89 lines
2.9 KiB
Scheme
89 lines
2.9 KiB
Scheme
(load "displaylib.scm")
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(title "Exercise 2.57")
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(print "
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Extend the differentiation program to handle sums and products of arbitrary numbers of (two or more) terms. Then the last example above could be expressed as
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(deriv '(* x y (+ x 3)) 'x)
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Try to do this by changing only the representation for sums and products, without changing the deriv procedure at all. For example, the addend of a sum would be the first term, and the augend would be the sum of the rest of the terms.
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")
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; -- Given library
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(define (variable? x) (symbol? x))
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(define (same-variable? v1 v2)
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(and (variable? v1) (variable? v2) (eq? v1 v2)))
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; Naive
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; (define (make-sum a1 a2) (list '+ a1 a2))
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; (define (make-product m1 m2) (list '* m1 m2))
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(define (=number? x v)
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(and (number? x) (= x v)))
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(define (make-sum a1 a2)
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(cond ((=number? a1 0) a2)
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((=number? a2 0) a1)
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((and (number? a1) (number? a2)) (+ a1 a2))
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(else (list '+ a1 a2))))
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(define (make-product m1 m2)
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(cond ((or (=number? m1 0) (=number? m2 0)) 0)
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((=number? m1 1) m2)
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((=number? m2 1) m1)
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((and (number? m1) (number? m2)) (* m1 m2))
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(else (list '* m1 m2))))
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; Added exponentation case
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(define (deriv exp var)
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(cond ((number? exp) 0)
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((variable? exp)
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(if (same-variable? exp var) 1 0))
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((sum? exp)
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(make-sum (deriv (addend exp) var)
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(deriv (augend exp) var)))
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((product? exp)
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(make-sum
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(make-product (multiplier exp)
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(deriv (multiplicand exp) var))
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(make-product (deriv (multiplier exp) var)
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(multiplicand exp))))
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((exponentiation? exp)
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(let ((u (base exp))
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(n (exponent exp)))
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(make-product
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(make-product n
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(make-exponentiation u (make-sum n -1)))
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(deriv u var))))
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(else
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(error "unknown expression type -- DERIV" exp))))
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(define (exponentiation? x) (and (pair? x) (eq? (car x) '**)))
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(define (base e) (cadr e))
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(define (exponent e) (caddr e))
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(define (make-exponentiation u n)
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(cond ((=number? n 0) 1)
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((=number? n 1) u)
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((and (number? u) (number? n)) (pow u n))
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(else (list '** u n))))
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; ---- END of given library -----
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; -- extend sum and prod
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(define (sum? x)
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(and (pair? x) (eq? (car x) '+)))
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(define (addend s) (cadr s))
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(define (augend s) (fold make-sum (caddr s) (cdddr s)))
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(define (product? x)
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(and (pair? x) (eq? (car x) '*)))
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(define (multiplier p) (cadr p))
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(define (multiplicand p) (fold make-product (caddr p) (cdddr p)))
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(display "(deriv '(+ x 3) 'x)")(newline)
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(display (deriv '(+ x 3) 'x))(newline)
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(display "(deriv '(* x y) 'x)")(newline)
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(display (deriv '(* x y) 'x))(newline)
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(display "(deriv '(* x y (+ x 3)) 'x)")(newline)
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(display (deriv '(* x y (+ x 3)) 'x))(newline)
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(display "(deriv '(* 3 x x) 'x)")(newline)
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(display (deriv '(* 3 x x) 'x))(newline)
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