define (reverse sequence) (fold-left (lambda (x y) <??>) nil sequence)) This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters. Learn more…

Exercise 2.38.  The accumulate procedure is also known as fold-right, because it combines the first element of the sequence with the result of combining all the elements to the right. There is also a fold-left, which is similar to fold-right, except that it combines elements working in the opposite direction: (define (fold-left op initial sequence) (define (iter result rest) (if (null? rest) result (iter (op result (car rest)) (cdr rest))))…

Exercise 2.37.  Suppose we represent vectors v = (vi) as sequences of numbers, and matrices m = (mij) as sequences of vectors (the rows of the matrix). For example, the matrix is represented as the sequence ((1 2 3 4) (4 5 6 6) (6 7 8 9)). With this representation, we can use sequence operations…

Exercise 2.36.  The procedure accumulate-n is similar to accumulate except that it takes as its third argument a sequence of sequences, which are all assumed to have the same number of elements. It applies the designated accumulation procedure to combine all the first elements of the sequences, all the second elements of the sequences, and so…

Exercise 2.35.  Redefine count-leaves from section 2.2.2 as an accumulation: (define (count-leaves t) (accumulate <??> <??> (map <??> <??>))) This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters. Learn more about bidirectional Unicode characters Show hidden characters (define nil '()) (define…

Exercise 2.34.  Evaluating a polynomial in x at a given value of x can be formulated as an accumulation. We evaluate the polynomial using a well-known algorithm called Horner’s rule, which structures the computation as In other words, we start with an, multiply by x, add an-1, multiply by x, and so on, until we reach…

Exercise 2.33.  Fill in the missing expressions to complete the following definitions of some basic list-manipulation operations as accumulations: (define (map p sequence) (accumulate (lambda (x y) <??>) nil sequence)) (define (append seq1 seq2) (accumulate cons <??> <??>)) (define (length sequence) (accumulate <??> 0 sequence)) This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode…