x + = 0
N = {,, 3....}, Z Q, b, b N c, d c, d N + b = c, b = d. N = =. < > P n P (n) P () n = P (n) P (n + ) n n + P (n) n P (n) n P n P (n) P (m) P (n) n m P (n + )
P (n) n m P n P (n) P () P (), P (),..., P (n) P (n + ) P (n) n n P (n) : n k = + + + n = k= n(n + ). ( + ) P () : =, n = n P (n) n + P (n + ) : + + + (n + ) = (n + )(n + ). + + + (n + ) = + + + n + (n + ) n(n + ) = + (n + ) ( n ) = (n + ) + = (n + )(n + ). + x = b,, b N, < b = b > b Z = {0, ±, ±, ±3,...}.
0 + = + 0 =, + ( ) = ( ) + = 0. < > = b, b Z = b +, b Z, b b 0 bx = b b Q = { m n : m Z, n N }. m n = m n m n = n m, m + m = m n + m n m, m = m m, n n n n n n n n m > m m n > m n m n m n > 0, m n m n N. n n q q = m n m n
6 = 3, 4 4 = 4 8 = 7 4. 3 7 4 4 6 4 x x =. q Q q = q = b b q = q = b = = b. = c c Z 4c = b c = b. b b b x = L R
l L r R l < r L R l,.4,.4,.44,.44,..., r l,.5,.4,.45,.443,....,.4,.4,.44,...,.5,.4,.45,... L R R Q A = A R A R x x A A
A R x x A A A, 3, 3 4,..., n n +,..., n N. x x 3 x < x < 3 = A R A R A x x x A A A R A A R A R A A
R A A A A A A A A A A A A A A, A = A, 3, 3 4,..., n n +,..., n N. < n n > n + > x x 3 x < x < 3 A = {x Q : x > 0 x < } A = 0 A =
A = {x Q : x > 0 x < } A Q A = Q A A = A R A R R = { : }., b c + b = c R, b, c R ( + b) + c = + (b + c) ( ). R, b R + b = b + ().
R 3 R R + 0 = 0 + = ( ). R 4 R R + ( ) = ( ) + = 0 ( )., b R b b b = + ( b)., b c b = c R 5, b, c R ( b) c = (b c) ( ). R 6, b R b = b (). R 7 R R = = ( ). R 8 R, 0 R = = ( )., b R, b 0 b b b = b.
R 9, b, c R (b + c) = b + c ( ). R < R 0, b R < b = b > b ( ). R, b, c R < b b < c < c R, b, c R < b + c < b + c R 3, b, c R < b c > 0 c < b c x R R + x = 0 + x = 0. R 3 x + 0 = x. R x = x + 0 = 0 + x = 0. R b, c R + b = + c = 0 R 3 R R R 3 c = c + 0 = c + ( + b) = (c + ) + b = ( + c) + b = 0 + b = b. R R 8 R R R 4 R R 5 R 8 R R R 9
R 0 R 3 (R, +,, <) (Q, +,, <) x =, x R x = R A R y A y A y A y A = { : A} A ( A) = s t = s A s A t = s A t A t A t A t A t A s t t s = t A R A = A > 0 x A x > A = A > 0 x A x < +
= A > 0 x A x A, 0, > 0 x A x > A = b < = b > 0 b A x A x b =, = A x, y R x > 0 n N nx > y nx y n N A = {nx : n N} A A = s x > 0 s x < s n 0 N n 0 x > s x s < (n 0 + )x s A x R Z x < + x [x] [x] x < [x] +.4 4.7 [.4] =, [ 4.7] = 5. x, y R x < y q Q x < q < y
y x > 0 n N n(y x) > nx + < ny nx < [nx] + nx + < ny, q = [nx] + n x < [nx] + n < y, x, y R x < y p x < p < y x < y x < y q Q x < q < y, x < q + < y. p = q + R { 0, = < 0. R b 0 b b b.
, b R + b + b ( ). b b b + b ( ). 0 < 0 b b b, ( + b ) + b + b, = b + b b + b b b. b = b + b + = b + b b. b b b, b b b = b + < < + R. + +
R + ( ) = ( ) + = (+ ) =, + (+ ) = (+ ) + = ( ) = +. > 0 ( ) = ( ) =, (+ ) = (+ ) = +. < 0 ( ) = ( ) = +, (+ ) = (+ ) =. ( ) + ( ) =, (+ ) + (+ ) = +, ( ) ( ) = +, (+ ) (+ ) = +, ( ) (+ ) = (+ ) ( ) =. ( ) + (+ ), (+ ) + ( ), 0 ( ), ( ) 0, 0 (+ ), (+ ) 0,, +, +, + +. A A = + A A =, b R < b (, b) = {x R : < x < b}, (, b] = {x R : < x b}, [, b) = {x R : x < b}, [, b] = {x R : x b}, (, ) = {x R : x > }, [, ) = {x R : x }, (, ) = {x R : x < }, (, ] = {x R : x }.
n k k= n (k ). k= n k = k= n(n + )(n + ). 6 n (k) k= n (k ). k= n (k + ) k = A + (B + Cn) n, k=0 A, B C n n (i) n > n (ii) n > n 3 x > ( + x) n + nx n N. x > ( + x) n + nx + + 3 n(n ) x n N. q Q q 3 = 6 3 6 p, q, r, s Q p + q = r + s, p = r, q = s q s
A = {x R : x > 0 0 < x } B = {x Q : x 0 0 < x } C = {0,, 3, 4,..., n },... D = {x R : x < 0 x + x < 0} { } E = n + ( )n : n N A R A = A A A, B R A B B A A B. A, B R A+B = {+b : A, b B} (A + B) = A + B. R x < y + > 0 x y x y + > 0 x y x y > 0 x = y < x < b < y < b x y < b
n n n n ( n ) ( n ) n 0 ( n ) n 0 {, } 3,..., n n +,... n =, n =,,... n n = ( )n n, n =,,... n = n, n =,,... n = ( ) n, n =,,...
=, n+ = n +, n =,,... n n n n n n n ε > 0 ( ε, + ε) ε ε n n n n ( ε, + ε) ε < n < + ε ε < n < ε n < ε ( n ) ( n ) ε > 0 N = N(ε) n n > N n < ε. n n n n = ε ε N ε N ε n ( ε, + ε)
N ( ε, + ε) ε > 0 ( ε, + ε) m ( n ) m = 5, 50 0 5 n n = n m +n = n m+n ( n ) (b n ) m n = b n n > m n n = n b n = ( n ) n =, n =,,... n,, 3,..., 0,..., 00,..., 000,..., n ε > 0 N n n > N n 0 = < ε nε >. n x = ε > 0 y = N Nε > n n > N nε > Nε > n = ( ) n, n =,,...,,,,,..., m =, m =,,... m =, m =,,... m m m ( m ) ( m ) n ( n )
n n R ε > 0 N n = N > N n n > N ( ) n < ε. ( ) N < ε < ε ε < < ε ε < < + ε. n = N + > N ( ) N+ < ε < ε ε < + < ε ε < < + ε. ε = 0 < < < < 0 ( n ) ( n ) ( n ) n = n = b n n n = b b ε = N b > 0 n n = n n > N n < ε = n n = b N b, n n > N n b < ε = b. N = {N, N } n n > N n > N n > N b = n + n b n + n b < = b b + b = b, n = n, n =,,... n n +
( n ) ( n ) + E > 0 N = N(E) n n > N n > E. n + n n n = + ( n ) E > 0 N = N(E) n n > N n < E. n n n n = E E N E N E n E n E n (E, ) (, E ( n ) ε + E ( n ) ± n = n, n =,,... + n E > 0 N n n > N n > E. x = > 0 y = E N N > E n n > N n > N > E ( n ), (b n ) (c n ) n b n c n n N
n n = n c n = n b n = ε > 0 n c n n N N n n > N n < ε ε < n < + ε n n > N c n < ε ε < c n < + ε. N = {N, N } n n > N n > N n > N ε < n b n c n < + ε b n < ε, b n n n b n c n n m m N b n =, n =,,... n, 4, 9,..., 00,..., 0000,..., 000000,..., n c n =, n =,,... n n = 0, n =,,... 0 < n n n N, n 0 = n n n = 0, n = 0 ( n ) M > 0 n M n N.
