This lab is also available in PDF.
Summary: This lab provides practice with simple user-defined
procedures.
a. Open the reading in a separate
window or tab.
b. Start DrScheme and make sure that you're in Pretty Big
Scheme mode.
Copy the fracpart procedure from
the reading on procedures
and make sure that it works as advertised.
;;; Procedure:
;;; fracpart
;;; Parameters:
;;; val, a real number
;;; Purpose:
;;; Express the fractional part of val as a fraction.
;;; Produces:
;;; rat, a rational number.
;;; Preconditions:
;;; val cannot be complex.
;;; Postconditions:
;;; 0 <= rat < 1.
;;; (- val rat) is a whole number (or a a close approximation).
;;; rat is exact.
;;; rat is approximately equal to the fractional part of val
;;; (within some unknown level of accuracy).
;;; rat is the ratio of two integers.
(define fracpart
(lambda (val)
(inexact->exact (- val (truncate val)))))
Write a Scheme procedure (addtwo a) that returns the sum
a+2.
In this exercise, and in the subsequent ones, you need not document the
procedure unless I explicitly tell you to do so.
a. Define a Scheme procedure, (feet->meters ft)
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).
c. How would you use this procedure to determine the number of feet in
a 1000-meter race? (And no, It depends on the number of runners
is not an acceptable answer.) Note that this is a thought question.
It does not ask you to compute that value, simply how you would
compute the value.
d. Add feet->meters to your library of procedures.
a. Define a procedure, (poly1 x), that corresponds to the
polynomial 5x2 - 8x + 2.
b. Test your procedure on the values 0, 1, 2, 3, 4.
In homework 2, you wrote a pair of
definitions for the roots of the quadratic polynomial
ax2+bx+c. The disadvantage of those definitions
was that you had to redefine a, b, and c whenever
you wanted to recompute roots. You can now make the root computations
a procedure.
a. Write a procedure (quadratic-root a b c)
that finds one root of the following quadratic equation by using the
quadratic formula. (I don't care which root you find.)
ax2+bx+c = 0
In case you've forgotten, the quadratic formula is
(-b +/- sqrt(b2 - 4ac))/2a
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)?
a. Write a procedure, (swap-first-two lst), 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, symbols,
empty lists, or lists with only one element.
b. Add this procedure to your library.
The volume of a sphere of radius r is 4/3 times pi times r3.
The circumference of a sphere of radius r is 2 times pi times r.
a. Write a procedure, (sphere-volume r), that
takes as its argument the radius of a sphere (in, say, centimeters)
and returns its volume (in, say, cubic centimeters).
b. Write a procedure, (sphere-circ->radius circ),
that converts the circumference of a sphere to its radius.
c. Use these procedures to compute the volume of a standard softball,
which has a circumference of 12 inches.
d. Use these procedures to compute the volume of a Chicago-style softball,
which has a circumference of 16 inches.
e. Compute the volumes of each kind of softball using centimeters instead of
inches.
a. 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.
b. Add snoc to your library.
If you find that you have extra time, you might want to attempt the
following tasks.
Write a procedure, (rotate lst), that,
given a nonempty list of elements (e.g., (a b c)),
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)
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.)
Write snoc (exercise 8) in two different ways. (You
should have already written it one way; find another way.)
Implement (round-to-n-places val p),
which rounds val to p places after the decimal point.
Note that you may have described the steps for round-to-n-places
in the lab on numeric values.
Write a procedure, (quadratic-roots a b
c) which computes both roots of the quadratic
equation and returns them in a list.
ax2+bx+c = 0
For example,
> (quadratic-roots 3 5 2)
(1 2/3)
;
