f is a function of one…1.Define a function that takes a list of reals and returns true iff they are in nondecreasing order.
(defun non-decrease (lst)
(if (null (cdr lst))
t
(and (<= (car lst) (cadr lst)) (non-decrease (cdr lst)))))
2.Define a function that takes an integer number of cents and returns four values showing how to make that number out of 25-, 10-, 5- and 1-cent pieces, using the smallest total number of coins.
(defun total-cents (cents)
(total-centss cents '(25 10 5 1)))
(defun total-centss (cents lst)
(if (null lst)
nil
(multiple-value-bind (x y) (floor cents (car lst))
(cons x (total-centss y (cdr lst))))))
3.A faraway planet is inhabited by two kinds of beings, wigglies and wobblies. Wigglies and wobblies are equally good at singing. Every year there is a great competition to chooses the ten best singers. Here are the results for the past ten years:
| YEAR | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| WIGGLIES | 6 | 5 | 6 | 4 | 5 | 5 | 4 | 5 | 6 | 5 |
| WOBBLIES | 4 | 5 | 4 | 6 | 5 | 5 | 6 | 5 | 4 | 5 |
Write a program to simulate such a contest. Do your results suggest that the committee is, in fact, choosing the ten best singers each year?
4.Define a function that takes 8 reals representing the endpoints of two segments in 2-space, and returns either nil if the segments do not intersect, or two values representing the x- and y-coordinates of the intersection if they do.
;;; solution origin: https://stackoverflow.com/questions/9043805/test-if-two-lines-intersect-javascript-function
(defun intersect-line (l1 l2 r1 r2)
(intersects
(car l1) (cdr l1) (car l2) (cdr l2)
(car r1) (cdr r1) (car r2) (cdr r2)))
(defun intersects (a b c d p q r s)
(let* ((det (- (* (- c a) (- s q))
(* (- r p) (- d b))))
(gamma (/ (+ (* (- s q) (- r a))
(* (- p r) (- s b)))
det))
(lambdap (/ (+ (* (- b d) (- r a))
(* (- c a) (- s b)))
det)))
(cond ((= det 0) nil)
((and (and (> lambdap 0) (< lambdap 1))
(and (> gamma 0) (< gamma 1)))
(cons (+ a (* lambdap (- c a)))
(+ b (* lambdap (- d b)))))
(t nil))))
;; (intersect-line '(1 . 4) '(3 . 2) '(2 . 2) '(4 . 4))
f is a function of one…5.Suppose f is a function of one (real) argument, and that min and max are nonzero reals with different signs such that f has a root (returns zero) for one argument i such that min < i < max. Define a function that takes four arguments, f, min, max, and epsilon, and returns an approximation of i accurate to within plus or minus epsilon.
(defun approximat-root (fn min max eps)
())
;; (approximat-root #'+ -2 2 .5)
6 Horner’s method is a trick for evaluating polynomials efficiently. To find ax^3+bx^2+cx+d you evaluate x(x(ax+b)+c)+d. Define a function that takes one or more arguments—the value of x followed by n reals representing the coefficients of an (n - l)th-degree polynomial—and calculates the value of the polynomial by Horner’s method.
7.How many bits would you estimate your implementation uses to represent fixnums?
(values most-positive-fixnum most-negative-fixnum)
Four types: short-float, single-float, double-float, long-float