( n ) n n ε > 0 N n n > N n < ε. n n < ε n < + ε n > N. M = {,,..., N, + ε} n M n N ( ). ( n ) ( n ) n n n 0 n n 0 n ( n ) n 0 n 0 n = 0
( n ) n n n n ε > 0 N n n > N n < ε. n n < ε, n n ( n ) (b n ) n b n b n +b n +b n ε > 0 n b n b ε = ε > 0 N N n n > N n < ε = ε, n n > N b n b < ε = ε. N = {N, N } n n > N n > N n > N n + b n ( + b) = n + b n b n + b n b < ε + ε = ε, n + b n + b n ( n ) (b n ) n 0 (b n ) n b n 0 n (b n ) M > 0 b n M n N.
ε > 0 n 0 ε = ε M n n > N > 0 N n < ε = ε M. n n > N n b n = n b n < ε M M = ε, n b n 0 n ( n ) c R n c n c n ( n ) (b n ) n b n b n b n b n n b n b 0 n b n b = n b n b n + b n b = ( n )b n + (b n b). n n 0 (b n ) ( n )b n 0. b n b b n b 0 (b n b) 0. n b n b = ( n )b n + (b n b) 0 n. ( n ) (b n ) b n 0 n N n b n b b 0 n b n b n b n b 0 b n b = b b n bb n. b n b
b n b ε = b > 0 N n n > N b n b b n b < b b n b > b b n > b b n < b, b n bb n b b n = (b n b) 0 b n b = b b n 0 n bb n n n = 0 n ( n ) = ( ) n n = ( ) 0 = 0, n ( ) n + n ( = + ) = n n + n n n = + 0 =, = n n n = 0 ( ) 4 4 7n 4 = n n 4 n 7 4 + n + = n 4 4 n n 4 + n ( 7) + n n = n 4 ( 7) = 7, = = 0 n n n n4 4 0 + ( 7) + 0 = 7, ( n )
( n ) n+ n n N ( n ) n+ > n n N ( n ) n+ n n N ( n ) n+ < n n N ( n ) ( n ) ( n ) n m n m. ( n ) A = { n : n N} A ( n ) A + ( n ) A = { n : n N} ( n ) n A n ( n ) n + n
( n ) n A n ( n ) n n ( n ) A = { n : n N} A A = n ε > 0 ε < ε A N ε < N, ( n ) A n n > N ε < N n < + ε. n > N n < ε n = A E > 0 ( n ) E A N N > E, ( n ) n n > N n N > E. n > N n > E ( n ) + n =! +! + +, n =,,.... n! n+ n = = (! + ) (! + + (n + )!! +! + + ) n! (n + )! > 0 n+ > n n N,
( n ) 0 < n =! +! + + n! + + + + n ( ) = n = <, n ( n ) ( n ) n > 0 n N ρ < n 0 n ρ > n + n n+ = ρ. n n n+ = ρ 0 ρ < ε > 0 n n ρ < ρ + ε = r < ε N n n N n+ ρ n+ n ρ n < ε n+ < ρ + ε n+ < r n. n n > N 0 < n < r n < r n < < r n N N = N r N rn ( ). r < r n 0 n 0 n n+ = ρ ρ > ε > 0 ρ > ρ ε = r > n n ε N n n N n+ n ρ > ε n+ n > ρ ε n+ > r n.
n > N n > r n > r n > > r n N N = N r N rn ( ). r > r n + n + n ( n ) ρ = ( n ) n 0 n N n ρ < n 0 n ρ > n + n n n = ρ. n n = ρ 0 ρ < ε > 0 n ρ < ρ + ε = r < ε N n n N n n ρ n n ρ < ε n n < ρ + ε n < r n. r < r n 0 n 0 n n n = ρ ρ > ε > 0 ρ > ρ ε = r > n ε N n n N n n ρ > ε n n > ρ ε n > r n. r > r n + n + n ( n ) ρ = R n = n, n =,,...
> ( n ) + n = n n < < n 0 n = ( n ) n n < ( n ) n + n > x > 0 = + x n = n = ( + x) n + nx n N. E > 0 y = E N Nx > E n > N ( n ) + n n = n + nx > + Nx > Nx > E, 0 < < > x > 0 = + x n = ( + x)n + nx > nx n N. 0 < n = n < nx n N. nx 0 n 0 n = 0 < < 0 0 < b = < b n 0 n n 0 n < b = > b n + n > 0 n = n n
> n = n > n = + x n x n 0 = n n = ( + x n ) n + nx n > nx n n N, 0 < x n < n n N, x n 0 n 0 < < > n n, n n n = n = n n n n = n n n = + x n x n 0 ( + x) n + nx + n(n ) x > n(n ) x n N ( n = + n) n n e ( + n) n = e. n q Q n ( + q n) n = e q. n = n, n =,,... n! n+ n = n+ (n + )! n n! = n + 0 < ( ),
n 0 n n = 4n, n =,,... n4 4 n n n 4 = 4 ( n n) 4 4 4 = 4 >, n + n ( n ) n+ = n +, n =,,..., =. n > 0, n =,,... = + = > = n n n + n + n + n + n+ n, =, =, 3 = +, 4 = + +,..., ( n ) = < n < n+ = n + < + = 3 <, ( n ) n n = n+ n + + ( ),
= + = + = 0 = 5 = + 5. n > 0, n =,,... 0 = + 5 ( n ) n = n+ n N n = n+ n N ( n ) ( n ) 0 < n < n N n N n = n n ( n ) ( n ) N N ( n ) n 0 n N n n n n n = n 3 3n + n = n 3 3n + n = n 3 3n + ( ) n 3 n = 3n + n = n n
n = n + n n = n n n = (n ) n n = n n n + ( n ) ( n ) ( n ) ( n ) ( n ) ( n ) n n = > 0 n > 0 n > N, N N ( n ) (b n ) b n < n n N n 0 b n 0 n n = 0 + 0 n n = 00 + ( )n n n = 00 n + ( )n n = + 4 +... + n n = + +... + n n ( ) ( n + n = ) 4n n ( n = ( ) n + ) n n + ( ) n n = 4n n = n + ( )n n n n = n + n n + n
( n ) n 0 n N n+ n n = ρ. ρ < n 0 n ( n ) n+ ρ n n N 0 < ρ < n 0 n n m, m N > 0 n+ ρ n n N ρ > n + n n N ( n ) n 0 n N n n ρ n N 0 < ρ < n 0 n n m, m N n n ρ n N ρ > n + n n m, m N n = 4n n! n = nn n! n = 4n n 8 n = n (n!) (n)! n = 4n n! n ( n n = + n) n ( n = + ) n n ( n = n + ) n n ( ) + n n n = n
( n = ) n n n = ( + n ) n n = n n + n ( n ) n+ = n +, n =,,..., = 0. ( n ) ( m ) ( m+ ) ( n ) m m+ m n n
