lehrstoff1

8/17/2019 lehrstoff1

1/14

a b a < b

R

• [a, b] = {x ∈R | a ≤ x ≤ b} • (a, b) = {x ∈R | a < x < b}

• [a, b) = {x ∈R | a ≤ x < b}

• (a, b] = {x ∈R | a < x ≤ b}

• (

−∞, b) =

{x ∈

R

| x < b

}• (−∞, b] = {x ∈R | x ≤ b}• (a, ∞) = {x ∈R | a < x}• [a, ∞) = {x ∈ R | a ≤ x}• (−∞, ∞) = R

a ∈ R ε ∈ R ε > 0 •

K ε(a) := (a − ε, a + ε) ε a

•

K ε(

∞) := ( 1

ε,

∞)

ε

∞

•

K ε(−∞) := (−∞, − 1ε ) ε −∞

x ∈ R y K ε(x)

|x − y| < ε

A

A ⊆ R

A

A

A

f

Df

Rf

f

x → f (x)

y = f (x)

y

x

x

y

f

{(x; f (x)) | x ∈ Df }

f


2/14

f : A → B

y ∈ B x ∈ A x x y = f (x)

f : A → B

y ∈ B

x ∈ A

y = f (x)

y ∈ B x ∈ A

f : A → B

f

y ∈ B

x ∈ A

f

g

c

cf Df f

c

f

f + g

f − g

f · g

f g

f

g

Df +g = Df −g =Df ·g = Df ∩ Dg

f g

f g

Df ∩ Dg g = 0 D f g

= (Df ∩Dg) \ {x | g(x) = 0}

g

f

f ◦ g x f (g(x)) f ◦ g x ∈ Dg g(x) ∈ Df

f

f

f

f̃

x f x

f̃ (x)

f ( f̃ (x)) = x

f

x ∈ Df −x ∈ Df f (x) = f (−x) f (x) = −f (−x)

f p > 0

x ∈ Df x + p ∈ Df x − p ∈ Df f (x + p) = f (x − p) = f (x)

f

I

I ⊆ Df k ∈ R K ∈ R f (x) ≥ k

f (x) ≤ K

x ∈ I

I ⊆ Df f


3/14

• x1, x2 ∈ I x1 < x2 f (x1) ≤ f (x2)

f

I

•

x1, x2 ∈ I x1 < x2 f (x1) < f (x2) f I

• x1, x2 ∈ I x1 < x2 f (x1) ≥ f (x2)

f

I

•

x1, x2 ∈ I x1 < x2 f (x1) > f (x2) f I

f (x) = ax + b

a

b

y

f (x) = ax2 + bx + c

(− b2a

; − D4a

) D = b2 − 4ac

f (x) = anxn + . . . a1x + a0 an = 0

n ak xk

f (x) = xn n ∈ N \ {0} x

n

xn

x

n

f (x) = n√

x

n

∈N

\ {0; 1

}

x

n

a a ∈ R a > 0 f (x) = expa(x) = a

x

a

a ∈ R

a > 0

a = 1

f (x) = loga(x)

x → sin x x → cos x x → x x →

x

x → |x|

D = R

R = [−1, 1]

sin(−x) = −sin x

sin(x + k · 2π) = sin x

k ∈ Z

2π

kπ

k ∈ Z

1

−1


4/14

[−π2

+ k · 2π, π2 + k · 2π] k ∈ Z

[π

2 + k · 2π, 3π

2 + k · 2π]k ∈ Z

−π2 ,

π2

arcsin

sin(arcsin x) = x

x ∈ [−1, 1],arcsin(sin x) = x

x ∈−π

2, π

2

.

[0, π]

arccos

cos(arccos x) = x x ∈ [−1, 1],arccos(cos x) = x x ∈ [0, π] .

−π2 ,

π2

( x) = x x ∈ (−∞, ∞),

( x) = x x ∈ −π2 , π2 . (0, π)

( x) = x x ∈ (−∞, ∞), ( x) = x x ∈ (0, π) .

α ∞ −∞ f α

f

α

β

ε > 0 δ > 0 f (x) ∈ K ε(β ) x ∈ K δ(α) x = α

limx→α

f (x) = β

f (x) → β

x → α

α ∈ R f

α

f

α β ε > 0 δ > 0 f (x) ∈ K ε(β ) x ∈ K δ−(α) x = α lim

x→α−f (x) = β f (x) → β x → α−


5/14

α ∈ R f

α

f

α β ε > 0 δ > 0

f (x) ∈ K ε(β ) x ∈ K δ+(α) x = α lim

x→α+f (x) = β f (x) → β x → α+

f α β

f

α

β f

α

f α

f g α

β f

β g

• c · f α c ∈R limx→α

c · f (x) = c · β g

• f ± g α limx→α

f (x) ± g(x) = β f ± β g

• f · g

α

limx→α

f (x) · g(x) = β f · β g

• f g

α

β g = 0 limx→α

f (x)g(x)

= βf βg

limx→∞

cx =

∞, c > 1,1, c = 1,

0, 0 < c 0

limx→∞

xn = ∞ limx→−∞

xn =

∞, n −∞,

n

limx→0

sin x

x = 1.

c ∈R limx→∞1 +

c

xx

= ec,

e e = 2.71828182 . . .

limx→0

loga(1 + x)

x =

1

ln a.

f α ∈ Df f

α

limx→α

f (x) = f (α).


