这个作业是用Racket自然数的阶乘、奇怪的双重立场等问题

CS 115 Spring 2020 Assignment 02

Due Wednesday, June 3 at 10:00 am (no late submissions)

Assignment Guidelines.

• This assignment covers material in Module 3.

• Submission details:

– Solutions to these questions must be placed in files a02q1.rkt, a02q2.rkt, a02q3.rkt, and

a02q4.rkt, respectively, and must be completed using Racket Intermediate Student.

– Unless otherwise indicated in the question you may use only the built-in functions and special

forms introduced in the lecture slides from CS115 up to and including the modules covered by

this assignment.

– Download the interface file from the course Web page to ensure that all function names are

spelled correctly and each function has the correct number and order of parameters.

– All solutions must be submitted to MarkUs. No solutions will be accepted through email, even

if you are having issues with MarkUs.

– Verify using MarkUs and your basic test results that your files were properly submitted and are

readable on MarkUs.

– For full style marks, your program must follow the CS115 Style Guide.

– Be sure to review the Academic Integrity policy on the Assignments page.

– For the design recipe, helper functions only require a purpose, a contract and an example.

• When a function returns an inexact answer, use a tolerance of 0.0001 in your tests.

• Restrictions:

– Unless the question specifically describes exceptions, you are restricted to using the functions

and special forms covered in or before Module 3.

– Read each question carefully for additional restrictions.

• The solutions you submit must be entirely your own work. Do not look up either full or partial

solutions on the Internet or in printed sources.

1

CS 115 Spring 2020 Assignment 02

Due Wednesday, June 3 at 10:00 am (no late submissions)

1. An Approximation. You no doubt have encountered the number π ≈ 3.14. π is irrational, so it is impossible to write down its exact decimal value. But we can approximate it. Here we will develop an

approximation for π which involves the factorial.

(a) Factorial. The factorial of a natural number is n! = 1 · 2 · 3 · 4 · …(n−1)· n. ExerciseWrite a function (factorial n) that consumes a Nat and computes n!.

For example,

(factorial 4) => 24

(factorial 0) => 1

(b) Odd-Double-Factorial. We can write a similar function d(n) that is the product of the first n odd

natural numbers. For example,

d(1) = 1 = 1

d(2) = 1 · 3 = 3

d(3) = 1 · 3 · 5 = 15

d(6) = 1 · 3 · 5 · 7 · 9 · 11 = 10395

(You can read probably more than you want about this function on the OEIS.)

Exercise

Write the function (odd-double-factorial n) that consumes a Nat returns the product of the first n odd

natural numbers.

( check-expect ( odd-double-factorial 6) 10395)

( check-expect (map odd-double-factorial (range 0 8 1))

(list 1 1 3 15 105 945 10395 135135))

(c) The upper crust number. It turns out that π may be approximated as

π = 2

1

1

+

1

1 · 3

+

1 · 2

1 · 3 · 5

+

1 · 2 · 3

1 · 3 · 5 · 7

+…

(Notice that each numerator is a factorial, and each denominator is an odd double factorial.)

Exercise

Write a function (approx-pi n) that consumes a Nat and returns the approximation of π using n terms.

For example, (approx-pi 1) => 2 and (approx-pi 2) => 2.6.

( check-expect ( approx-pi 1) 2)

( check-expect ( approx-pi 2) (* 2 (+ 1 (/ 1 3))))

( check-expect ( approx-pi 3) (* 2 (+ 1 (/ 1 3) (/ 2 15))))

( check-within ( approx-pi 20) 3.14159 0.00001)

2

CS 115 Spring 2020 Assignment 02

Due Wednesday, June 3 at 10:00 am (no late submissions)

2. Area under a Curve. You likely know the formula for the area of a rectangle: it is the product of the width

and height, b · h.

There are formulas for many other shapes, but some shapes don’t have an area formula. Sometimes it is

enough to approximate the area of a figure.

The area between a curve and the x-axis may be approximated by summing the areas of a set of rectangles.

For example, the area under the curve y = x

2 +1 between x = 2 and x = 4, shown in Figure 1, is approximately

5 · 0.5+7.25 · 0.5+10 · 0.5+13.25 · 0.5 = 17.75

since the widths are all 0.5, and the heights are 5, 7.25, 10, and 13.25.

We can get a better approximation using more rectangles, and we can get as close as we like to the exact

answer just by using more and more rectangles.

(This method of approximating area is called a Riemann sum.)

1 2 3 4 5

5

10

x

y

FIGURE 1. A plot of y = x

2 +1

Exercise Write a function (approx-area F xmin xmax nsteps) that uses this technique to approximate the area

under a function F, using nsteps steps between xmin and xmax.

Here are a few sample functions for testing. Make your solution work for these, but also for other

functions. You might try y = 2x, for example. Pick some of your own.

(define ( square-plus-one x) (+ (* x x) 1))

(define ( constant-function-3 x) 3)

Use the following contract for approx-area:

;; approx-area: Function Num Num Nat -> Num

;; Requires: F has contract (Num -> Num).

Here are some tests using these functions:

( check-expect ( approx-area square-plus-one 2 4 4) 17.75)

( check-expect ( approx-area constant-function-3 1 5 100) 12)

3

CS 115 Spring 2020 Assignment 02

Due Wednesday, June 3 at 10:00 am (no late submissions)

3. Inquiry: next question up is recursive, yes?. Exercise

Write a function acronymize that consumes a (listof Str), where each Str is of length at least 1, and

returns a Str containing the first letter of each item in the list.

( check-expect ( acronymize (list “Portable” “Network” “Graphics”)) “PNG”)

( check-expect ( acronymize (list “GNU’s” “Not” “UNIX”)) “GNU”)

( check-expect ( acronymize (list “Inquiry:” “next” “question” “up”

“is” “recursive ,” “yes?”))

“Inquiry”)

4. An Average Question. An average, or mean, is a number that is in some way near the “middle” of the set

of values. But different “middles” make sense in different circumstances.

(a) Average. Usually when we say “mean” or “average” we mean the arithmetic mean, which is the sum

of the items divided by the number of items. For example, the arithmetic mean of [4,7,13] is 4+7+13

3 = 8;

the arithmetic mean of [5,5,5,5] is 5+5+5+5

4 = 5.

Exercise Write a function (mean L) that returns the arithmetic mean of a non-empty (listof Num).

(mean (list 16 4 2 2)) => 6

(b) Root Mean Square. The Root Mean Square of a series of measurements is of great importance to

Physicists, Electrical Engineers, Biologists, and others.

RMS is calculated as the square root of the mean of the square of the values.

So for example, the RMS of [1,2,−2,3] is

r

1

4

((1)

2 + (2)

2 + (−2)

2 + (3)

2)

=

r

1

4

(1+4+4+9)

=

r

18

4

=

√

4.5

Exercise

Write a function (root-mean-square L) that consumes a (listof Num) and returns the Root Mean

Square of the values.

( check-expect ( root-mean-square (list 1 1 -1 1)) 1)

( check-expect ( root-mean-square (list 2 2 -2 2)) 2)

( check-expect ( root-mean-square (list -5 -1 -1)) 3)

( check-within ( root-mean-square (list 2 1 2)) (sqrt 3) 0.00001)

( check-within ( root-mean-square (list 1 2 -2 3)) (sqrt 4.5) 0.00001)

4

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