1- masala. Quyidagi jadvalda keltirilgan ma’lumotlar asosida

54-masala. Viloyat bo‘yicha ishlab chiqarilgan sabzavot maxsulotlari xajmi to‘g‘risida quyidagi ma’lumotlar berilgan (min tonna):

Yillar	2013	2014	2015	2016	2017	2018	2019	2020
Sabzavot	175,4	186,8	192,6	206,5	215,1	222,2	233,8	242,7

Berilgan ma’lumotlar asosida:
1) to‘rt yillik sirg‘anchiq o‘rtachalarni aniqlang;
2) darajali funktsiya bo‘yicha analitik tekislang;
3) approksimatsiya o‘rtacha xatosini aniqlang;

1. 
   To'rt yillik sirg'anchiq o'rtachalarni aniqlang:
   

Yillar: 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020
Sabzavot hajmi (min tonna): 175.4, 186.8, 192.6, 206.5, 215.1, 222.2, 233.8, 242.7
To'rt yillik sirg'anchiq o'rtacha formulasi:
((Q4 - Q1) + (Q3 - Q2)) / 2
Q1 = 175.4
Q2 = 186.8
Q3 = 192.6
Q4 = 206.5
((206.5 - 175.4) + (192.6 - 186.8)) / 2 = (31.1 + 5.8) / 2 = 36.9 / 2 = 18.45
To'rt yillik sirg'anchiq o'rtacha: 18.45 min tonna

2. 
   Darajali funktsiya bo'yicha analitik tekislang:
   

Darajali funktsiya formulasi:
Q = a * b^t
Q - Sabzavot hajmi (min tonna)
t - Yillar
Bizning ma'lumotlarimizga ko'ra:
Q1 = 175.4, t1 = 2013
Q2 = 186.8, t2 = 2014
Q3 = 192.6, t3 = 2015
Q4 = 206.5, t4 = 2016
Q5 = 215.1, t5 = 2017
Q6 = 222.2, t6 = 2018
Q7 = 233.8, t7 = 2019
Q8 = 242.7, t8 = 2020
Uchun tekisliklarni topish uchun, logarifmni olib darajalarni hisoblashimiz kerak:
ln(Q1) = ln(a * b^t1) = ln(a) + t1 * ln(b)
ln(Q2) = ln(a * b^t2) = ln(a) + t2 * ln(b)
ln(Q3) = ln(a * b^t3) = ln(a) + t3 * ln(b)
ln(Q4) = ln(a * b^t4) = ln(a) + t4 * ln(b)
ln(Q5) = ln(a * b^t5) = ln(a) + t5 * ln(b)
ln(Q6) = ln(a * b^t6) = ln(a) + t6 * ln(b)
ln(Q7) = ln(a * b^t7) = ln(a) + t7 * ln(b)
ln(Q8) = ln(a * b^t8) = ln(a) + t8 * ln(b)
Bu sistemani yechish uchun, yuqoridagi tenglamalarni foydalanib, yechimlarini topamiz:
ln(Q1) = ln(175.4) ≈ 5.166
ln(Q2) = ln(186.8) ≈ 5.231
ln(Q3) = ln(192.6) ≈ 5.262
ln(Q4) = ln(206.5) ≈ 5.332
ln(Q5) = ln(215.1) ≈ 5.371
ln(Q6) = ln(222.2) ≈ 5.405
ln(Q7) = ln(233.8) ≈ 5.453
ln(Q8) = ln(242.7) ≈ 5.493
Bizga ikki tenglamali sistemni yechish uchun, "ln(a)" va "ln(b)" deb hisoblashimiz kerak. Buning uchun, yuqoridagi yechimlardan foydalanib, sistemni yechib olishimiz mumkin.
ln(a) - ln(Q1) = -0.166
ln(b) ≈ 0.027
Shunday qilib, darajali funksiya formulasi:
Q = exp(ln(a)) * exp(ln(b))^t
Q = a * b^t
Aldangan yechimlarga asosan:
a ≈ exp(ln(a)) ≈ exp(-0.166) ≈ 0.847
b ≈ exp(ln(b))≈ exp(0.027) ≈ 1.027
Shuningdek, darajali funksiya formulasi:
Q ≈ 0.847 * 1.027^t

3. 
   Approksimatsiya o'rtacha xatosini aniqlang:
   

Approksimatsiya o'rtacha xatosi, faktik hajm (berilgan ma'lumotlar) va aproximatsiya o'rtacha (darajali funksiya formulasi) orasidagi farqni ifodalaydi.
Yillar: 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020
Sabzavot hajmi (min tonna): 175.4, 186.8, 192.6, 206.5, 215.1, 222.2, 233.8, 242.7
Darajali funksiya formulasi:
Q ≈ 0.847 * 1.027^t
Aproksimatsiya o'rtachalarini hisoblash uchun darajali funksiya formulasi orqali qiymatlarni topib, faktik hajmlar bilan solishtirishimiz kerak:
Yillar: 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020
Aproksimatsiya o'rtachalari (min tonna): Q1, Q2, Q3, Q4, Q5, Q6, Q7, Q8
Q1 ≈ 0.847 * 1.027^2013 ≈ 175.348
Q2 ≈ 0.847 * 1.027^2014 ≈ 187.096
Q3 ≈ 0.847 * 1.027^2015 ≈ 198.949
Q4 ≈ 0.847 * 1.027^2016 ≈ 211.921
Q5 ≈ 0.847 * 1.027^2017 ≈ 226.027
Q6 ≈ 0.847 * 1.027^2018 ≈ 241.285
Q7 ≈ 0.847 * 1.027^2019 ≈ 257.713
Q8 ≈ 0.847 * 1.027^2020 ≈ 275.33
Faktik hajm va aproximatsiya o'rtachalari orasidagi farqlarni topish uchun, har bir yil uchun faktik hajmdan aproximatsiya o'rtachasini ayirib olishimiz kerak:
Farqlar: (Q1 - 175.4), (Q2 - 186.8), (Q3 - 192.6), (Q4 - 206.5), (Q5 - 215.1), (Q6 - 222.2), (Q7 - 233.8), (Q8 - 242.7)
Farqlarni hisoblash:
(Q1 - 175.4) ≈ 0.948
(Q2 - 186.8) ≈ 0.296
(Q3 - 192.6) ≈ 6.349
(Q4 - 206.5) ≈ 5.421
(Q5 - 215.1) ≈ 10.927
(Q6 - 222.2) ≈ 19.085
(Q7 - 233.8) ≈ 23.913
(Q8 - 242.7) ≈ 32.63
Shunday qilib, faktik hajm va aproximatsiya o'rtachalari orasidagi farqlar:
2013: 0.948
2014: 0.296
2015: 6.349
2016: 5.421
2017: 10.927
2018: 19.085
2019: 23.913
2020: 32.63
Approksimatsiya o'rtacha xatosi:
2013: 0.948 min tonna
2014: 0.296 min tonna
2015: 6.349 min tonna
2016: 5.421 min tonna
2017: 10.927 min tonna
2018: 19.085 min tonna
2019: 23.913 min tonna
2020: 32.63 min tonna