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1.1.3 · MAVZU

Matematik formulalar yordamida hisoblashlar

45 ta masala. Masala shartini ko‘rib, yechishni boshlang.

Namuna masalalar 2 ta yechim

Masalalar 45 ta

1.1.60-masala

O‘rganilayotgan dasturlash tili qoidalariga muvofiq quyidagi ifodalarni yozing:

1)t = (2*cos(x - π/6) / (0.5 + sin^2(y))) * (1 + z^2 / (3 - z^2 / 5))
2)u = ((8 + |x - y|^2 + 1)^(1/3)) / (x^2 + y^2 + 2) - e^(|x - y|) * (tg^2(z) + 1)^x
3)v = ((1 + sin^2(x + y)) / |x - 2y / (1 + x^2*y^2)|) * x^(|y|) + cos^2(arctg(1/z))
4)w = |cos(x) - cos(y)|^(1 + 2*sin^2(y)) * (1 + z + z^2/2 + z^3/3 + z^4/4)
5)α = ln(y^(-sqrt(|x|))) * (x - y/2) + sin^2(arctg(z))
6)β = sqrt(10*(x^(1/3) + x^(y+2))) * (arcsin^2(z) - |x - y|)
7)γ = 5*arctg(x) - (1/4)*arccos(x) * (x + 3*|x - y| + x^2) / (|x - y|*z + x^2)
8)φ = (e^(|x - y|) * |x - y|^(x+y)) / (arctg(x) + arctg(z)) + (x^6 + ln^2(y))^(1/3)
9)ψ = |x^(y/x) - (y/x)^(1/3)| + (y - x) * (cos(y) - z/(y - x)) / (1 + (y - x)^2)
10)a = 2^(-x) * sqrt(x + (y)^(1/4)) * (e^(x - 1/sin(z)))^(1/3)
11)b = (y^(|x|^(1/3)) + cos^3(y)) * (|x - y| * (1 + sin^2(z) / sqrt(x + y))) / (e^(|x - y|) + x/2)
12)c = 2^(y^x) + (3^x)^y - (y * (arctg(z) - π/6)) / (|x| + 1 / (y^2 + 1))
13)f = ((y + (x - 1)^(1/3))^(1/4)) / (|x - y| * (sin^2(z) + tg(z)))
14)g = (y^(x+1) / (|y - 2|^(1/3) + 3)) + ((x + y/2) / |2x + y|) * (x + 1)^(-1/sin(z))
15)h = ((x^(y+1) + e^(y-1)) / (1 + x*|y - tg(z)|)) * (1 + |y - x|) + |y - x|^2 / 2 - |y - x|^3 / 3
Kod yozish

Namuna masalalar

Ushbu namunalarda matematik formulalarni dasturlash tillarida (Python va C++) ifodalash, koordinatalar orqali geometrik kattaliklarni (perimetr va yuza) hisoblash hamda yordamchi o‘zgaruvchilar yordamida darajalarni samarali hisoblash usullari ko‘rib chiqiladi.

1-NAMUNAVIY MASALA

Uchburchakning perimetri va yuzasini topish

Uchburchak uning uchlari koordinatalari bilan aniqlanadi. Uchburchakning perimetri va yuzasini topish dasturini tuzing.

Bosqichma-bosqich yechim

  1. 1. Uchburchak uchlarining koordinatalari kiritiladi: (x1, y1), (x2, y2), (x3, y3).

    A(x1, y1), B(x2, y2), C(x3, y3)

  2. 2. Ikki nuqta orasidagi masofa formulasi orqali uchburchak tomonlari (a, b, c) hisoblanadi.

    a = √((x2 - x1)² + (y2 - y1)²)
b = √((x3 - x2)² + (y3 - y2)²)
c = √((x1 - x3)² + (y1 - y3)²)

  3. 3. Uchburchakning perimetri tomonlar yig‘indisiga teng: perimeter = a + b + c. Yarim perimetr: s = perimeter / 2.

    perimeter = a + b + c,   s = perimeter / 2

  4. 4. Geron formulasi orqali uchburchak yuzasi topiladi.

    area = √(s * (s - a) * (s - b) * (s - c))

Python yechimi

math kutubxonasidan sqrt funksiyasi va f-string orqali natijalarni chiqarish:

import math

# Uchburchakning uchlari koordinatalarini kiritish
x1, y1 = map(float, input("Birinchi nuqtaning koordinatalarini kiritish (x1, y1): ").split())
x2, y2 = map(float, input("Ikkinchi nuqtaning koordinatalarini kiritish (x2, y2): ").split())
x3, y3 = map(float, input("Uchinchi nuqtaning koordinatalarini kiritish (x3, y3): ").split())

# Uchburchakning uch tomonining uzunliklarini hisoblash
a = math.sqrt((x2 - x1) ** 2 + (y2 - y1) ** 2)
b = math.sqrt((x3 - x2) ** 2 + (y3 - y2) ** 2)
c = math.sqrt((x1 - x3) ** 2 + (y1 - y3) ** 2)

