Z c dx=cx+C • Df

Konstans f¨uggv´eny f(x) =c,c∈R f(x) = 3

• D_f = (−∞,∞), R_f ={c}

• f minden¨utt folytonos

• lim

x→−∞f(x) =c

• lim

x→∞f(x) =c

• f⁰(x) = 0

• Df⁰ = (−∞,∞), Rf⁰ ={0}

• Z

c dx=cx+C

• D_f = (−∞,∞), R_f ={3}

• f minden¨utt folytonos

• lim

x→−∞f(x) = 3,

• lim

x→∞f(x) = 3

• f⁰(x) = 0

• Df⁰ = (−∞,∞), Rf⁰ ={0}

• Z

3dx= 3x+C

Line´aris f¨uggv´eny f(x) =ax+b,a >0,b∈R f(x) = 2x+ 1

• D_f = (−∞,∞), Rf = (−∞,∞)

• f minden¨utt folytonos

• lim

x→−∞f(x) =−∞

• lim

x→∞f(x) =∞

• f⁰(x) =a

• Df⁰ = (−∞,∞), Rf⁰ ={a}

• Z

(ax+b)dx=ax²

2 +bx+C

• D_f = (−∞,∞), Rf = (−∞,∞)

• f minden¨utt folytonos

• lim

x→−∞f(x) =−∞

• lim

x→∞f(x) =∞

• f⁰(x) = 2

• Df⁰ = (−∞,∞), Rf⁰ ={2}

• Z

(2x+ 1)dx=x²+x+C

Line´aris f¨uggv´eny f(x) =ax+b,a <0,b∈R f(x) =−2x−1

• D_f = (−∞,∞), R_f = (−∞,∞)

• f minden¨utt folytonos

• lim

x→−∞f(x) =∞

• lim

x→∞f(x) =−∞

• f⁰(x) =a

• D_f⁰ = (−∞,∞), Rf⁰ ={a}

• Z

(ax+b)dx=ax²

2 +bx+C

• D_f = (−∞,∞), R_f = (−∞,∞)

• f minden¨utt folytonos

• lim

x→−∞f(x) =∞

• lim

x→∞f(x) =−∞

• f⁰(x) =−2

• D_f⁰ = (−∞,∞), Rf⁰ ={−2}

• Z

(−2x−1)dx=−x²−x+C

Hatv´anyf¨uggv´eny f(x) =xⁿ,np´aros pozit´ıv eg´esz f(x) =x²

• D_f = (−∞,∞), R_f = [0,∞)

• f minden¨utt folytonos

• lim

x→−∞f(x) =∞

• lim

x→∞f(x) =∞

• f⁰(x) =nxⁿ⁻¹

• Df⁰ = (−∞,∞), Rf⁰ = (−∞,∞)

• Z

xⁿdx= xⁿ⁺¹ n+ 1+C

• Df = (−∞,∞), Rf = [0,∞)

• f minden¨utt folytonos

• lim

x→−∞f(x) =∞

• lim

x→∞f(x) =∞

• f⁰(x) = 2x,

• Df⁰ = (−∞,∞), Rf⁰ = (−∞,∞)

• Z

x²dx= x³ 3 +C

Hatv´anyf¨uggv´eny f(x) =xⁿ,np´aratlan pozit´ıv eg´esz f(x) =x³

• D_f = (−∞,∞), R_f = (−∞,∞)

• f minden¨utt folytonos

• lim

x→−∞f(x) =−∞

• lim

x→∞f(x) =∞,

• f⁰(x) =nxⁿ⁻¹

• D_f⁰ = (−∞,∞), R_f⁰ = [0,∞)

• Z

xⁿdx= xⁿ⁺¹ n+ 1+C

• D_f = (−∞,∞), R_f = (−∞,∞)

• f minden¨utt folytonos

• lim

x−→−∞f(x) =−∞

• lim

x−→∞f(x) =∞

• f⁰(x) = 3x²

• D_f⁰ = (−∞,∞), R_f⁰ = [0,∞)

