Description
1.1 Aims
The aim of this lab is to introduce you to Scheme. After completing this lab,
you should have some familiarity with the following topics:
• Doing arithmetic within Scheme.
• Scheme lists.
• Writing simple functions in Scheme.
• Writing recursive functions in Scheme.
1.2 Exercises
1.2.1 Starting up
Follow the provided directions for starting up this lab in a new git lab2 branch
and a new submit/lab2 directory. Note that this lab does not have an exercises directory.
1.2.2 Exercise 1: Arithmetic in Scheme
Fire up racket in the terminal in which you are running the script program:
$ racket
Let’s use Scheme as a calculator. Let’s calculate the sum of the rst few terms
of the harmonic series
1 +
1
2
+
1
3
+
1
4
+
1
5
1
using the Scheme expression:
(+ 1 (/ 1 2) (/ 1 3) (/ 1 4) (/ 1 5))
Notice that Scheme leaves the result as an exact rational number. It should be
easy to verify the result.
Add the next term to the harmonic series and verify the result. Note that
Scheme has automatically reduced the fraction.
Now change the rst 1 in the series to a oating point number; note that the
result is now a oating point number.
Now write Scheme expressions to:
1. Evaluate the polynomial 3x
3 − 2x
2 + 4x − 1 at 4. (note that (expt b e)
yields b
e
.
2. Determine the total amount which results after investing a principal of
$1000.00 for 5 years at 5% interest compounded anually. The usual compound interest formula is:
A = P(1 + r
n
)
nt
where A is the amount resulting from the initial principal P, invested at
interest rate r which is compounded n times per time period and t is the
total number of time periods. In our case, we have P = 1000, r = 0.05,
n = 1 and t = 5.
Verify the answer returned by Scheme by comparing it with the answer
returned by one of the many compound interet calculators available on
the web.
1.3 Exercise 2
In this exercise, we will play with Scheme lists and some list functions:
Let’s de ne some lists:
(define list1 ‘(1 2 3 4 5))
(define list2 ‘(a b c d e))
Run car and cdr on each list:
(car list1)
(cdr list1)
(car list2)
(cdr list2)
You can make repeated use of car and cdr to extract elements from each list.
For example to extract the 3 from list1, you would use:
2
(car (cdr (cdr list1)))
which as discussed in class can be expressed as (caddr list1).
Append the two lists together:
(append list1 list2)
Now look at both list1 and list2 and note that they are unchanged; i.e.
append is non-destructive.
1. Write an expression to extract the b from list2.
2. Write an expression which results in a list containing the last two elements
of list2.
3. Write an expression which results in appending the last two elements of
list2 to the last 3 elements of list1.
1.3.1 Exercise 3: De ning Simple Scheme Functions
This exercise familiarizes you with the syntax used for de ning functions
in Scheme.
Let’s de ne a function to calculate the area of a rectangle:
(define (rect-area width height) (* width height))
Run the function on a 4×5 rectangle.
(a) Write a function to calculate the perimeter of a rectangle and use
that function to calculate the perimeter of a 4×5 rectangle.
(b) Write a function line-length to calculate the length of a line given
the 2D coordinates of its end-points. Note that the length of a line
between (x1, y1) and (x2, y2) is p
(x1 − x2)
2 + (y1 − y2)
2.
i. Write a version of the function which requires 4 separate parameters x1, y1, x2 and y2.
(line-length 7 4 10 8) => 5
Note that Scheme provides a sqrt function.
ii. Write a second version which takes 2 parameters, each of which
is a pair giving the x and y coordinates of a point:
(line-length ‘(7 4) ‘(10 8)) => 5
1.3.2 Exercise 4: Recursive Scheme Functions
In the subset of Scheme we are using for this lab, we do not use detructive
assigment. Hence the only way to do repetition is to use any built-in
3
repetition operations or use recursion. We concentrate on the latter in
this exercise.
Let’s write a function to multiply all the numbers in a list:
(de ne (mult args)
(if (null? args)
1
(* (car args) (mult (cdr args)))))
(mult ‘(2 4 8 16)) => 1024
Note that we cdr- down the list until we nd the () at the end of the
input list and return mult of () as 1. On the way back up the recursion,
we multiply the result of the recursive call with each element.
(a) Write a function sum to sum all the elements of its single list argument. What should be the return value when the list is empty?
(sum ‘(1 2 3 4 5)) => 15
(b) Two lists having the same length can be zip’d togther to result in a
list of 2-element lists where the rst element of the pair is an element
of the rst list and the second element of the pair is the corresponding
element of the second list.
Write a function zip to zip the contents of two lists together. You
may assume that both lists have the same length.
(zip ‘(1 2 3 4) ‘(a b c d)) =>
‘((1 a) (2 b) (3 c) (4 d))
4
