Hm

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HM Tobias M. B¨olz

1 Allgemeines

3 Ableitungen

• Bernoullische Ungleichung: (1 + x)n ≥ 1 + nx (x ≥ 1)  n! • nk = k!(n−k)!

3.1 Differentiationsregeln • Produktregel:

• log(x  ·y) = log(x) + log(y) log xy = log(x) − log(y) (x, y > 0)

(u · v)0 (uvw)0

• Quotientenregel:

 u 0 v

= = =

u0 · v + u · v 0 u0 vw + uv 0 w + uvw0

u0 v−uv 0 v2

• Kettenregel: (u(v(x)))0 = u0 (v(x)) · v 0 (x)

2 Grenzwerte 3.2 Wichtige Ableitungen f¨ ur n → ∞: √ • na→1 √ • nn→1 √ • n n! → ∞ n • 1 + nx → ex n • 1 − nx → e−x √ • n1 n n! → 1e •

an n!

→0



nn n!

→∞

 a



n



an nk

f xn

f0 nxn−1

1

−n xn+1 1 √ 2 x 1 √ n n xn−1 x

n

x √ x √ n x ex ln x ax loga x sin x cos x tan x

1 cos2 x

1 cosh2 x −1 cosh2 x √1 x+ 1 √ 1 ,x > x2 −1 1 , |x| < 1−x2

−1 sin2 x

arcothx

1 1−x2 , |x|

e

1 x

ax ln a 1 1 ln a x

cos x − sin x = 1 + tan2 x

cot x

→ 0 (a > −1) n (a>1) → 0∞ (|a|<1)

sinx x

√ 1 1−x2 √ −1 1−x2 1 1+x2 −1 1+x2

cosh x sinh x

4 Integartion • Partielle Integration:

f¨ ur x → 0: •

f0

f arcsin x arccos x arctan x arccotx sinh x cosh x tanh x coth x arsinhx arcoshx artanhx



→1

• x log x → 0

R

f0 f

= log f

4.1 Polarkoordinaten x =R r cos ϕ y = r sin ϕ ⇒ f · r dϕdr (ϕ ∈ [0, 2π])

2.1 L’Hospital 0

lim fg = lim fg0 wenn f, g → 0 oder g → ±∞

1

R

u0 v = uv −

R

uv 0

1 1

>1

7 Differenzierbarkeit

4.2 Zylinderkoordiaten x =R r cos ϕ y = r sin ϕ ⇒ f · r dϕdrdz (ϕ ∈ [0, 2π])

f (x0 + a) − f (x0 ) − a · f 0 (x0 ) ! =0 a→0 ||a||

z=z

lim

8 Extremwerte mit Nebenbedingungen 4.3 Kugelkoordinaten

• D beschr¨ankt und abgeschlossen, also kompakt

x =R r cos ϕ cos ϑ y = r sin ϕ cos ϑ ⇒ f · r2 cos ϑ dϕdϑdr (ϕ ∈ [0, 2π], ϑ ∈ [− π2 , π2 ])

z = r sin ϑ

• f stetig ⇒ f nimmt auf D Minimum und Maximum an Rang h0 (x0 ) minimal ⇒ Multiplikatorenregel von Larange: F (x) = f (x) + λh(x) F 0 (x) = 0 ⇒ Extremstellen.

5 e, Sinus, Kosinus,... 5.1 in R =

∞ P

=

n=0 ∞ P

cos x

=

n=0 ∞ P

sinh x

=

n=0 ∞ P

cosh x

=

n=0 ∞ P

=

n=0 ∞ P

ea sin x

arctan x

n=0

9 Implizit definierte Funktionen

1 n n! a

• f (x, g(x)) stetig differenzierbar

(−1)n 2n+1 (2n+1)! x

• f (x0 , g(x0 )) = c u uft ¨berpr¨

(−1)n 2n (2n)! x

• det(fy (x0 , y0 )) 6= 0 (bzw. fy (x0 , y0 ) 6= 0)

1 2n+1 (2n+1)! x

=

1 x 2 (e

− e−x )

1 2n (2n)! x

=

1 x 2 (e

+ e−x )

⇒ Laut Satz u ¨ber implizit definierte Funktionen existiert eine Umgebung Uε (x0 ) und ein g : U → R mit g(x0 ) = y0 und f (x0 , g(x0 )) = c und

(−1)n 2n+1 2n+1 x

g 0 (x0 ) = −

5.2 in C: ez sin z cos z sinh z cosh z

=

∞ P

=

n=0 ∞ P

=

n=0 ∞ P

= =

n=0 1 z 2 (e 1 z 2 (e

zn n! 2n+1

z (−1)n (2n+1)! 2n

z (−1)n (2n)!

=

ex (cos y + i sin y)

=

1 iz 2i (e

=

1 iz 2 (e

− e−iz ) + e−iz )

− e−z ) + e−z )

5.3 Additionstheoreme • sin(x + y) = sin x cos y + sin y cos x • cos(x + y) = cos x cos y − sin x sin y • sin2 x + cos2 x = 1 • cosh2 x − sinh2 x = 1

6 Richtungsableitung lim

t→0

f (x0 + ta) − f (x0 ) t

(||a|| = 1)

2

1 fx (x0 , g(x0 )) fy (x0 , g(x0 ))

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