• Z

x³dx= x⁴ 4 +C

A hatv´anyf¨uggv´enyek ¨osszehasonl´ıt´asa

Gy¨okf¨uggv´eny f(x) = √ⁿ

x=x¹ⁿ,np´aros pozit´ıv eg´esz f(x) =√ x

• D_f = [0,∞), R_f = [0,∞)

• f a 0-ban balr´ol folytonos, min- den¨utt m´ashol folytonos

• lim

x→0+f(x) = 0

• lim

x→∞f(x) =∞

• f⁰(x) = _n¹·xⁿ¹⁻¹= ¹

n·ⁿ√ xⁿ⁻¹

• D_f⁰ = (0,∞), R_f⁰ = (0,∞)

• Z

√n

x dx= Z

x¹ⁿdx= xⁿ¹⁺¹

1

n + 1+C

• Df = [0,∞), Rf = [0,∞)

• f a 0-ban balr´ol folytonos, min- den¨utt m´ashol folytonos

• lim

x→0+f(x) = 0

• lim

x→∞f(x) =∞

• f⁰(x) = ₂^√1_x

• D_f⁰ = (0,∞), R_f⁰ = (0,∞)

• Z √

x dx=2 3x³² +C

Gy¨okf¨uggv´eny f(x) = √ⁿ

x=xⁿ¹,np´aratlan pozit´ıv eg´esz f(x) =√³ x

• D_f = (−∞,∞), R_f = (−∞,∞)

• f minden¨utt folytonos

• lim

x→−∞f(x) =−∞

• lim

x→∞f(x) =∞

• f⁰(x) = _n¹·xⁿ¹⁻¹= ¹

n·ⁿ√ xⁿ⁻¹

• D_f⁰ = (−∞,0)∪(0,∞)

• R_f⁰ = (0,∞)

• Z

√n

x dx= Z

x¹ⁿdx= xⁿ¹⁺¹

1

n + 1+C

• Df = (−∞,∞), Rf = (−∞,∞)

• f minden¨utt folytonos

• lim

x→−∞f(x) =−∞

• lim

x→∞f(x) =∞

• f⁰(x) = ¹

3√³ x²

• D_f⁰ = (−∞,0)∪(0,∞)

• R_f⁰ = (0,∞)

• Z

√3

xdx=3 4x⁴³ +C

A gy¨okf¨uggv´enyek ¨osszehasonl´ıt´asa

Hiperbola f(x) =_x¹_n,np´aratlan pozit´ıv eg´esz f(x) =_x¹

• Df = (−∞,0)∪(0,∞)

• R_f = (−∞,0)∪(0,∞)

• f minden¨utt folytonos

• lim

x→−∞f(x) = 0, lim

x→0−f(x) =−∞

• lim

x→0+f(x) =∞, lim

x→∞f(x) = 0

• f⁰(x) =−_xn+1ⁿ

• D_f⁰ = (−∞,0)∪(0,∞)

• R_f⁰ = (−∞,0)

• Z 1

xⁿ dx= x⁻ⁿ⁺¹

−n+ 1 +C, n6= 1

• Df = (−∞,0)∪(0,∞)

• Rf = (−∞,0)∪(0,∞)

• f minden¨utt folytonos

• lim

x→−∞f(x) = 0, lim

x→0−f(x) =−∞

• lim

x→0+f(x) =∞, lim

x→∞f(x) = 0

• f⁰(x) =−_x¹₂

• Df⁰ = (−∞,0)∪(0,∞)

• Rf⁰ = (−∞,0)

• Z 1

xdx= ln|x|+C

Hiperbola f(x) =_x¹n,np´aros pozit´ıv eg´esz f(x) =_x¹2

• Df = (−∞,0)∪(0,∞)

• Rf = (0,∞)

• f minden¨utt folytonos

• lim

x→−∞f(x) = 0, lim

x→0−f(x) =∞

• lim

x→0+f(x) =∞, lim

x→∞f(x) = 0

• f⁰(x) =−_xn+1ⁿ

• Df⁰ = (−∞,0)∪(0,∞)

• R_f0 = (−∞,0)∪(0,∞)

• Z 1

xⁿ dx= x⁻ⁿ⁺¹

−n+ 1 +C

• Df = (−∞,0)∪(0,∞)

• Rf = (0,∞)

