University of Guadalajara Functional Programming with Scheme Code

HomeworkInclude display statements and function calls as necessary to completely demonstrate that your code works for each of the following problems. Email your racket file to me when done (normally a .rkt extension).
Problem 1
Given the implementation of f1 as below, implement f2 and f3 such that:
f2 uses let instead of let*
f3 uses lambda instead of let*
(define (f1 a b)
(let* ((x (* 2 a))
(y (* 3 x)))
(* x y)))
(f1 4 5)
Ensure that f1 and f2 produce the same results as f1. Define no external variables.
Problem 2
Rewrite each of the lambdas using a let.
(define (bam a)
((lambda (x) (* x 5))
(+ a 20)))
(bam 3)
(define (wam a b c)
((lambda (w x y) (* w x y))
(+ a b) (* a b) (- a b)))
(wam 4 5 6)
(define (zam a b c)
((lambda (x)
((lambda (y) (expt y c))
(* x 3)))
(+ a b)))
(zam 5 6 7)
Remember to hit CTRL+i in drracket to correct indentation anytime things don’t seem to be working correctly, and then look for things that don’t line up properly.
Problem 3
Redefine wacky so that the procedure and all subproce- dures have the same behavior, but each has unique names (i.e. x1, x2, . . . , y1, y2, . . . ) for its local variables.
(define (wacky x y)
(define (tacky x z)
(define (snacky y z)
(+ x y z))
(snacky z x))
(tacky y x))
(wacky 3 4)
Do the reverse process of the previous problem. Rewrite f1, g and h so that no more than 3 variable names (a, b, c) are needed for the arguments.
(define (f1 a b)
(define (g x y) (* x y))
(define (h m n) (- m n))
(+ (g a b) (h b a)))
Do the same with f2.
(define (f2 a b)
(define (g x y) (+ a x y))
(define (h m n) (* b m n))
(+ (g a b) (h b a)))
(f2 3 4)
Rewrite f3 using only the function names foo and bar, and the variables x and y.
(define (f3 a)
(define (g1 b)
(define (h1 c) (expt c a))
(define (h2) (- a b))
(+ (h1 b) (h2)))
(+ a (g1 a)))
(f3 3)
Problem 4
A univariate polynomial of order n is given by the following equation.
Pn (x) = anxn + . . . + a2x2 + a1x + a0
Here, ai are the coefficients of the polynomial, and x is its unique variable. You might implement a procedure poly-3 for computing the polynomial of order 3 of x as follows.
(define (poly-3 x a0 a1 a2 a3)
(+ a0 (* a1 x) (* a2 x x) (* a3 x x x)))
In poly-3, the coefficients and the variable are bundled together as arguments; and you would have to specify the coefficients each time you want to compute the same polynomial with different values of x. Instead, implement the procedure make-poly-3 that generates a procedure that computes the polynomial for an arbitrary x.
(define (make-poly-3 a0 a1 a2 a3)
…)
(define my-poly-3
(make-poly-3 1 2 3 4))
(my-poly-3 2)
Next, write a function sum-poly-3-range which will sum up the results for calling my-poly-3 for the values in a range:
(define (sum-poly-3-range from to)
…)
(sum-poly-3-range 1 50)