6/14

f α ∈ Df

f

α

α

limx→α−

f (x) = f (α).

f

α ∈ Df f α

α

limx→α+

f (x) = f (α).

f

g

α

f ± g

f ·

g g(α) = 0 f

g α

g

α

f

g(α)

f ◦ g α

f

x

f x

limh→0

f (x + h) − f (x)h

f x

f (x)

f

x

f (x+h)−f (x)h

f

f : x → f (x),

x ∈ Df f f (x) f

f x f

x

f

x

f x x + h

h → 0

f (x + h) =

f (x + h) − f (x)

h − f (x)

→0

· h →0

+ f (x) · h →0

+f (x) → f (x).

f

x

f

g

I


7/14

• c · f c ∈R I (c · f ) = c · f • f ± g I (f ± g) = f ± g • f ± g

I

(f · g) = f · g + f · g

• f g

I

g(x) = 0

x ∈ I

f g

= f ·g−f ·g

g2

g

I f g(x) x ∈ I

f ◦ g

I

(f ◦ g) = (f ◦ g) · g.

f

I f̃ x ∈ I f f̃ f ( f̃ (x)) = 0

f̃

I

f̃ = 1

f ◦ f̃ g = f̃

1 = (f ( f̃ (x))) = f ( f̃ (x)) · f̃ (x), (f ( f̃ (x))) = (x) = 1

f

x0 (x0; f (x0))

f (x0) f (x0; f (x0))

y

−f (x0) = f

(x0)·

(x−

x0).

f

I

f (x) = 0 x ∈ I

E f (x) = x · f (x)

f (x)

f I f I


8/14

f

g

α

α

g(x) = 0 limx→α

f (x) =

limx→α

g(x) = 0(∞ − ∞)

limx→α

f (x)

g(x) = β

limx→α

f (x)

g(x) = β.

f

I

f

I

f

I

f x0

K ε(x0) ⊂ Df x0 f (x0) ≤ f (x) f (x0) ≥ f (x) x ∈ K ε(x0)

f x0

I ⊂ Df f (x0) ≤f (x) f (x0) ≥ f (x) x ∈ I

f x0 f x0

f (x0) = 0.

x f (x) = 0

f x0

x0 x f (x) < 0

x < x0 f (x) > 0 x > x0

x0 f

f x0 f x0

x0 f f (x0) = f

(x0) =· · · = f (m−1)(x0) = 0 f (m)(x0) = 0 f

• m x0

•

m

x0 f (m)(x0) > 0

x0 f (m)(x0) < 0

[a, b]

[a, b]


9/14

I

I

I

I

x0 f

x0

(x0; f (x0))

(x0; f (x0))

±∞

f

x0 n

n

T n(x) = f (x0) + f (x0) · (x − x0) + f

(x0)

2! · (x − x0)2 + · · · + f

(n)(x0)

n! · (x − x0)n

n

f

x0

f

x0 n

T n(x0) = f (x0), T

n(x0) = f (x0), T

n (x0) = f (x0), . . . , T

(n)n (x0) = f

(n)(x0).

f

x0 n + 1 x1 x x0

f (x) − T n(x) = f (n+1)(x1)

(n + 1)! (x − x0)n+1


10/14

(an) K a ∈ R K f ∈ R an ≥ K a

an ≤ K f n ∈ N (an)

K a ∈ R K f ∈ R

(an)

(an)

• an ≤ an+1 n ∈N

• an < an+1 n ∈N•

an ≥ an+1 n ∈N•

an > an+1 n ∈N

β ∞ −∞

(an) β ε > 0 δ > 0 an ∈ K ε(β ) n ∈ K δ(∞)

f ∞

(an) an = f (n) β f ∞ β (an)

T P n

T n n r = 1 + P 100

T n = T + P 100T n = T

1 + P 100T n

T n = T n−1

1 + P

100

n = 1, 2, . . . T 0 = T

T n = T · rn T n = T · e Pn100 = T · en(r−1)

T n = A ·r · rn−1r−1 A


11/14

B · rn = A · rn−1r−1 B

A

n

f

K a ∈ R K f ∈ R f (x, y) ≥ K a f (x, y) ≤ K f (x, y) ∈ Df

(x, y)

∈ R2

ε

∈ R

ε > 0

ε (x, y) K ε(x, y)