• f minden¨utt folytonos

• lim

x→−∞f(x) = 0, lim

x→0−f(x) =∞

• lim

x→0+f(x) =∞, lim

x→∞f(x) = 0

• f⁰(x) =−_x²3

• D_f⁰ = (−∞,0)∪(0,∞)

• R_f⁰ = (−∞,0)∪(0,∞)

• Z 1

x²dx=−1 x+C A hiperbol´ak ¨osszehasonl´ıt´asa

Exponenci´alis f¨uggv´eny f(x) =a^x,a >1 f(x) =e^x

• D_f = (−∞,∞)

• R_f = (0,∞)

• f minden¨utt folytonos

• lim

x→−∞f(x) = 0, lim

x→∞f(x) =∞

• f⁰(x) =a^xlna

• D_f⁰ = (−∞,∞)

• R_f⁰ = (0,∞)

• Z

a^xdx= a^x lna+C

• D_f = (−∞,∞)

• R_f = (0,∞)

• f minden¨utt folytonos

• lim

x→−∞f(x) = 0, lim

x→∞f(x) =∞

• f⁰(x) =e^x

• D_f⁰ = (−∞,∞)

• R_f⁰ = (0,∞)

• Z

e^xdx=e^x+C

Exponenci´alis f¨uggv´eny f(x) =a^x,0< a <1 f(x) = ¹_e^x

• D_f = (−∞,∞)

• R_f = (0,∞)

• f minden¨utt folytonos

• lim

x→−∞f(x) =∞, lim

x→∞f(x) = 0

• f⁰(x) =a^xlna

• D_f⁰ = (−∞,∞)

• R_f⁰ = (−∞,0)

• Z

a^xdx= a^x lna+C

• Df = (−∞,∞)

• Rf = (0,∞)

• f minden¨utt folytonos

• lim

x→−∞f(x) =∞, lim

x→∞f(x) = 0

• f⁰(x) = ¹_e^x ln ¹_e

=− ¹_e^x

• D_f⁰ = (−∞,∞)

• Rf⁰ = (−∞,0)

• R ₁

e

x

dx= (¹_e)^x

ln(¹e) =− ¹_ex

+C

Exponenci´alisok ¨osszehasonl´ıt´asa

Logaritmus f¨uggv´eny f(x) = log_a(x),a >1 f(x) = lnx

• D_f = (0,∞)

• R_f = (−∞,∞)

• f minden¨utt folytonos

• lim

x→0+f(x) =−∞, lim

x→∞f(x) =∞

• f⁰(x) = _xln¹_a

• Df⁰ = (0,∞)

• Rf⁰ = (0,∞)

• D_f = (0,∞)

• R_f = (−∞,∞)

• f minden¨utt folytonos

• lim

x→0+f(x) =−∞, lim

x→∞f(x) =∞

• f⁰(x) = ¹_x

• Df⁰ = (0,∞)

• Rf⁰ = (0,∞)

Logaritmus f¨uggv´eny f(x) = log_a(x),0< a <1 f(x) = log¹

e(x)

• D_f = (0,∞)

• R_f = (−∞,∞)

• f minden¨utt folytonos

• lim

x→0+f(x) =∞, lim

x→∞f(x) =−∞

• f⁰(x) = _xln¹_a

• Df⁰ = (0,∞)

• Rf⁰ = (−∞,0)

• Df = (0,∞)

• Rf = (−∞,∞)

• f minden¨utt folytonos

• lim

x→0+f(x) =∞, lim

x→∞f(x) =−∞

• f⁰(x) = ¹

xln(¹_e) =−_x¹

• D_f⁰ = (0,∞)

• R_f⁰ = (−∞,0) Logaritmusok ¨osszehasonl´ıt´asa

Szinuszf¨uggv´eny f(x) = sinx

• D_f = (−∞,∞)

• R_f = [−1,1]

• f minden¨utt folytonos,

• lim

x→−∞f(x)´es lim

x→∞f(x)nem l´etezik

• f⁰(x) = cosx

• D_f⁰ = (−∞,∞)

• R_f⁰ = [−1,1]

• Z

sinx dx=−cosx+C

Koszinuszf¨uggv´eny f(x) = cosx

• D_f = (−∞,∞)

