Kalkuluaren oinarrizko teorema (edo Kalkulu inegralaren oinarrizko teorema ) funtzio baten deribazioa eta integrazioa alderantzizko eragiketak direla baieztatzean datza. Baieztapen horrek edozein funtzio jarraitu integragarrirako egiaztatzen du haren integralaren deribatua hura bera dela. Teorema hori funtsezkoa da matematikaren adarretako bat den analisi matematiko edo kalkulu deiturikoan.
Teorema horren ondorio zuzena Barrowren erregela da, Kalkuluaren bigarren oinarrizko teorema ere deiturikoa. Bigarren teorema horren bidez funtzio baten integrala kalkula dezakegu integratu beharreko funtzioaren jatorrizkoa erabiliz.
Arkimedesek eta antzinako beste matematikariek bolumenen, azaleren eta luzera makurren gutxi gorabeherako kalkulua egiteko metodoak bazituzten ere, hasiera batean Isaac Barrow matematikari ingelesak garatutako ideia bati esker eta gero Isaac Newtonen eta Gottfried Leibnizen ekarpenei esker teorema hori enuntziatu eta frogatu ahal izan zuten.
Lehenengo oinarrizko teorema
Aldatu
Izan bitez
[
a
,
b
]
{\displaystyle \,[a,b]}
tartean integragarria den
f
{\displaystyle \,f}
funtzio bat eta
F
{\displaystyle \,F}
F
(
x
)
=
∫
α
x
f
(
t
)
d
t
{\displaystyle F(x)={\int _{\alpha }^{x}f(t)dt}}
α
∈
[
a
,
b
]
{\displaystyle \alpha \in [a,b]}
Teoremak hau esaten du:
f
{\displaystyle \,f}
c
∈
[
a
,
b
]
{\displaystyle c\in [a,b]}
jarraitua bada , orduan
F
{\displaystyle \,F}
c
{\displaystyle \,c}
deribagarria da eta
F
′
(
c
)
=
f
(
c
)
{\displaystyle \,F'(c)=f(c)}
Lema garrantzitsua:
Demagun
f
{\displaystyle f}
[
a
,
b
]
{\displaystyle [a,b]}
m
≤
f
(
x
)
≤
M
∀
x
∈
[
a
,
b
]
{\displaystyle m\leq f(x)\leq M\forall x\in [a,b]}
m
(
b
−
a
)
≤
∫
a
b
f
(
t
)
d
t
≤
M
(
b
−
a
)
{\displaystyle m(b-a)\leq {\int _{a}^{b}f(t)dt}\leq M(b-a)}
Frogapenaren hasiera
Hipotesia:
Biz
c
∈
(
a
,
b
)
{\displaystyle c\in (a,b)}
Biz
[
a
,
b
]
{\displaystyle [a,b]}
c puntuan jarraitua den
f
{\displaystyle f}
Biz
[
a
,
b
]
{\displaystyle [a,b]}
F
{\displaystyle F}
F
(
x
)
=
∫
α
x
f
(
t
)
d
t
{\displaystyle F(x)=\int _{\alpha }^{x}f(t)dt}
α
∈
[
a
,
b
]
{\displaystyle \alpha \in [a,b]}
Tesia:
F'(c)=f(c) Definizioz hau daukagu:
F
′
(
c
)
=
lim
h
→
0
F
(
c
+
h
)
−
F
(
c
)
h
{\displaystyle F'(c)={\lim _{h\rightarrow 0}{\frac {F(c+h)-F(c)}{h}}}}
Demagun h>0 dela, orduan
F
(
c
+
h
)
−
F
(
c
)
=
∫
c
c
+
h
f
(
t
)
d
t
