Problems should be submitted using the D2L dropbox by the beginning of class on Wednesday.

Do all four problems. Be sure to use good recursive style.

1. One way to study a sequence of numbers is to compute the differences between successive terms. If you reach all zeros after k steps, it means the original sequence can be generated by a degree k polynomial. Write a function differences which takes a list of numbers as a parameter and returns the list of differences. That is, a list containing the second minus the first, third minus the second, etc.

Examples:

  (differences '(2 7 10 3 6))
 (5 3 -7 3)

  (differences '(1 3))
 (2)

  (differences '())
 ()

  (differences '(1 2 -3 -4 7 12))
 (1 -5 -1 11 5)

2. Write a function zip which takes two lists as parameters and returns a list of pairs where the first pair consists of the first elements of the two parameters, the second pair has the second elements of the parameters, etc. If the lists are not the same length, you should use the builtin function error to report an error. The function error takes two parameters, an atom for the name of the function, and a string for error message. Thus, you might use something like

 (error 'zip "first list too short")

Examples:

  (zip '(a b c d e) '(32 7 10 3 1) )
 ((a 32) (b 7) (c 10) (d 3) (e 1))

  (zip '() '() )
 ()

  (zip '((a b) c (d e f)) '(c (a) (b c)) )
 (((a b) c) (c (a)) ((d e f) (b c)))

  (zip '(1 2 -3) '(50 40 30 20))
 x x zip: first list too short

3. Write a function deep-add which takes a list as a parameter and computes the sum of all numbers which appear anywhere in the list or in sublists. You may find it useful to know that Scheme has a function number? which returns #t if its argument is numeric atom and #f otherwise.

Examples:

  (deep-add '(5 a b 8 2))
 15

  (deep-add '((4 (6 1)) 2 3 (4)))
 20

  (deep-add '(these (aren't 77) (all 32 (numbers 93 here))))
 202

  (deep-add '())
 0

  (deep-add '(no numbers here))
 0

4. Write a function drop-parens which takes a list as a parameter and returns a list in which all nested parentheses have been removed. [Scheme has a built-in function flatten which does this same thing. Don't use flatten in your solution to this problem.]

Examples:

  (drop-parens '((a 34)(b 77)(g 6)) )
 (a 34 b 77 g 6)

  (drop-parens '(a b c) )
 (a b c)

  (drop-parens '() )
 ()

  (drop-parens '(()((() x)())) )
 (x)