( n ) + + + n + n n ( n ) n s n = + + + n. (s n ) s n s + + + n + = n = s. (s n ) + + n = + n = (s n ) n= n= n=
x x n. x n+ s n = + x + + x n x, = x n + x = n=0 n n=0 n, m x n < x < x n 0 s n n x n=0 x n = x. x > x n + s n + n + x < x = s n + n + x = (s n ) (s n ) n=m
n = + +... + m + n. n= n= n= n=m+,,..., m n = s b n = t λ, µ R (λ n + µb n ) = λ n + µ b n = λs + µt. n= n= n= n n n s n = k, t n = b k u n = (λ k + µb k ) k= k= u n = λs n + µt n. n = s b n = t s n s t n t n n= n= u n λs + µt n (λ n + µb n ) = λs + µt n= k= n = s n 0 n n= n s n = n n n s n s s n n= s n n= n = s n s n s s = 0 n. n n n n n=
n n = n + n n= ( ) n+ n = ( ) n+ n n= n= n n + n = n n + 0 n n 0 n n n n= n= n n 0 n N n= n n 0 n N n= (s n ) (s n ) n 0 n N n= n n = + s n+ = + + + n + n+ = s n + n+ s n n N, (s n ) (s n ) (s n ) n n= n=
(s n ) s n + n n = + n= n= n p = p + p + + n p +, p p > p p = n = + + 3 + 4 + 5 + 6 + 7 + 8 +, n= (s n ) (s n ) 3 + 4 > 4 =, 5 + 6 + 7 + 8 > 4 8 =, 9 + 0 + + 6 > 8 6 =,, 4, 8,... / / + + 4 + 8 + + n = n s n + n s n + n n N. n = s = + n n + s n+ = s n + n + + n + + + ( ) n + n = s n + n + + n + + + n+ s n + n n+ = s n +,
s n+ s n + + n + = + (n + ), M > 0 n 0 + n 0 > M n = n 0 n n n s n s n = s n 0 + n 0 > M, (s n ) n = + n= p < (s(p) np n ) (s (p) n M > 0 n n n= s (p) n = p + p + + n p > + + + n = s n > M, (s (p) n ) ) n= n p = + p > np n= p =, 4, 8,... + p 3 < p = p, p 4 p + 5 p + 6 p + 7 p < 4 4 p = 4 p, 8 p + 9 p + + 5 p < 8 8 p = 8 p. + p + 4 p + 8 p + = + p + n=0 < p ) n = ( p ( ) ( + ) 3 + = p p p = p p. n=0 ( ) n p
(s (p) n ) (s (p) n ) s (p) n p p n N, (s (p) n ) n= n= p > np n p p > np n= n n 0 n N M > 0 n Mb n n N b n n n= M > 0 M n b n b n 0 n N b n = + n = + n= (s n ) n= n= n= n (s n ) b n = t s n Mt n N (s n ) n= (s n ) n = s (t n ) b n n= n= n=
t n Ms n N (t n ) b n (s n ) n= n = + n= n n= b n n b n > 0 n N n = ρ > 0, n b n ( ) n M > 0 b n n= n b n M n Mb n n N. b n n= n = ρ > 0 n b n b n = n n ρ > 0 ( n b n > 0 ) n M > 0 n= M n b n n N n n= b n n= n n 3 + n + n= n n = > 0 n N n 3 + n + n n n 3 b n = n=
n n = 3 n n= n= n= n n 3 + n + n n b n n N, n n= n M = n 3 + n + n= n n + n= n n n n + > 0 n > 0 n N n n n + n n n= n n + n= = n N ρ < ρ > n n= n n= n+ = ρ. n n n n > 0 n+ = ρ 0 ρ < ε > 0 n n ρ < ρ+ε = r < ε n=
N n n N n = + + + N + n= n r n N N. n + + + N + n=n = + + + N + N n=0 r < n n= r n r n N N n=n r n n+ = ρ ρ > ε > 0 ρ > ρ ε = n n r > ε N n n N n = + + + N + n= n=0 n r n N N. n + + + N + n=n = + + + N + N n=0 r > n n= r n r n N N n=n r n ρ = n n 0 n N n n=0 n n = ρ. n=
ρ < ρ > n n= n n= n n = ρ 0 ρ < ε > 0 n ρ < ρ+ε = r < ε N n n N n = + + + N + n= n < r n. n < + + + N + n=n = + + + N + r N r < n n= n=0 r n n=n r n n n = ρ ρ > ε > 0 ρ > ρ ε = r > n ε n=0 N n n N n = + + + N + n= n > r n. n > + + + N + n=n = + + + N + r N n=0 r > n n= r n n=n r n r n r n n=0
ρ = n= n (n!) (n)! n+ n = n+ ((n + )!) (n + )! n (n!) (n)! = (n + ) (n + )(n + ) 4 = < n n= n= n (n!) (n)! n n n = n n = ( n n) = > n, n n n= n+ n = n+ + 5 n+ n + 5 n = n+ + 5 n= n n n + 5 n n + 5 = + 5 n + 5 + 5 0 = n, n + 5 0 n n n= n=
n n= n n= n n n 0 n + n n n N. n n= n= n n ( n + n ) n n= n = n= ( n + n n ) = n= n= ( n + n ) n= n n= n, n= ( n + n ) n= n= n ( ) n n n= n n= ( ) n n n= n n 4 n= n n 4 n= n n N, n 4 n4 n=
n4 n= n n 4 n= n n 4 n= ( ) n+ n n= + n n= ( ) n+ n n ( n ) n 0 n n= ( ) n+ n n+ n ( ) n n= n= ( n ) n 0 s n = + + n n 0 n n=
n 0 ( ) n n n= n > 0 n N n > 0 n N + 3 + + 3 + 3 + 3 3 + 4 + 3 4 +. n n= n= n n= n= n n + + 8 + 4 + 3 + 6 + 8 + 64 +. n b n = b (b n b n+ ) = b b n= (n )(n + ) = n= n + n = n + n n= n= n(n + )(n + ). ( n + n). n= + n n. n= n + n. n n= ( n n ) n. n=
n= ( ) n+ n + n. n= n= n= n + n n 3. n n. n! n n. ( n n ) n= θ n = n n n 3 ( + n) n < e < 3 n = ( + θ n ) n n 0 n N n > 0 n N n= n= n= n n + n n + n n= n + n n n p, q R n= ( + ) n. n p n n p (p > 0). n= n=
n= n= (0 < q < p). n p nq + ( ) n n. n= ( ) n + n. ( n ( ) 0) n+. n= ( ) n n n +. n= n+ n ( ) n. n= n n x n. n= n= n x n n.
