dokazi bez rije ci - matematika.hr

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Dokazi bez rijeci

doc.dr.sc. Julije Jakseticjulije@math.hr

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Sume

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Suma prvih n neparnih brojeva

1 + 3 + · · · + (2n− 1) = n2

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Suma prvih n neparnih brojeva

1 + 3 = 22

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Suma prvih n neparnih brojeva

1 + 3 + 5 = 32

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Identitet za sumu prvih n brojeva

Sn = 1 + 2 + · · · + n

8Sn + 1 = (2n + 1)2

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Identitet za sumu prvih n brojeva

8 · 1 + 1 = 32

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Identitet za sumu prvih n brojeva

8 · (1 + 2) + 1 = 52

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Identitet za sumu prvih n brojeva

8 · (1 + 2 + 3) + 1 = 72

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Suma geometrijskog reda I

1/2

1/2

1/2

1/2

1/4

1/4

1/8

1/8

1

4+

1

16+

1

64+ · · · = 1

3

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Suma geometrijskog reda II

P

S T

Q

R1

1

r

r

r2

r2

r3

r3. . .

4PRQ ∼ 4STP

1 + r + r2 + · · · = 1

1− r

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Trisekcija kuta u beskonacnokoraka

1

2

34

1

3=

1

2− 1

4+

1

8− 1

16+ · · ·

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Ekvipotentnost

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

#(a, b) = #R

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Identiteti za trokut

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Pitagorin poucak

c

a

b

c c− a

b

c + a=c− ab⇒ c2 = a2 + b2

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Kosinusov poucak

aa

a

2a cos(γ)− b

bc

a−c

γ

(a + c)(a− c) = (2a cos γ − b)b⇒ c2 = a2 + b2 − 2ab cos γ

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Suma arkus tangensa I

arctg1

2+ arctg

1

3=π

4

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Suma arkus tangensa II

arctg 1 + arctg 2 + arctg 3 = π

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Nejednakosti

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

A-G nejednakost I

a b

√ab

a+b2

√ab ≤ a + b

2

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Nejednakost sredina

a b

HS

AS KS

GS

21a + 1

b

≤√ab ≤ a + b

2≤√a2 + b2

2

Povrsina trokuta = ab

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

A-G nejednakost II-Lema

abbc

ac

⊆ a2

b2c2

ab + bc + ac ≤ a2 + b2 + c2

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

A-G nejednakost II

abc

abc

abc

a b c

bc

ac

ab

⊆a3

b3

c3

a b c

a2

b2

c2

3abc ≤ a3 + b3 + c3

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Napierova nejednakost-prvi dokaz

0 < a < b⇒ 1

b<

ln b− ln a

b− a <1

ay

xa b

y = lnx

p1

p2

p3

n(p3) < n(p2) < n(p1)

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Napierova nejednakost-drugidokaz

y

xa b

y = 1x

1

b(b− a) <

b∫

a

1

xdx <

1

a(b− a)

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Eksponencijalna nejednakostAB > BA za e ≤ A < B

y = nAx

y = nBx

1 e A B

y = lnx

nA > nB ⇒lnA

A>

lnB

B⇒ AB > BA

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Limesi

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

limn→∞

(1 +

1

n

)n= e

y

x1 1 + 1n

y = 1x

n

n + 1

1

n< ln

(1 +

1

n

)<

1

nn

n + 1< n ln

(1 +

1

n

)< 1

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

√

2 +

√2 +

√2 +√· · · = 2

y

x2

√2

2 +√2

√2 +

√2

2 +√2 +

√2

√2 +

√2 +

√2

4

2

y=x− 2

y =√x

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Parcijalna integracija

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Parcijalna integracijav

up = f(a) q = f(b)

r = g(a)

s = g(b)u = f(x)

v = g(x)

s∫

r

udv +

q∫

p

vdu = uv∣∣∣(q,s)

(p,r)

b∫

a

f (x)g′(x)dx = f (x)g(x)∣∣∣b

a−

b∫

a

g(x)f ′(x)dx

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Dandelinove kugle

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Dandelinove kugle

•First •Prev •Next •Last •Go Back •Full Screen •Close •Quit

Hvala na paznji!