Media geometrica Lesson

7 Slides • 0 Questions

1

Media geometrica

a doua numere reale pozitive

2

Media geometrica sau media proporțională a două numere reale pozitive

este egală cu rădăcina pătrată din produsul lor.

3

Exemplu

1. Calculați media geometrică a numerelor a= 4 și b=16.

Mg=\sqrt{a\cdot b}=\sqrt{4\cdot16}=\sqrt{64}=8

4

Exemplu 2

1. Calculați media geometrică a numerelor

a=5\sqrt{2}

si

b=10\sqrt{2}

Mg=\sqrt{a\cdot b}=\sqrt{5\sqrt{2}\cdot10\sqrt{2}}=\sqrt{50\cdot2}=\sqrt{100}=10

5

Obseravție

Dacă

0<a<b

atunci

a\le\sqrt{a\cdot b}\le b

6

Inegalitatea mediilor

Dacă x si y sunt numere reale pozitive , atunci are loc inegalitatea mediilor:

M_h\le M_g\le M_a

unde

M_h=\frac{2xy}{x+y}

(media armonica)

7

Exemplu

Verificați inegalitatea mediilor pentru numerele x=6 si y=54.

Calculăm media arminică, media geometrică si media aritmetică pentru numerele 6 și 54.

M_h=\frac{2xy}{x+y}=\frac{2\cdot6\cdot54}{6+54}=\frac{648}{60}=10,8

M_g=\sqrt{x\cdot y}=\sqrt{6\cdot54}=\sqrt{324}=18

M_a=\frac{x+y}{2}=\frac{6+64}{2}=30

10,8<18<30

Deci

M_h\le M_g\le M_a

Media geometrica

a doua numere reale pozitive

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