X, Y X Y x X y Y f : X Y f X Y y = f(x) f x y f x f(x) y f(x) f f(x) f x x y y = f(x) x x X y Y x X y Y x X y Y f : X Y X f f(x) = {y Y : x X y = f(x)} = {f(x) : x X} f
f : X Y Y f : X Y f Y Y X R c R f : X R f(x) = c X = R f(x) = {c} X R f : X R f(x) = x X = R f(x) = R X R f : X R f(x) = x + X = R f(x) = [, ) X R f : X R { 0 x, f(x) = 0 x < 0 x >, X = R f(x) = {0, } X R f : X R f(x) = x(x ), X = R {0, } A B A B A B f(x) = (, 4] (0, ) A B A B X R f : X R { x, f(x) = 0 x, X = R f(x) = {0, } y = x x y x 0 x > 0
y y = x y x y t y t y(t) f : X Y f x, x X x x f(x ) f(x ) x, x X f(x ) = f(x ) x = x f f(x) = Y y Y x X y = f(x) x X y Y y Y x X f : R R f(x) = x f : R R f(x) = x + f : X Y g : U V f(x) U g f : X V (g f)(x) = g(f(x)) f g g f f(x) U x X f(x) U A A R f : A R g : A R
f + g : A R (f + g)(x) = f(x) + g(x) R f : A R ( f)(x) = f(x) f g : A R (f g)(x) = f(x)g(x) g(x) 0 x A f g : A R ( f g f(x) g(x) f g f(x) g(x) x A A A R f : A R f x, x A x < x f(x ) f(x ) ) (x) = f x, x A x < x f(x ) < f(x ) f x, x A x < x f(x ) f(x ) f x, x A x < x f(x ) > f(x ) f f A A R f : A R f M R x A f(x) M f m R x A f(x) m f M R x A f(x) M A A R f : A R f x A f( x) = f(x) f x A f( x) = f(x)
f : X Y f : X f(x) y f(x) x X y = f(x) f : f(x) X f (y) = x f(x) = y. f f f : X Y f f : X X f f : f(x) f(x) (f f)(x) = x x X (f f )(y) = y y f(x) f : R R f(x) = x 3 f R f R y = x 3 x f : R R f (y) = 3 y +. y x x + f : R R f (x) = 3. (x, f(x)) f (f(x), x) f (x, f(x)) x X OX f(x) Y OY (f(x), x) y = x f(x) X OX f(x) Y OY X OX y = x Y OY X OX X OX f(x)
x Y OY y = x (f(x), x) (x, f(x)) f f y = x ( n ) f : N R N f(n) = n p : R R p(x) = 0 p(x) = n x n + n x n + + x + 0, n N {0} 0,,..., n R n 0 n n = 0 p(x) = 0 p(x) = 0 p n = p(x) = x + 0 p n n n ρ p p(x) = (x ρ)p (x) p n f : A R f(x) = p(x) q(x), p, q q(x) 0 f A = {x R : q(x) 0} x x, x, x x X OX Y OY (t, s) (, 0), (0, ), (, 0) (0, ) π x = 0 (, 0) x = π/ (0, ) x = π
(, 0) x x (, 0) x > 0 x < 0 x (t, s) x = s, x = t, x = x x, x = x x. x x / {kπ + π/ : k Z} x x x / {kπ : k Z} x (t, s) x x x x π x x = x x3 3! + x5 5! + + ( )n x n+ + = (n + )! n=0 ( ) n x n+. (n + )! ( ) n x n+ x R (n + )! n = ( )n x n+ (n + )! n=0 n+ n = ( ) n+ x n+3 (n + 3)! ( ) n x n+ (n + )! x = (n + )(n + 3) 0 n x R. x x R x x = x! + x4 4! + + ( )n x n + = (n)! ( ) n x n. (n)! x x R ( ) n x n n = ( )n x n (n)! (n)! n=0 n=0 n+ n = ( ) n+ x n+ (n + )! ( ) n x n (n)! x = (n + )(n + ) 0 n x R.
x x R x x x x x x x = = 3 3! + 5 5! + 3 3! + 5 5! = 6 + 0 = 0 0 0.8467. = 0.8447... x R x, x. x + x =. ( π ) ( π ) x = x, x = x. x x. x ( π/, π/) x x x. : R [, ] : R [, ] π, b R ( + b) = b + b. ( b) = b b.
( + b) = b b. ( b) = b + b. () =. () = = =. + b = + b b = b + b = + b b = + b b. + b. b. b. : R [, ] [ π/, π/] : [ π/, π/] [, ] : [, ] [ π/, π/] (y) = x x = y x [ π/, π/] : R [, ] [0, π] : [0, π] [, ] : [, ] [0, π] (y) = x x = y x [0, π] : ( π/, π/) R : R ( π/, π/) (y) = x x = y x ( π/, π/) : (0, π) R : R (0, π) (y) = x x = y x (0, π) > 0 n = n
Z Q R ( e x e = + n n n) e x = + x + x! + + xn n! + = x x x R x n n! n = x n n! n=0 n+ n = x n+ (n + )! x n n! = x n + e x x R n=0 x n n!. 0 n x R. e x +x = e x e x x, x R e x = x R ex e x > 0 x R e x R e x R e x (0, + ) x = x x = x x > 0 e x x < 0 e x = e x x > 0 x = 0 e0 = x, x R x < x x x > 0 e x x e x x > ex e x > ex > e x e x < e x,
e x R x > 0 e x > + x x ex = x e x = 0 x, > 0 x + ex = + e x : R (0, ) : (0, ) R y = x e x = y x, > 0 x = e x., b > 0 x, y R x+y = x y. ( x ) y = xy. x =. x (b) x = x b x. > 0 x : R (0, + ) 0 < < = > x e x
A A R f : A R f(x) l f l x x 0 ε > 0 δ = δ(ε) > 0 x A 0 < x x 0 < δ f(x) l < ε. f(x) l x x 0 x x 0 f(x) = l ε ε δ ε f(x) l δ x x 0 0 < x x 0 < δ x 0 δ < x < x 0 + δ x x 0 f(x) l < ε l ε < f(x) < l + ε f(x) (l ε, l + ε) x (x 0 δ, x 0 + δ) {x 0 } 0 < x x 0 x x 0 x x 0 x = x 0 x 0 A f(x 0 ) x = x 0 x x 0 f(x) f x 0 x 0 A x 0 δ > 0 (x 0 δ, x 0 + δ) {x 0 } A x 0 δ > 0 x A 0 < x x 0 < δ x (x 0 δ, x 0 + δ) {x 0 } x 0 A A = (0, ) A [0, ] A 0, x 0 A A δ > 0 (x 0 δ, x 0 + δ) A x 0 A = (0, ) {} A x 0 A
f : R R f(x) = x + x f(x) = 5 ε > 0 δ > 0 x R 0 < x < δ f(x) 5 < ε x 4 < ε. x 4 = x x + = x x + 4 x ( x + 4) < δ(δ + 4), δ(δ + 4) ε δ + 4δ ε 0 4 + ε δ + 4 + ε. δ 0 < δ 4 + ε δ ε f : [, ] R { x + x 0, f(x) = 0 x = 0. x 0 f(x) = δ > 0 ε > 0 x 0 f(x) f(0) A A R f : A R x x 0 f(x) = l x x 0 f(x) = m l = m l m ε = δ > 0 l m x A 0 < x x 0 < δ f(x) l < ε = x x 0 f(x) = m δ > 0 x A 0 < x x 0 < δ f(x) m < ε = > 0 x x 0 f(x) = l l m, l m. δ = {δ, δ } x A 0 < x x 0 < δ 0 < x x 0 < δ 0 < x x 0 < δ l m = l f(x) + f(x) m f(x) l + f(x) m < l = m l m + l m = l m,