• Rf = [−1,1]

• f minden¨utt folytonos

• lim

x→−∞f(x)´es lim

x→∞f(x)nem l´etezik

• f⁰(x) =−sinx

• D_f⁰ = (−∞,∞)

• Rf⁰ = [−1,1]

• Z

cosx dx= sinx+C

Tangensf¨uggv´eny f(x) =tgx

f(x) =tgx= sinx cosx

• Df = (−∞,∞)\ {^π₂ +kπ|k∈Z}

• Rf = (−∞,∞)

• f minden¨utt folytonos

• lim

x→^π₂+kπ−f(x) =∞, lim

x→^π₂+kπ+f(x) =−∞

• lim

x→−∞f(x)´es lim

x→∞f(x)nem l´etezik

• f⁰(x) = _cos¹2x

• Df⁰ = (−∞,∞)\ {^π₂ +kπ|k∈Z}

• Rf⁰ = [1,∞)

Kotangensf¨uggv´eny f(x) =ctgx

f(x) =ctgx= cosx sinx

• Df = (−∞,∞)\ {kπ|k∈Z}

• Rf = (−∞,∞)

• f minden¨utt folytonos

• lim

x→kπ−f(x) =−∞, lim

x→kπ+f(x) =∞

• lim

x→−∞f(x)´es lim

x→∞f(x)nem l´etezik

• f⁰(x) =−_sin¹2x

• Df⁰ = (−∞,∞)\ {kπ|k∈Z}

• R_f⁰ = (−∞,−1]

Arkuszszinusz f¨uggv´eny f(x) =arcsinx

• D_f = [−1,1]

• R_f =

−^π₂,^π₂

• fa−1-ben jobbr´ol, az1-ben balr´ol folytonos, min- den¨utt m´ashol folytonos

• lim

x→−1+f(x) =−^π₂, lim

x→1−f(x) =^π₂

• f⁰(x) = ^{√ 1}

1−x²

• Df⁰ = (−1,1)

• Rf⁰ = [1,∞)

Arkuszkoszinusz f¨uggv´eny f(x) =arccosx

• D_f = [−1,1]

• R_f = [0, π]

• fa−1-ben jobbr´ol, az1-ben balr´ol folytonos, min- den¨utt m´ashol folytonos

• lim

x→−1+f(x) =π, lim

x→1−f(x) = 0

• f⁰(x) =−^√1

1−x²

• Df⁰ = (−1,1)

• Rf⁰ = (−∞,−1]

Arkusztangens f¨uggv´eny f(x) =arctgx

• D_f = (−∞,∞)

• R_f = −^π₂,^π₂

• f minden¨utt folytonos

• lim

x→−∞f(x) =−^π₂, lim

x→∞f(x) = ^π₂

• f⁰(x) = _1+x¹2

• Df⁰ = (−∞,∞)

• Rf⁰ = (0,1]

Arkuszkotangens f¨uggv´eny f(x) =arcctgx

• D_f = (−∞,∞)

• Rf = (0, π)

• f minden¨utt folytonos

• lim

x→−∞f(x) =π, lim

x→∞f(x) = 0

• f⁰(x) =−_1+x¹2

• D_f0 = (−∞,∞)

• R_f⁰ = [−1,0)

Szinusz hiperbolikusz f¨uggv´eny f(x) =shx

f(x) =shx= e^x−e^−x 2

• Df = (−∞,∞)

• Rf = (−∞,∞)

• f minden¨utt folytonos

• lim

x→−∞f(x) =−∞, lim

x→∞f(x) =∞

• f⁰(x) =chx

• Df⁰ = (−∞,∞)

• R_f⁰ = [1,∞)

• Z

shx dx=chx+C

Koszinusz hiperbolikusz f¨uggv´eny f(x) =chx

f(x) =chx= e^x+e^−x 2

• Df = (−∞,∞)

• Rf = [1,∞)

• f minden¨utt folytonos

• lim

x→−∞f(x) =∞, lim

x→∞f(x) =∞

• f⁰(x) =shx

• Df⁰ = (−∞,∞)

• Rf⁰ = (−∞,∞)