(x, y)

ε

K ε(x, y) = {(a, b) ∈ R2 |

(a − x)2 + (b − y)2 < ε}

f (x0, y0) ∈Df

K ε(x0, y0) ⊂ Df (x0, y0) f (x0, y0) ≤ f (x, y)

f (x0, y0) ≥ f (x, y) (x, y) ∈ K ε(x0, y0)

f

H ⊆ Df ⊆ R2 f H (x0, y0) ∈ H f (x0, y0)

≤f (x, y) f (x0, y0) ≥ f (x, y) (x, y) ∈ H

β ∞ −∞ f

(x0, y0) (x0, y0) f (x0, y0) β ε > 0

δ > 0

f (x, y) ∈ K ε(β ) (x, y) ∈ K δ(x0, y0)(x, y) = (x0, y0) lim

(x,y)→(x0,y0)f (x, y) = β

f (x, y) → β (x, y) → (x0, y0)

f (x0, y0) ∈R2 (x0, y0)

∈ Df f (x0, y0)

lim(x,y)→(x0,y0)

f (x, y) = f (x0, y0) .

H ⊆ R2 H

f

(x0, y0) ∈R2 f x(x) = f (x, y0) f y(y) = f (x0, y)

f

x

y

(x0, y0)


12/14

f

f

x

y

x0

y0

f (x0, y0)

x

y

f x(x0) f

y(y0) f x y

(x0, y0) f

x(x0, y0) ∂f ∂x

(x0, y0)

f y(x0, y0) ∂f ∂y

(x0, y0)

f f x

f y (x, y) f x y (x, y)

f x(x, y) f

y(x, y) f x

y

f

(x0, y0) [f

x(x0, y0), f

y(x0, y0)] f (x0, y0) ( f )(x0, y0)

(x, y) f

(x, y)

f f

(x0, y0)

(x0, y0)

(x0, y0) f

f

(x0, y0)

D(x, y) = f xx(x, y) · f yy(x, y) − f xy(x, y) · f yx(x, y).•

D(x0, y0) > 0 (x0, y0) f f xx(x0, y0) > 0 (x0, y0)

f xx(x0, y0) < 0 (x0, y0)

•

D(x0, y0) < 0 f (x0, y0)

(x1, y1), . . . , (xn, yn)

y = mx + b

σ(m, b) =ni=1

(mxi + b − yi)2 → min!

(m0, b0) σ

m0 =

ni=1(xi · yi) − n · x̄ · ȳni=1(xi)

2 − n · (x̄)2 , b0 = ȳ − m0 · x̄.


13/14

f (x0, y0) ∈Df

ϕ(x, y) = 0 K ε(x0, y0) ⊂ Df

(x0, y0) f (x0, y0) ≤ f (x, y) f (x0, y0) ≥f (x, y) (x, y) ∈ K ε(x0, y0) ϕ(x, y) = 0

f (x, y)

ϕ(x, y) = 0 λ

L(x ,y,λ) = f (x, y) + λ · ϕ(x, y).

λ

f (x0, y0)

ϕ(x, y) = 0

L(x ,y,λ) λ0 (x0, y0, λ0)

Lx(x ,y,λ) = 0,

Ly(x ,y,λ) = 0,

Lλ(x ,y,λ) = ϕ(x, y) = 0.

f

ϕ(x, y) = 0 (x0, y0, λ0) L(x ,y,λ) = f (x, y) + λ

· ϕ(x, y)

D(x ,y,λ) = 2 · ϕx(x, y) · ϕy(x, y) · Lxy(x ,y,λ)−(ϕx(x, y))2 · Lyy(x ,y,λ) − (ϕy(x, y))2 · Lxx(x ,y,λ).

•

D(x0, y0, λ0) > 0 (x0, y0) f ϕ(x, y) = 0

•

D(x0, y0, λ0) < 0 (x0, y0) f ϕ(x, y) = 0

F f

I ⊂ R F

(x) = f (x)

x ∈ I

f

I ⊂ R f I

g I ⊂ R f g

F

(f ◦ g ) · g

I f (g(x)) · g(x) x = F (g(x)) + C.


14/14

g I ⊂ R

f

g F

(f ◦ g) · g

I

f (x) x =

f (g(t)) · g(t) x.

u

v

I ⊂ R u · v u · v

u(x) · v(x)

x = u(x) · v(x) −

u(x) · v(x)

x.

f [a, b] x

f

[a, b]

ba

f (x) x a b

x

x

ba

f (x)

x

f (x)

a

b

f

[a, b] F f

ba

f (x) x = F (b) − F (a).

F (b) − F (a) = [F (x)]ba

f

[a, b]

x0 ∈ [a, b]

F (x) =

xx0

f (t) t

[a, b]

F (x) =f (x) x ∈ (a, b)

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