A A R f : A R f(x) + x x 0 E > 0 δ = δ(e) > 0 x A 0 < x x 0 < δ f(x) > E. f(x) + x x 0 x x 0 f(x) = + f(x) x x 0 E > 0 δ = δ(e) > 0 x A 0 < x x 0 < δ f(x) < E. f(x) x x 0 x x 0 f(x) = E E δ E δ x x 0 f(x) l ε f(x) + E A f : R {} R f(x) = f(x) = + (x ) x E > 0 δ > 0 x R {} x < δ f(x) > E (x ) > E x < δ 0 < x < δ (x ) > δ, δ E δ E 0 δ. E E
δ 0 < δ E δ ε g : R { } R g(x) = g(x) = (x + ) x δ > 0 E > 0 x x 0 f(x) f (x 0 δ, x 0 + δ) (x 0 δ, x 0 ) (x 0, x 0 + δ) f(x) f(x) f : (0, ) R x x 0 A A R f : A R f(x) l f l x x 0 ε > 0 δ = δ(ε) > 0 x A x 0 δ < x < x 0 f(x) l < ε. f(x) l x x 0 x x 0 f(x) = l f(x) l f l x x 0 ε > 0 δ = δ(ε) > 0 x A x 0 < x < x 0 + δ f(x) l < ε. f(x) l x x + 0 f(x) = l x x + 0 + f(x) = +, x x 0 f(x) = x x + 0 f(x) = +, x x + 0 x x 0 f(x) = x 0 A A R f : A R f(x) x x 0 x x 0 f(x) f(x) x x + 0
f : R R { x < 0, f(x) = x 0. f(x) = x 0 ε > 0 δ > 0 x R δ < x < 0 f(x) ( ) < ε ( ) < ε, δ f(x) = x 0 + f(x) x + A A R f : A R f(x) l f l x + ε > 0 M = M(ε) > 0 f(x) l x + x A x > M f(x) l < ε. f(x) = l x + f(x) l f l x ε > 0 M = M(ε) > 0 f(x) l x x A x < M f(x) l < ε. f(x) = l x M > 0 x A x > M x < M M > 0 x A x > M x < M + A f(x) = x + +, f(x) = +, f(x) = f(x) = x x + x
x x 0 f(x) = l ε > 0 δ > 0 x A (x 0 δ, x 0 + δ) {x 0 } f(x) l < ε ε > 0 δ > 0 x A 0 < x x 0 < δ f(x) l ε. x x 0 f(x) = l x A x 0 f(x) l f : R R { x < 0, f(x) = x 0. x 0 f(x) = 0 ε 0 < ε < δ > 0 x R δ < x < 0 f(x) 0 = 0 = ε x R 0 < x < δ f(x) 0 = 0 = ε f(x) = f(x) = x 0 x 0 A A R f : A R x x 0 f(x) = l (x n ) A x n x 0 x n x 0 f(x n ) l n x x 0 f(x) = l ε > 0 δ > 0 x A 0 < x x 0 < δ f(x) l < ε. δ > 0 x n x 0 x n x 0 n N n n > N 0 < x n x 0 < δ f(x n ) l < ε,
ε > 0 N n n > N f(x n ) l < ε n f(x n) = l (x n ) A x n x 0 x n x 0 n (f(x n )) l x x 0 f(x) = l ε > 0 δ > 0 x A 0 < x x 0 < δ f(x) l ε. δ > 0 δ = n x n A 0 < x n x 0 < n f(x n) l ε. > 0, n N 0 < x n x 0 < n x n x 0 x n x 0 f(x n ) l ε (f(x n )) l n f(x) = l f(x) = l x x 0 x x 0 (x n ) A x n x 0 x n x 0 (f(x n )) l n A A R f, g : A R f(x) = l g(x) = m x x 0 x x 0 x x 0 (f(x) + g(x)) = l + m x x 0 f(x)g(x) = lm g(x) 0 x A m 0 x x 0 f(x) g(x) = l m
A A R f : A R f x 0 A f(x) f(x) = f(x 0 ) x x 0 x x 0 ε > 0 δ = δ(ε) > 0 x 0 f x A x x 0 < δ f(x) f(x 0 ) < ε. A A R f : A R f A x 0 A f x 0 x x 0 f(x) f(x 0 ) x x 0 f(x) x 0 A A f A f x 0 A δ > 0 (x 0 δ, x 0 +δ) A x 0 x A x x 0 < δ x = x 0 ε > 0 δ > 0 x A x x 0 < δ x = x 0 f(x) f(x 0 ) = f(x 0 ) f(x 0 ) = 0 < ε,
x x 0 f(x) = f(x 0 ) f x 0 f(x) = f(x 0 ) x x 0 x x 0 f(x) x x + 0 f(x) f(x) = f(x 0 ) = f(x) f x x + 0 x 0 x x 0 f(x) = l = f(x) l f(x 0 ) x x 0 x x + 0 f(x 0 ) = l x x 0 f(x) = l l + = f(x) x x + 0 x x 0 f(x) f(x) x x + 0 f : A R x 0 A ε > 0 δ > 0 x A x x 0 < δ f(x) f(x 0 ) ε. f : R R f(x) = c c R f x 0 R x 0 R x 0 ε > 0 δ > 0 x R x x 0 < δ f(x) f(x 0 ) = c c = 0 < ε, δ > 0 f : R R f(x) = x f x 0 R x 0 R x 0 ε > 0 δ > 0 δ = ε x R x x 0 < δ f(x) f(x 0 ) = x x 0 < ε,
f : R R f(x) = x 3x + f x 0 R x 0 R x 0 ε > 0 δ > 0 x R x x 0 < δ f(x) f(x 0 ) = x 3x x 0 + 3x 0 < ε. x 3x x 0 + 3x 0 = (x x 0 )(x + x 0 ) 3(x x 0 ) = x x 0 x + x 0 3 = x x 0 x x 0 + x 0 3 x x 0 ( x x 0 + x 0 3 ) < δ(δ + x 0 3 ), δ(δ + x 0 3 ) ε δ + x 0 3 δ ε 0 x 0 3 x 0 3 + 4ε δ x 0 3 + x 0 3 + 4ε. x0 3 δ 0 < δ + 4ε x 0 3 δ ε A A R f : A R f x 0 A (x n ) A x n x 0 f(x n ) f(x 0 ) n A A R f, g : A R f, g x 0 A f + g x 0 f g x 0
g(x) 0 x A f g x 0 A, B A, B R f : A R, g : B R f(a) B f x 0 A g f(x 0 ) B g f : A R x 0 f x 0 A (x n ) A x n x 0 f(x n ) f(x 0 ) n g f(x 0 ) B (f(x n )) B f(x n ) f(x 0 ) g(f(x n )) g(f(x 0 )) (g f)(x n ) (g f)(x 0 ) n (x n ) A x n x 0 (g f)(x n ) (g f)(x 0 ) n g f x 0 I I R f : I R I f I I I R f : I R I f : f(i) R f(i) n N f : R R f(x) = x n R
n R n = f(x) = x f(x) = x n R f(x) = x n+ = x n x R p : R R R p p(x) = n x n + n x n + + x + 0, n N {0} 0,,..., n R n 0 p R f : A R A f f(x) = p(x) q(x), p, q q(x) 0 f A, : R [, ] R : A R : B R A B x 0 R ε > 0 δ > 0 x R x x 0 < δ x x 0 < ε. x x 0 = x x 0 x + x 0 = x x 0 x + x 0 x x 0 = x x 0 < δ, δ = ε R
x x 0 = x 0 x x = x x x + x 0 x A = R {kπ + π/ : k Z} x = x x B = R {kπ : k Z}, x A B : [, ] [ π/, π/] [, ] : [, ] [0, π] [, ] : R ( π/, π/) R : R (0, π) R : [ π/, π/] [, ], : [0, π] [, ], : ( π/, π/) R : (0, π) R e x : R (0, + ) R : (0, + ) R (0, + ) > 0 x : R (0, + ) R x 0 R e x x 0 ε > 0 δ > 0 x R x x 0 < δ e x e x 0 < ε, x = x 0 + h h < δ e x 0+h e x 0 < ε.