# Perimetrni hisoblash
perimeter = a + b + c

# Yuzani hisoblash uchun Geron formulasi
s = perimeter / 2
area = math.sqrt(s * (s - a) * (s - b) * (s - c))

# Natijalarni chiqarish
print(f"Uchburchakning perimetri: {perimeter}")
print(f"Uchburchakning yuzi: {area}")
Namuna kirishi
0 0
4 0
0 3
Dastur natijasi
Uchburchakning perimetri: 12.0
Uchburchakning yuzi: 6.0

C++ yechimi

<cmath> kutubxonasining sqrt va pow funksiyalaridan foydalanib yechish:

#include <iostream>
#include <cmath>
using namespace std;

int main() {
    double x1, y1, x2, y2, x3, y3;
    // Uchburchakning uchlari koordinatalarini kiritish
    cout << "Birinchi nuqtaning koordinatalarini kiritish (x1, y1): ";
    cin >> x1 >> y1;
    cout << "Ikkinchi nuqtaning koordinatalarini kiritish (x2, y2): ";
    cin >> x2 >> y2;
    cout << "Uchinchi nuqtaning koordinatalarini kiritish (x3, y3): ";
    cin >> x3 >> y3;

    // Uchburchakning uch tomonining uzunliklarini hisoblash
    double a = sqrt(pow(x2 - x1, 2) + pow(y2 - y1, 2));
    double b = sqrt(pow(x3 - x2, 2) + pow(y3 - y2, 2));
    double c = sqrt(pow(x1 - x3, 2) + pow(y1 - y3, 2));

    // Perimetrni hisoblash
    double perimeter = a + b + c;

    // Yuzani hisoblash uchun Heron formulasi
    double s = perimeter / 2;
    double area = sqrt(s * (s - a) * (s - b) * (s - c));

    // Natijalarni chiqarish
    cout << "Uchburchakning perimetri: " << perimeter << endl;
    cout << "Uchburchakning yuzi: " << area << endl;
    return 0;
}
Namuna kirishi
0 0
4 0
0 3
Dastur natijasi
Uchburchakning perimetri: 12
Uchburchakning yuzi: 6

Yakuniy javob

Geron formulasi orqali perimetr va yuza hisoblanadi.

2-NAMUNAVIY MASALA

X ning ko‘rsatilgan darajalarini hisoblash

X soni berilgan. X², X⁴, X⁵ ketma-ketlik uchun yordamchi o‘zgaruvchilardan foydalanib 3 ta ko‘paytirish amalini bajarib, X⁵ hisoblansin va barcha X ning ko‘rsatilgan darajalari ekranga chiqarish dasturini tuzing.

Bosqichma-bosqich yechim

  1. 1-ko‘paytirish: X2 = X * X (X² hisoblanadi).

    X² = X * X

  2. 2-ko‘paytirish: X4 = X2 * X2 (X⁴ hisoblanadi, ya’ni X² ning kvadrati).

    X⁴ = X² * X²

  3. 3-ko‘paytirish: X5 = X * X4 (X⁵ hisoblanadi). Jami 3 ta ko‘paytirish amali bajarildi.

    X⁵ = X * X⁴

Python yechimi

Faqat 3 ta ko‘paytirish amali yordamida X², X⁴, X⁵ ni topish:

# X qiymatini kiritish
X = float(input("X sonini kiriting: "))

# Yordamchi o'zgaruvchilar
X2 = X * X  # X² (1-ko'paytirish)
X4 = X2 * X2  # X⁴ (2-ko'paytirish)
X5 = X * X4  # X⁵ (3-ko'paytirish)

# Natijalarni chiqarish
print(f"X^2: {X2}")
print(f"X^4: {X4}")
print(f"X^5: {X5}")
Namuna kirishi
2
Dastur natijasi
X^2: 4.0
X^4: 16.0
X^5: 32.0

C++ yechimi

C++ da yordamchi o‘zgaruvchilar bilan 3 ta ko‘paytirish amalini bajarish:

#include <iostream>
using namespace std;

int main() {
    double X;
    // X qiymatini kiritish
    cout << "X sonini kiriting: ";
    cin >> X;

    // Yordamchi o'zgaruvchilar
    double X2, X4, X5;
    // X² ni hisoblash (1-ko'paytirish)
    X2 = X * X;
    // X⁴ ni hisoblash (2-ko'paytirish: X² ning kvadrati)
    X4 = X2 * X2;
    // X⁵ ni hisoblash (3-ko'paytirish: X * X⁴)
    X5 = X * X4;

    // Natijalarni chiqarish
    cout << "X^2: " << X2 << endl;
    cout << "X^4: " << X4 << endl;
    cout << "X^5: " << X5 << endl;
    return 0;
}
Namuna kirishi
2
Dastur natijasi
X^2: 4
X^4: 16
X^5: 32

Yakuniy javob

X² = X·X, X⁴ = X²·X², X⁵ = X·X⁴ (jami 3 ta ko‘paytirish amali).