{\displaystyle F(c+h)-F(c)={\int _{c}^{c+h}f(t)dt}}
m
h
{\displaystyle m_{h}}
M
h
{\displaystyle M_{h}}
m
h
=
inf
{
f
(
x
)
|
c
≤
x
≤
c
+
h
}
{\displaystyle m_{h}=\inf\{f(x)|c\leq x\leq c+h\}}
M
h
=
sup
{
f
(
x
)
|
c
≤
x
≤
c
+
h
}
{\displaystyle M_{h}=\sup\{f(x)|c\leq x\leq c+h\}}
'Lema' aplikatuta, hau daukagu:
m
h
⋅
h
≤
∫
c
c
+
h
f
(
t
)
d
t
≤
M
h
⋅
h
{\displaystyle m_{h}\cdot h\leq {\int _{c}^{c+h}f(t)dt}\leq M_{h}\cdot h}
Beraz,
m
h
≤
F
(
c
+
h
)
−
F
(
c
)
h
≤
M
h
{\displaystyle m_{h}\leq {\frac {F(c+h)-F(c)}{h}}\leq M_{h}}
Orain demagun
h
<
0
{\displaystyle h<0}
m
∗
h
=
inf
{
f
(
x
)
|
c
+
h
≤
x
≤
c
}
{\displaystyle {m^{*}}_{h}=\inf\{f(x)|c+h\leq x\leq c\}}
M
∗
h
=
sup
{
f
(
x
)
|
c
+
h
≤
x
≤
c
}
{\displaystyle {M^{*}}_{h}=\sup\{f(x)|c+h\leq x\leq c\}}
'Lema' aplikatuta, hau daukagu:
m
∗
h
⋅
(
−
h
)
≤
∫
c
+
h
c
f
(
t
)
d
t
≤
M
∗
h
⋅
(
−
h
)
{\displaystyle {m^{*}}_{h}\cdot (-h)\leq {\int _{c+h}^{c}f(t)dt}\leq {M^{*}}_{h}\cdot (-h)}
Honako hau betetzen denez:
F
(
c
+
h
)
−
F
(
c
)
=
∫
c
c
+
h
f
(
t
)
d
t
=
−
∫
c
+
h
c
f
(
t
)
d
t
{\displaystyle F(c+h)-F(c)={\int _{c}^{c+h}f(t)dt}=-{\int _{c+h}^{c}f(t)dt}}
Orduan:
m
∗
h
⋅
h
≥
F
(
c
+
h
)
−
F
(
c
)
≥
M
∗
h
⋅
h
{\displaystyle {m^{*}}_{h}\cdot h\geq F(c+h)-F(c)\geq {M^{*}}_{h}\cdot h}
h
<
0
{\displaystyle h<0}
m
∗
h
≤
F
(
c
+
h
)
−
F
(
c
)
h
≤
M
∗
h
{\displaystyle {m^{*}}_{h}\leq {\frac {F(c+h)-F(c)}{h}}\leq {M^{*}}_{h}}
Eta
f
{\displaystyle f}
c puntuan jarraitua denez, orduan:
lim
h
→
0
m
h
=
lim
h
→
0
M
h
=
lim
h
→
0
m
∗
h
=
lim
h
→
0
M
∗
h
=
f
(
c
)
{\displaystyle \lim _{h\rightarrow 0}m_{h}=\lim _{h\rightarrow 0}M_{h}=\lim _{h\rightarrow 0}{m^{*}}_{h}=\lim _{h\rightarrow 0}{M^{*}}_{h}=f(c)}
Azkenean, guzti horrek teorema frogatzera garamatza:
F
′
(
c
)
=
lim
h
→
0
F
(
c
+
h
)
−
F
(
c
)
h
=
f
(
c
)
{\displaystyle F'(c)={\lim _{h\rightarrow 0}{\frac {F(c+h)-F(c)}{h}}}=f(c)}
F
(
x
)
=
∫
0
x
t
2
d
t
⇒
F
′
(
x
)
=
x
2
{\displaystyle F(x)=\int _{0}^{x}t^{2}dt\Rightarrow F'(x)=x^{2}}
H
(
x
)
=
∫
10
exp
3
x
sin
(
t
)
d
t
⇒
H
′
(
x
)
=
sin
(
e
3
x
)
e
3
x
3
{\displaystyle H(x)=\int _{10}^{\exp {3x}}\sin(t)dt\Rightarrow H'(x)=\sin(e^{3x})e^{3x}3}
G
(
x
)
=
∫
0
x
2
arcsin
(
t
)
d
t
⇒
G
′
(
x
)
=
arcsin
(
x
2
)
2
x
{\displaystyle G(x)=\int _{0}^{x^{2}}\arcsin(t)dt\Rightarrow G'(x)=\arcsin(x^{2})2x}
Bigarren oinarrizko teorema
Aldatu
Barrow ren erregelaIsaac Barrowren omenez.