e x 0+h e x 0 = e x 0 e h e x 0 = e x 0 e h = e x 0 + h + h! + h3 3! + + hn n! + ) e x 0 ( h + h + h 3 + + h n +! 3! n! ( = e x 0 h + h )! + h + + h n + h n 3! n! (n + )! + ( e x 0 h + h ) + h + + h n + n = e x h 0 h < δ ex 0 δ, = e x 0 h h h < δ < δ e x δ 0 δ ε δ ε, e x 0 + ε δ < e x 0+h e x 0 < ε, e x e x R x x = e x f(x) = x e x R x R
f : [, b] R [, b] f m, M R m f(x) M x [, b]. f [, b] x [, b] I I n = [, b ] = [, b] = [ n, b n ] f x [ n, b n ] f x I I + b = + b I [ f x, ] + b I = [, b ] = [, ] [ + b f x, ] + b [ ] + b M f x, b [ + b I = [, b ] =, b ] I = [, b ] = b b = b b = b = b f x I + b I I n = [ n, b n ] = n b n b b = b n N
b n n = b n N n f x I n n N I n+ I n I n c I n c d c d c d I n n c b n n d b n n N 0 d c b n n = b n b 0 n 0 d c 0 n d = c c I = [, b] c = c (, b) c = b c (, b) f c ε > 0 δ > 0 x [, b] x c < δ f(x) f(c) < ε x (c δ, c + δ) f(x) < f(c) + ε f x (c δ, c + δ) n N I n (c δ, c + δ) f x I n c = c = b f : [, b] R [, b] f x, x [, b] f(x ) f(x) f(x ) x [, b]. f f x [, b] f([, b]) = s x [, b] s = f(x ) f(x) < s x [, b] g : [, b] R g(x) = s f(x). g [, b] g x [, b] M > 0 g(x) M x [, b] n N x n [, b] s n < f(x n) < s,
n < s f(x n ) = g(x n) M n N, N M f x f(x ) f(x) x [, b] [, b] f : [, b] R f() < 0 f(b) > 0 (, b) ρ (, b) f(ρ) = 0 f : [, b] R [, b] f() < 0 f(b) > 0 ρ (, b) f(ρ) = 0 I I n = [ n, b n ] f( n ) < 0 f(b n ) > 0 I = [, b ] = [, b] f( ) = f() < 0 f(b ) = f(b) > 0 I + b = + b ( ) + b I f = 0 ρ = ( ) + b + b f > 0 [ I = [, b ] =, ] ( ) + b + b f < 0 [ + b I = [, b ] =, b ] I = [, b ] = b b = b b = b f( ) < 0 < f(b ) = b + b I I n = [ n, b n ] = n b n b b = b n N b n n = b n N n
f( n ) < 0 < f(b n ) n N ( n ) (b n ) b n n = b 0 n n n = b n = ρ n n ρ [, b] f ρ f(ρ) = n f( n) 0 n f(b n) = f(ρ), f(ρ) = 0 f() > 0 f(b) < 0 f : [, b] R [, b] f() < ξ < f(b) c (, b) f(c) = ξ g : [, b] R g(x) = f(x) ξ f [, b] g [, b] g() = f() ξ < 0 g(b) = f(b) ξ > 0 c (, b) g(c) = 0 f(c) = ξ f(b) < ξ < f() f : [, b] [, b] [, b] c [, b] f(c) = c f() = f(b) = b f [, b] < f() f(b) < b g : [, b] R g(x) = x f(x) f [, b] g [, b] g() = f() < 0 g(b) = b f(b) > 0 c (, b) g(c) = 0 f(c) = c
f(x) = x 7 f(x) = x x + f(x) = x x + f(x) = x + x f(x) = x + x + x x =. x 0 x x x x + x + x + x x 3x + x 3 3x + x 3 8 x x x( x + x) x + x + x 0 x x x x + x x + x x 0 x x 0 x x x x + x x + x x + x f : R R f(0) = δ > 0 x ( δ, δ) 6 7 < f(x) < 8 7
x x 0, f : R R f(x) = x 0 x = 0. f : R R f(x) = x x 0, x 0 x = 0. b R x + x, f : R R f(x) = x b x (, 3), x + x 3. 3x + 6 x <, f : R R f(x) = x b x [, 3], x b x > 3. ( ) f : A R 0 A f = ( ) n n N f n f : A R 0 f(x ) f(x ) x x x, x A f A f : R R f(x) x x R f f : R R R R n+ = f( n ), n =,,... n n = f() = f : R R R f(0) = f() x 0 [0, ] f(x 0 ) = 0 3x x = 0 (0, ) α, β, γ > 0 λ < µ < ν α x λ + β x µ + γ x ν = 0 (λ, µ) (µ, ν)
f : [0, ] R [0, ] f(0) = f() x 0 [0, ] f(x 0 + ) = f(x 0 ) f : (, b) R (, b) f (, b) f : [, b] R [, b] f(x) > 0 x [, b] m > 0 f(x) m x [, b] f, g : [, b] R [, b] f(x) > g(x) x [, b] m > 0 f(x) > g(x) + m x [, b] [0, π] = n+ = ( n ), n =,,... n n = 0 f : [, b] R [, b] x, x,..., x n [, b] x 0 [, b] f(x 0 ) = f(x ) + f(x ) + + f(x n ). n
A A R f : A R x 0 A f x 0 f(x 0 + h) f(x 0 ) h 0 h f (x 0 ) f x 0 x 0 + h = x f(x) f(x 0 ). x x 0 x x 0 f x 0 f (x 0 ) A A R f : A R f A x 0 A f : R R f(x) = c c R f x 0 R f (x 0 ) = 0 f(x 0 + h) f(x 0 ) h 0 h = h 0 c c h 0 = h 0 h = 0 = 0. h 0 f : R R f(x) = x f x 0 R f (x 0 ) =
f(x 0 + h) f(x 0 ) h 0 h = h 0 x 0 + h x 0 h h = h 0 h = =. h 0 f : R R f(x) = x 3x + f x 0 R f (x 0 ) = x 0 3 f(x 0 + h) f(x 0 ) h 0 h = h 0 (x 0 + h) 3(x 0 + h) + x 0 + 3x 0 h = h 0 x 0 + x 0 h + h 3x 0 3h x 0 + 3x 0 h = h 0 x 0 h + h 3h h = h 0 (x 0 + h 3) = x 0 3. f : R R f(x) = x f(x) = { x x 0, x x < 0, f (x 0 ) = x 0 > 0 f (x 0 ) = x 0 < 0 x 0 = 0 f(0 + h) f(0) h 0 + h f(0 + h) f(0) h 0 h f(0 + h) f(0) h 0 h h = h 0 + h = h h 0 + h = = h 0 + h = h 0 h = h h 0 h = h 0 ( ) =, f x 0 = 0 f : R R { x x 0, f(x) = x x < 0. f (x 0 ) = x 0 > 0 f (x 0 ) = x 0 x 0 < 0 x 0 = 0 f(0 + h) f(0) h 0 + h f(0 + h) f(0) h 0 h = h 0 + h 0 h = h 0 h 0 h h = h 0 + h = = h 0 + = h 0 h = 0,
f(0 + h) f(0) f x 0 = 0 h 0 h : R [, ] x 0 R (x 0 ) = x 0 (x 0 + h) x 0 h 0 h = h h 0 x 0 x x h h = h 0 h ( x 0 + h ) ( x 0 + h ) = x 0 = x 0, h 0 = A A R f : A R f x 0 A f x 0 f x 0 f(x) f(x 0 ) = f (x 0 ). x x 0 x x 0 x x 0 (f(x) f(x 0 )) = x x 0 f(x) f(x 0 ) x x 0 (x x 0 ) f(x) f(x 0 ) = (x x 0 ) = f (x 0 ) 0 = 0, x x 0 x x 0 x x 0 x x 0 f(x) = f(x 0 ), f x 0 f x 0 f x 0 f : R R f(x) = x A A R f : A R f x 0 A f x 0
