User-Defined Procedures

Summary: This lab provides practice with simple user-defined procedures.

You may want to glance over the corresponding reading before beginning this lab.

• Exercises
  • Exercise 0: Preparation
  • Exercise 1: Values as Fractions
  • Exercise 2: Adding 2
  • Exercise 3: Converting Feet to Meters
  • Exercise 4: A Quadratic Polynomial
  • Exercise 5: Quadratic Roots
  • Exercise 6: Swapping List Elements
  • Exercise 7: Spherical Volumes
  • Exercise 8: snoc
  • Exercise 9: Rotate
  • Exercise 10: Scoring Figure Skating

Exercises

Exercise 0: Preparation

Start DrScheme and make sure that you're in full Scheme mode.

Exercise 1: Values as Fractions

Copy the frac procedure from the reading on procedures and make sure that it works as advertised.

Exercise 2: Adding 2

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

Exercise 3: Converting Feet to Meters

a. Define a procedure named feet->meters that takes one argument, a real number representing a length measured in feet, and returns the number that represents the same length as measured in meters. Note that one foot is equal to exactly 761/2500 meters.

b. Use this procedure to determine the number of meters in one mile (5280 feet).

Exercise 4: A Quadratic Polynomial

a. Define a procedure, poly1, that corresponds to the polynomial 5x² - 8x + 2.

b. Test your procedure on the values 0, 1, 2, 3, 4.

Exercise 5: Quadratic Roots

a. 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. (I don't care which root you find.)

b. Test your procedure by computing

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

c. Use algebra to check these answers.

d. What are (quadratic-root 1 0 1) and (quadratic-root 1 0 2)?

Exercise 6: Swapping List Elements

Write a procedure, swap-first-two, that, given a list as an argument, creates a new list that interchanges the first two elements of the original list, leaving the rest of the list unchanged. Thus,

> (swap-first-two (list 'a 'b 'c 'd 'e))
(b a c d e)

In this problem, assume that the list given to swap-first-two has at least two elements; do not worry about the possibility that swap-first-two might be applied to numbers, empty lists, or lists with only one element.

Exercise 7: Spherical Volumes

The volume of a sphere of radius r is 4/3 times pi times r³.

a. Write a procedure named sphere-volume that takes as its argument the radius of a sphere (in, say, centimeters) and returns its volume (in, say, cubic centimeters).

b. Use this procedure to compute the volume of a softball (radius: eight centimeters).

Exercise 8: snoc

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

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

Exercise 9: Rotate

Given a nonempty list of elements (e.g., (a b c)), write a procedure, rotate, that creates a new list with the original first element moved to the end .

For example,

> (rotate (list 'a 'b 'c))
(b c a)
> (rotate (list 1 2))
(2 1)
> (rotate (rotate '(first second third fourth)))
(third fourth first second)

Exercise 10: Scoring Figure Skating

In a figure-skating competition, judges have observed the competitors' performances and awarded three separate scores to each competitor: one for accuracy, one for style, and one for the difficulty of the chosen routine. Each score is in the range from 0 to 10. The rules of the competition specify that a competitor's three scores are to be combined into a weighted average, in which accuracy counts three times as much as difficulty and style counts twice as much as difficulty. The overall result should be a single number in the range from 0 to 10.

a. Write a comment in which you describe the nature and purpose of a procedure that takes three arguments -- a competitor's accuracy, style, and difficulty scores -- and returns their weighted average.

b. Define the procedure that you have described.

c. Test your procedure, looking for cases in which the weighted average is computed incorrectly. (If you find any, make corrections in your definition.)