шамин с сдо

f(x)

c

ε

c

f(x)

f(x) > f(c),

x < c

f(x) < f(c),

x > c

f(x)

c

ε

c

x

U(c, ε)

x 6= c

f(x) < f(c).

f(x)

c

ε

c

x

U(c, ε)

f(x) > f(c), x < c

f(x) > f(c), x > c.

f(x)

f(x) = x2

c−1 = −1

f(−1) = 1

x U(c−1, 1)

x < c−1 f(x) > 1

x > c−1

f(x) < 1

c1

f(x)

c0 = 0

f(c0) = 0

x 6= 0

f(x) > 0

f(x)

c

c

f′(c) = lim

f(x) − f(c)

,

x→c

x − c

ε = f′(c) > 0

c

x

− c

− f′(c) < f′(c).

f(xx

)

f(c)

−

x

c

c

f(x) − f(c)

> 0.

x − c

f(x) > f(c)

x > c f(x) < f(c)

x < c

f(x)

c

c

c

c

f(x)

c f′(c) = 0

f(x)

c

f′(c)

f′(c) = 0

f(x) = x2

f′(x) = 2x

x < 0

x > 0

x = 0

f(x)

[a, b]

(a, b)

f(a) = f(b)

ξ (a, b)

[a, b]

M

M = f(a)

M = f(b)

f(a) = f(b)

f(x) = M

x

[a, b]

f′(x) = 0

F (a) = F (b) = 0

ξ

(a, b)

F ′(x) = 0

f(x) = sin x

[0, π/2]

f(x)

g(x)

[a, b]

(a, b)

g(x)

(a, b)

g(a) 6= g(b)

ξ (a, b)

f(b) − f(a)

=

f′(x)

.

g(b) − g(a)

g′(x)

f(x)

g(x)

Vε

g′(x)

Vε

lim f(x) = lim g(x) = 0.

x→a

x→a

lim

f′(x)

,

x a g′(x)

→

lim

f(x)

= lim

f′(x)

.

x a g(x)

x a

g′

(x)

→

→

xn Vε

xn → a

f(x) g(x)

x = a

U(ε, a)

xn Vε

f(x)

g(x)

[xn, a]

[a, xn]

f(xn) − f(a)

=

f′(ξn)

,

g(xn) − g(a)

g′(ξn)

ξn (xn, a)

ξn (a, xn)

f(a) = g(a) = 0

f(xn)

=

f′(ξn)

.

g(xn)

g′(ξn)

n → ∞

xn → a

ξn

xn

a

ξn → a

n → ∞

{xn}

f′(x)

lim

f(x)

= lim

.

x a g(x)

x a

g′

(x)

→

→

lim

ex − 1 − x

.

x→0

x2

lim

ex − 1 − x

= lim

ex − 1

= lim

ex

=

1

.

2x

2

2

x→0

x2

x→0

x→0

lim

ecos x − e

.

x→0

x

lim

ecos x − e

= lim

− sin x · ecos x

= 0.

x→0

x

x→0

1