f : R R { x 0, f(x) = x < 0, f A A R f, g : A R f, g x 0 A f + g x 0 (f + g) (x 0 ) = f (x 0 ) + g (x 0 ) R f x 0 ( f) (x 0 ) = f (x 0 ) f g x 0 (f g) (x 0 ) = f (x 0 )g(x 0 ) + f(x 0 )g (x 0 ) g(x) 0 x A f g x 0 ( ) f (x 0 ) = f (x 0 )g(x 0 ) f(x 0 )g (x 0 ). g (g(x 0 )) f, g x 0 f(x 0 + h) f(x 0 ) h 0 h = f (x 0 ) g(x 0 + h) g(x 0 ) = g (x 0 ). h 0 h (f + g)(x 0 + h) (f + g)(x 0 ) h 0 h = h 0 f(x 0 + h) f(x 0 ) + g(x 0 + h) g(x 0 ) h f(x 0 + h) f(x 0 ) g(x 0 + h) g(x 0 ) = + h 0 h h 0 h = h 0 f(x 0 + h) + g(x 0 + h) (f(x 0 ) + g(x 0 )) h = f (x 0 ) + g (x 0 ),
( f)(x 0 + h) ( f)(x 0 ) h 0 h = h 0 f(x 0 + h) f(x 0 ) h (f g)(x 0 + h) (f g)(x 0 ) h 0 h = h 0 f(x 0 + h) f(x 0 ) h = f (x 0 ) ( ), = h 0 f(x 0 + h)g(x 0 + h) f(x 0 )g(x 0 ) h = h 0 f(x 0 + h)g(x 0 + h) f(x 0 )g(x 0 + h) + f(x 0 )g(x 0 + h) f(x 0 )g(x 0 ) h = h 0 (f(x 0 + h) f(x 0 ))g(x 0 + h) + f(x 0 )(g(x 0 + h) g(x 0 )) h f(x 0 + h) f(x 0 ) = h 0 h g(x g(x 0 + h) g(x 0 ) 0 + h) + f(x 0 ) h 0 h 0 h = f (x 0 )g(x 0 ) + f(x 0 )g (x 0 ), h 0 g(x 0 + h) = g(x 0 ) g x 0 ( h 0 h g(x 0 + h) ) g(x 0 ) g(x 0 + h) g(x 0 ) = h 0 h g(x 0 )g(x 0 + h) = g(x 0 ) g(x 0 + h) g(x 0 ) h 0 h g(x 0 + h) = g (x 0 ) (g(x 0 )), h 0 ( ) (x 0 ) = g (x 0 ) g (g(x 0 )), h 0 g(x 0 + h) = g(x 0 ) g x 0 /g
A, B A, B R f : A R, g : B R f(a) B f x 0 A g f(x 0 ) B g f : A R x 0 (g f) (x 0 ) = g (f(x 0 ))f (x 0 ) (g f)(x) (g f)(x 0 ) x x 0 = g(f(x)) g(f(x 0)) f(x) f(x 0 ) = g(f(x)) g(f(x 0)) x x 0 f(x) f(x 0 ) f(x) f(x 0 ) x x 0, (g f)(x) (g f)(x 0 ) x x 0 x x 0 g(f(x)) g(f(x 0 )) x x 0 f(x) f(x 0 ) f(x) f(x 0 ) x x 0 x x 0 g(y) g(f(x 0 )) y f(x 0 ), y f(x ψ(y) = 0 ) g (f(x 0 )) y = f(x 0 ). g(f(x)) g(f(x 0 )) x x 0 = ψ(f(x)) f(x) f(x 0) x x 0, x A {x 0 } x f(x) = f(x 0 ) f(x) f(x 0 ) f (x 0 ) x x 0 x x 0 f x 0 f x 0 g f(x 0 ) ψ f(x 0 ) ψ f x 0 x x 0 ψ(f(x)) ψ(f(x 0 )) = g (f(x 0 )) g(f(x)) g(f(x 0 )) x x 0 x x 0 = x x 0 ψ(f(x)) x x 0 f(x) f(x 0 ) x x 0 = g (f(x 0 ))f (x 0 ), f : R R f(x) = x n, n N R f (x) = nx n
n n = f(x) = x R f (x) = f(x) = x n R f (x) = nx n f(x) = x n+ = x n x = g(x)h(x) f (x) = g (x)h(x) + g(x)h (x) = nx n x + x n = (n + )x n, f : R R f(x) = x n, n N R f (x) = nx n f(x) = x n = x = n g(x) f (x) = g (x) (g(x)) = nxn (x n ) = nxn x n = nx n. p : R R p(x) = n x n + n x n + + x + 0 n N {0} 0,,..., n R n 0 p R p (x) = n n x n + (n ) n x n + + f : A R f(x) = p(x) p, q q(x) q(x) 0 f A f (x) = p (x)q(x) p(x)q (x) (q(x)) : R [, ] R (x) = x : R [, ] R (x) = x : A R x = x x x A = R {kπ + π/ : k Z}, A (x) = x x x( x) x = x + x x = x.
: B R B = R {kπ : k Z} (x) = x. : [, ] [ π/, π/] (y) = x x = y x [ π/, π/] [ π/, π/] x = x 0 x ( π/, π/) (, ) (y) = x = x. x = y x ( π/, π/) x + x = x = x = y (y) =, y (, ). y y x (x) =, x (, ). x : [, ] [0, π] (y) = x x = y x [0, π] [0, π] x = x 0 x (0, π) (, ) (y) = x = x. x = y x (0, π) x + x = x = x = y (y) =, y (, ). y y x (x) =, x (, ). x
: R ( π/, π/) (y) = x x = y x ( π/, π/) ( π/, π/) x = 0 x ( π/, π/) x R (y) = x = x. x = y x ( π/, π/) x + x = x = + y y x (y) = + y, y R. (x) = + x, x R. : R (0, π) (y) = x x = y x (0, π) (0, π) x = 0 x (0, π) R x (y) = x = x. x = y x (0, π) x + x = x = + y y x (y) = + y, y R. (x) = + x, x R. e x : R (0, + ) R (e x ) = e x e x R : (0, ) R e x y = x e x = y e x R (e x ) = e x 0 x R (0, ) (y) = e =, y (0, ). x y
y x (x) =, x (0, ). x x x = e x f(x) = x e x R x R f : R R f(x) = (x ) g(x) = x h(x) = x R f = h g R f (x) = h (g(x))g (x) = (x ) (x) = x (x ) f : R R f(x) = ( (x )) g(x) = x h (x) = x h (x) = x R f = h (h g) R f (x) = h ((h g)(x))(h g) (x) = h (h (g(x)))h (g(x))g (x) = ( (x )) ( (x )) (x) = x (x ) ( (x )) A A R f : A R x A f : A R x f (x) f f A f f : A R x f (x) = (f ) (x) f n f (n) : A R, n N, n > f A f (n) f (n+) : A R x f (n+) (x) = (f (n) ) (x) (n + ) f n n f : A R x 0 A f (n) (x 0 ) n N
p : R R p(x) = n x n + n x n + + x+ 0 n N {0} 0,,..., n R n 0 x R p (m) (x) = 0 m n + f : (, b) R f x 0 (, b) x 0 f (x 0 ) = 0 f x 0 h > 0 h < 0 f(x 0 + h) f(x 0 ) h f(x 0 + h) f(x 0 ) h f x 0 f (x 0 ) = 0 0,, f +(x 0 ) = h 0 + f(x 0 + h) f(x 0 ) h 0,, f (x 0 ) = h 0 f(x 0 + h) f(x 0 ) h 0 f (x 0 ) = f (x 0 ) = f +(x 0 ) 0, f x 0 f : (, b) R x 0 (, b) f x 0 δ > 0 x (, b) x x 0 < δ f(x) f(x 0 ). 0, 0. f x 0 δ > 0 x (, b) x x 0 < δ f(x) f(x 0 ).
