шамин с сдо
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M = sup f(x)
[a,b]
[a, b]
xε [a, b]
f(x) |
[a, b] |
1
ϕ(x) = M − f(x) .
ε > 0
0 ≤ M − f(xε) < ε.
ϕ(xε) = |
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M − f(xε) |
ε |
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ϕ |
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[a, b] |
xM [a, b]
f(xM ) = M.
m = inf f(x) −m = sup(−f(x))
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[a,b] |
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xm [a, b] |
−f(xm) = |
−m
f(xm) = m.
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f(x) |
[a, b] |
f(a) f(b) |
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(a, b) |
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c (a, b) |
f(c) = 0 |
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f |
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f(x) |
[a1, b1] |
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x0 = (b1 − a1)/2 |
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[a2, b2]
f(x)
[ak+1, bk+1] [ak, bk], k = 1, 2, . . . .
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k − |
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2k |
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c |
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f |
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ε |
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c |
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f(x) |
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ε |
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U(c, ε) |
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f(c) = 0 |
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f(x) |
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[a, b] |
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f(a) = A f(b) = B |
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A 6= B |
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C |
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A B |
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c (a, b) |
f(c) = C |
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F (x) = f(x) − C |
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f(x) |
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F (x) |
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F (x) |
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(a, b) |
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c |
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F (c) = 0 |
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f(c) = C |
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f(x)
I
Yf = {f(x) : x I}
f(x)
x I
Yf
f−1
Yf
[a, b]
f−1
f(x)
y Yf |
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y = f(x) |
f(x) |
I
f−1(x)
x = f−1(f(x)), x = f(f−1(x)).
(x)
I
f(x)
[A, B] |
A = f(a) B = f(b) |
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y [A, B] |
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x |
[a, b] |
x |
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x′ |
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[a, b] |
f(x′) = y |
x′ = x |
x′ |
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x |
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x′ < x |
x′ > x |
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f(x′) < f(x) |
f(x′) > f(x) |
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f(x′) = y = f(x)
f(x)
y0
(A, B) |
x0 = f−1(y0) (a, b) |
ε > 0 |
[x0 − ε, x0 + ε] [a, b].
ya = f(x0 − ε) yb = f(x0 + ε)
y0 (ya, yb) |
δ > 0 |
(y0 − δ, y0 + δ) (ya, yb). |
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f−1((y0 − δ, y0 + δ)) f−1((ya, yb)) = (x0 − ε, x0 + ε). y0 = A y0 = B
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f(x) |
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I R |
g(x) |
J R |
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g(x) |
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Yg I, |
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h(x) = f(g(x)) |
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J |
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f(x) |
[a, b] |
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g(x) |
[c, d] |
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g(x) |
[a, b] |
h(x) = f(g(x))
[c, d]
x0 [c, d]
xx0+Δx [c, d]
g(x) |
g |
g → 0, |
x → 0. |
g |
f(x) |
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f(x) |
f → 0, |
g → 0. |
h → 0, x → 0.
h(x)
f(x)
x0
f(x)
x0 |
x0 |
x0
x0
f(x)
x0
c = lim f(x) − lim f(x)
x→x0+0 x→x0−0
x0
x2 
c
x1
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f(x) |
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x0 |
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x0 |
x0 |
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lim f(x) |
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x→x0+0 |
lim |
0 f(x) |
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±∞ |
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x x0 |
− |
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→ |
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f(x) = c |
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x |
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y = c − c = 0. |
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f(x) = x |
R |
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y = |
x → 0, |
x → 0. |
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Pn(x) |
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sin x |
cos x |
tg x ctg x |
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sin x |
| sin x| < |x|.
| sin x| |
1 |
|x| < |
π/2 |
|x| |
|x| | sin x|
x −x
lim sin x = 0.
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x→0 |
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x0 |
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x |
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sin x |
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sin |
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· cos x0 |
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x→0 |
− |
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x→0 |
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2 |
+ 2 |
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= 0. |
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lim [sin(x + x) |
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sin x] = 2 |
lim |
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x |
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x |
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cos x |
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sin x |
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R |
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cos x |
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cos x = sin x + |
π |
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2 |
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tg x |
ctg x |
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sin x < x < tg x, x 0, |
π |
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2 |
cos x < sin x < 1. x
cos x |
sin x |
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x |
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0 < |x| < π2 |
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x → 0 |
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lim cos x = 1 |
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x 0 |
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cos x |
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→ |
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lim |
sin x |
= 1. |
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x |
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x→0 |
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arcsin x, arccos x, arctg x