Post on 06-Jul-2018
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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
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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
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• 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
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[−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−
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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).
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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
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• 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
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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]
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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
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(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
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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)
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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.
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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.
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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)