Teorema hau askotan erabiltzen da integral mugatuak ebaluatzeko.
Hipotesia:
Biz
[
a
,
b
]
{\displaystyle [a,b]}
f
{\displaystyle f}
Biz
[
a
,
b
]
{\displaystyle [a,b]}
diferentziagarria den
g
{\displaystyle g}
g
′
(
x
)
=
f
(
x
)
∀
x
∈
[
a
,
b
]
{\displaystyle g'(x)=f(x){\ }\forall x\in [a,b]}
Tesia:
∫
a
b
f
(
x
)
d
x
=
g
(
b
)
−
g
(
a
)
{\displaystyle \int _{a}^{b}f(x)dx=g(b)-g(a)}
Frogapena:
Izan bedi
F
(
x
)
=
∫
a
x
f
(
t
)
d
t
{\displaystyle F(x)=\int _{a}^{x}f(t)dt}
Kalkuluaren lehenengo oinarrizko teorema dela medio, hau daukagu:
F
′
(
x
)
=
f
(
x
)
=
g
′
(
x
)
∀
x
∈
[
a
,
b
]
{\displaystyle F'(x)=f(x)=g'(x){\ }\forall x\in [a,b]}
Beraz:
∃
K
∈
R
{\displaystyle \exists K\in \mathbb {R} {\ }}
∀
x
∈
[
a
,
b
]
,
F
(
x
)
=
g
(
x
)
+
K
{\displaystyle \forall x\in [a,b],F(x)=g(x)+K}
Hau aintzat hartuta:
0
=
F
(
a
)
=
g
(
a
)
+
K
{\displaystyle 0=F(a)=g(a)+K}
Eta hortik segitzen denez
c
=
−
g
(
a
)
{\displaystyle c=-g(a)}
F
(
x
)
=
g
(
x
)
−
g
(
a
)
{\displaystyle F(x)=g(x)-g(a)}
Bereziki, baldin
x
=
b
{\displaystyle x=b}
∫
a
b
f
(
t
)
d
t
=
F
(
b
)
=
g
(
b
)
−
g
(
a
)
{\displaystyle \int _{a}^{b}f(t)dt=F(b)=g(b)-g(a)}
∫
0
π
cos
(
x
)
d
x
=
sin
(
π
)
−
sin
(
0
)
=
0
{\displaystyle \int _{0}^{\pi }\cos(x)dx=\sin(\pi )-\sin(0)=0}
∫
1
e
d
x
x
=
ln
(
e
)
−
ln
(
1
)
=
1
{\displaystyle \int _{1}^{e}{\frac {dx}{x}}=\ln(e)-\ln(1)=1}
![{\displaystyle \,[a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3481ed60d2bbdaaa1c3cbb8fc27f260c21615f89)
![{\displaystyle \alpha \in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0039890b40418e3f7fec101e69abe60369ebebe)
![{\displaystyle c\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/997256364b06acf0710e5d24da39e8c42991a249)
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
![{\displaystyle m\leq f(x)\leq M\forall x\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdc624c92b473a3e6bd188c8b5d7f432baf479ed)
![{\displaystyle g'(x)=f(x){\ }\forall x\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ec09903c9ad92ed894f0ec4db223ab7d07e1ceb)
![{\displaystyle F'(x)=f(x)=g'(x){\ }\forall x\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbad2647046ee93e7deeee358961b8c35d85726a)
![{\displaystyle \forall x\in [a,b],F(x)=g(x)+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23f13b52dd0331395663fb78fe682485a40a5c7c)
Datuak: Q1217677
Multimedia: Fundamental theorem of calculus