• Z

chx dx=shx+C

Tangens hiperbolikusz f¨uggv´eny f(x) =thx

f(x) =thx= shx chx

• Df = (−∞,∞)

• Rf = (−1,1)

• f minden¨utt folytonos

• lim

x→−∞f(x) =−1, lim

x→∞f(x) = 1

• f⁰(x) = ¹ ch²x

• Df⁰ = (−∞,∞)

• Rf⁰ = (0,1]

Kotangens hiperbolikusz f¨uggv´eny f(x) =cthx

f(x) =cthx=chx shx

• Df = (−∞,0)∪(0,∞)

• Rf = (−∞,−1)∪(1,∞)

• f minden¨utt folytonos

• lim

x→−∞f(x) =−1, lim

x→0−f(x) =−∞

• lim

x→0+f(x) =∞, lim

x→∞f(x) = 1

• f⁰(x) =− ¹ sh²x

• Df⁰ = (−∞,0)∪(0,∞)

• R_f⁰ = (−∞,0)

Area szinusz hiperbolikusz f¨uggv´eny f(x) =arshx

• Df = (−∞,∞)

• Rf = (−∞,∞)

• fminden¨utt folytonos

• lim

x→−∞f(x) =−∞, lim

x→∞f(x) =∞

• f⁰(x) = ^√1

x²+1

• D_f⁰ = (−∞,∞)

• Rf⁰ = (0,1]

Area koszinusz hiperbolikusz f(x) =archx

f¨uggv´eny

• D_f = [1,∞)

• Rf = [0,∞)

• f az 1-ben jobbr´ol folytonos, minden¨utt m´ashol folytonos

• lim

x→1+f(x) = 0, lim

x→∞f(x) =∞

• f⁰(x) = ^{√ 1}

x²−1

• Df⁰ = (1,∞)

• R_f⁰ = (0,∞)

Area tangens hiperbolikusz f¨uggv´eny f(x) =arthx

• D_f = (−1,1)

• R_f = (−∞,∞)

• f minden¨utt folytonos

• lim

x→−1+f(x) =−∞, lim

x→1−f(x) =∞

• f⁰(x) =_1−x¹2

• Df⁰ = (−1,1)

• Rf⁰ = [1,∞)

Area kotangens hiperbolikusz f(x) =arcthx

f¨uggv´eny

• D_f = (−∞,−1)∪(1,∞)

• Rf = (−∞,0)∪(0,∞)

• f minden¨utt folytonos

• lim

x→−∞f(x) = 0, lim

x→−1−f(x) =−∞

• lim

x→1+f(x) =∞, lim

x→∞f(x) = 0

• f⁰(x) = _1−x¹₂

• D_f⁰ = (−∞,−1)∪(1,∞)

• R_f⁰ = (−∞,0)

Abszol´ut ´ert´ek f¨uggv´eny f(x) =|x|

f(x) =|x|=







−x hax <0 0 hax= 0 x hax >0

• D_f = (−∞,∞)

• Rf = [0,∞)

• f minden¨utt folytonos,

• lim

x→−∞f(x) =∞, lim

x→∞f(x) =∞

• f⁰(x) =

−1 hax <0 1 hax >0

• D_f0 = (−∞,0)∪(0,∞)

• R_f⁰ ={−1,1}

Altal´anos hatv´any f¨uggv´eny´ f(x) =x^α,αirracion´alis f(x) =x

√

5, x^√15, x⁻

√

5, x^−√15

• D_f = [0,∞), haα >0,D_f = (0,∞), haα <0

• R_f = [0,∞), haα >0,R_f = (0,∞), haα <0

• f a null´aban balr´ol folytonos, minden¨utt m´ashol folytonos, haα >0

• f minden¨utt folytonos, haα <0

• lim

x→0+f(x) = 0, lim

x→∞f(x) =∞, haα >0

• lim

x→0+f(x) =∞, lim

x→∞f(x) = 0, haα <0

• f⁰(x) =αx^α−1

• Df⁰ = (0,∞)

• Rf⁰ = (0,∞),haα >0

• Rf⁰ = (−∞,0),haα <0

• Z

x^αdx= x^α+1 α+ 1 +C