User-Defined Procedures

As section 2.2 of the textbook explains, a programmer can have Scheme construct a new procedure, not equivalent to any of the built-in ones, by writing a lambda-expression. For instance, Scheme recognizes the expression

(lambda (n)
  (* n n))

as a description of a procedure that, when called, squares its argument. Like any other value, this procedure can be given a name by means of a definition:

(define square
  (lambda (n)
    (* n n)))

Such a procedure can be called, just as if it were a built-in procedure:

> (square 12)
144

1. Write a Scheme procedure (add2 a) that returns the sum a+2.

2. Define a procedure f(x) = x² - 3x + 2 and evaluate it for x = 0, 1, 2, 3, 4 .

3. Write a procedure (quadratic-root a b c) that finds one root of a quadratic equation
   ax² + bx + c = 0 using the quadratic formula. Use it to find a root of the above equation.

   Test your procedure by computing

    
        (quadratic-root 1 -5 6)
        (quadratic-root 2 -10 12)
        (quadratic-root 1 4 4).
   

   In each case, use algebra to check your answers.

   What are (quadratic-root 1 0 1) and (quadratic-root 1 0 2)?
   [Would these two examples work in other programming languages that you know?]

4. Write a procedure swap that interchanges the first two elements on a list, leaving the rest of the list unchanged. Thus, (swap '(a b c d e)) should return (b a c d e). In this problem, assume that the list given to swap has at least two elements; do not worry about the possibility that swap might be applied to numbers, empty lists, or lists with only one element.

5. The volume of a sphere of radius r is 4/3 times pi times r3. Write a procedure named sphere-volume that takes as its argument the radius of a sphere (in, say, centimeters) and returns its volume (in cubic centimeters). Use this procedure to compute the volume of a softball (radius: eight centimeters).

6. Define a procedure snoc (``cons backwards'') that takes two arguments, of which the second should be a list. snoc should return a list just like its second argument, except that the first argument has been added at the right end:

   
   > (snoc 'alpha '(beta gamma delta))
   (beta gamma delta alpha)
   > (snoc 1 (list 2 3 4 5 6))
   (2 3 4 5 6 1)
   > (snoc "first" '())
   ("first")
   

   Hint: There are two ways to define this procedure. One uses calls to reverse and cons; the other uses calls to append and list.

7. Textbook Exercises: As you have time, work on exercises 2.1-2.5 from the textbook.

• Answers to parts 1-6.
  
• As in the previous required lab, your write-up should use the format given in http://www.math.grin.edu/~walker/courses/151.sp99/lab-format.html

This document is available on the World Wide Web as

http://www.math.grin.edu/~walker/courses/151.sp99/lab-procedures.html