f x 0 f x 0 f : (, b) R f x 0 (, b) x 0 f (x 0 ) = 0 h < δ f : R R f(x) = x 3 f (0) = 0 f I I R f : I R x 0 I f f (x 0 ) = 0 f : [, b] R f [, b] x 0 [, b] f(x 0 ) x 0 = x 0 = b x 0 (, b) f (x 0 ) = 0 x 0 (, b) f x 0 x 0 f : R R f(x) = x 3 3x + x [0, ] f (x) = 3x 3 x = x = f x = 0, x = x = f f(0) =, f() = f() = 3,
f [0, ] f() = 3 f() = f : [, b] R f [, b] (, b) f() = f(b) x 0 (, b) f (x 0 ) = 0 f [, b] f [, b] (, b) b f x 0 f (x 0 ) = 0 f x 0 f (x 0 ) = 0 (, b) (, b) f b f() = f(b) f f(x) = f() = f(b) x [, b] f (x) = 0 x [, b] x 0 (, b) f (x 0, f(x 0 )) X OX f : [, b] R f [, b] (, b) x 0 (, b) f (x 0 ) = f(b) f(). b g : [, b] R g(x) = f(x) f(b) f() (x ). b g [, b] (, b) g() = f() f(b) f() ( ) = f(), b f(b) f() g(b) = f(b) (b ) = f(), b g() = g(b) x 0 (, b) g (x 0 ) = 0 f f(b) f() (x 0 ) = b
L (, f()) (b, f(b)) f(b) f() b x 0 (, b) f f(b) f() (x 0 ) = f (x 0 ) b T f (x 0, f(x 0 )) T L f : (, b) R f (x) = 0 x (, b) f (, b) x, x (, b) x x x < x f [x, x ] (x, x ) x 0 (x, x ) f (x 0 ) = f(x ) f(x ) x x. f (x 0 ) = 0 f(x ) = f(x ) f (, b) f (, b) f : R R f(x ) f(x ) (x x ) x, x R f f(x + h) f(x) h (x + h x) h = h h = h, f(x + h) f(x) = 0 h 0 h f (x) = 0 x R f R f, g : (, b) R f (x) = g (x) x (, b) f = g + c c R f g : (, b) R (f g) (x) = f (x) g (x) = 0 x (, b) f g = c f = g +c c R
f : (, b) R (, b) f (x) 0 x (, b) f (, b) f (x) > 0 x (, b) f (, b) f (x) 0 x (, b) f (, b) f (x) < 0 x (, b) f (, b) < x < x < b f [x, x ] (x, x ) x 0 (x, x ) f (x 0 ) = f(x ) f(x ) x x f(x ) f(x ) = f (x 0 )(x x ). f (x 0 ) 0 x x > 0 f(x ) f(x ) 0 f(x ) f(x ) x, x (, b) x < x f(x ) f(x ) f (, b) f : (, b) R (, b) x 0 (, b) f (x 0 ) = 0 f (x 0 ) f (x 0 ) > 0 f x 0 f (x 0 ) < 0 f x 0 f (x 0 ) = 0 δ > 0 0 < f (x 0 ) = x x 0 f (x) f (x 0 ) x x 0 f (x) =. x x 0 x x 0 x (, b) x 0 < x < x 0 + δ f (x) > 0 x (, b) x 0 δ < x < x 0 f (x) < 0 x 0 < x < x 0 + δ f [x 0, x] (x 0, x) x (x 0, x) f(x) f(x 0 ) = f (x )(x x 0 ) > 0 f(x 0 ) < f(x),
x 0 δ < x < x 0 f [x, x 0 ] (x, x 0 ) x (x, x 0 ) f(x 0 ) f(x) = f (x )(x 0 x) < 0 f(x 0 ) < f(x), f(x 0 ) < f(x) x (x 0 δ, x 0 + δ) f x 0 f (x 0 ) f (x 0 ) = 0 x 0 f : (, b) R (, b) x, x (, b) (x, f(x )) (x, f(x )) f f (x, f(x )) (x, f(x )) f (, b) x 0 (, b) f (x 0, f(x 0 )) f f f (x 0, f(x 0 )) (x 0, f(x 0 )) f (, b) x 0 (, b) f (x 0, f(x 0 )) y = f(x 0 ) + f (x 0 )(x x 0 ). f (, b) x 0 (, b) f(x) > f(x 0 ) + f (x 0 )(x x 0 ) x (, b) x x 0 f (, b) x 0 (, b) f(x) < f(x 0 ) + f (x 0 )(x x 0 ) x (, b) x x 0 f (, b) x 0 (, b) x (, b) x 0 < x f
[x 0, x] (x 0, x) x (x 0, x) f(x) f(x 0 ) = f (x )(x x 0 ) > f (x 0 )(x x 0 ) f(x) > f(x 0 ) + f (x 0 )(x x 0 ), f (, b) x (, b) > x 0 f f(x) < f(x 0 ) + f (x 0 )(x x 0 ). f (, b) f (, b) f (, b) f (, b) f (, b) f : (, b) R (, b) f (x) > 0 x (, b) f (, b) f (x) < 0 x (, b) f (, b) x 0 (, b) f f f f : R R f(x) = x + f R f (x) = x (x + ). f (, 0) (0, ) f(0) = R f (x) > 0 x (, 3 f(x) = 0 f(x) = 0 f x + x f (x) = (3x ) (x + ) 3. (, ) 3 3 ) f ( 3, ( ) 3, x (, ) 3 f (x) < 0 x ( ) 3, 3 ) f
f(x) x x 0 g(x) f, g x 0 g(x) 0 x x 0 x x 0 f(x) = g(x) = 0 x x 0 x x 0 f(x) = g(x) = ±. x x 0 x x 0 0 + 0 + f, g f, g : (, x 0 ) (x 0, b) R (, x 0 ) (x 0, b) g(x) 0 g (x) 0 x (, x 0 ) (x 0, b) x x 0 f(x) = f (x) g(x) = 0 f(x) = g(x) = ± x x 0 x x 0 x x 0 x x 0 g (x) f(x) x x 0 g(x) = f (x) x x 0 g (x). x x 0 f(x) g(x) f (x) f(x) = g(x) = 0 x x + 0 x x + 0 x x + g (x) 0 x x 0 f (x) f(x) = g(x) = 0 x x 0 x x 0 x x 0 g (x) = l x x + 0 f(x) g(x) = l x x+ 0 = + x x 0 f(x) g(x) = + f(x) = g(x) = 0 f (x) = l x + x + x + g (x) f(x) x + g(x) = l x + x f(x) = x + f(x) = x + g(x) = 0 x + g(x) = + x + f (x) x + g (x) f (x) x + g (x) = + = l f(x) x + g(x) = + f(x) x + g(x) = l
f(x) = { x + x +, x, 4x, x < x = x = f x = R f(x) = (x 4 + x + ) 4 ( ) x f(x) = x + f(x) = (x + x + ) 3 f(x) = (x) (x ) f(x) = ( (x )) f(x) = (x + x + ) ( f(x) = x + ) x + ( ) f(x) = x + u (x) u(y) = y y + y(x) = (x + ) n n f(x) = x m f(x) = x + R R ( ) x 0, f(x) = x 0 x = 0. ( ) x x 0, f(x) = x 0 x = 0.
( ) x x 0, f(x) = x 0 x = 0. x x 0, f(x) = x x = 0. ( f ): R R f f(0) = f (0) = 0 nf = 0 n n f : [0, ] R [0, ] f(0) = f() = f() = 0 x 0 (0, ) f (x 0 ) = 0 f : R R f δ > 0 f ( δ, δ) f, g : R R x 0 R f(x 0 ) = 0 f x 0 g x 0 f g x 0 f, g : [, b] R [, b] f() = g() f(b) = g(b) x 0 (, b) f g (x 0, f(x 0 )) (x 0, g(x 0 )) f(x) = x 3 x + [ 3, 3] f(x) = x 3 + x + [, ] f(x) = x 5 5x 3 0x [, ] f(x) = 4x 3 + 3x + = 0 f(x) = 4x 4 x + = 0 f(x) = x 3 3x x + = 0 R
f(x) = x 3 3x + f(x) = x + f(x) = x x + f(x) = x x f(x) = x + x p : R R p(x) = (x ρ )(x ρ ) (x ρ n ) n N, ρ, ρ,..., ρ n R ρ < ρ < < ρ n p (x) = 0 n n x n + n x n + + x = 0 n N,,..., n R n 0 ρ n n x n + (n ) n x n + + = 0 r 0 < r < ρ p : R R p(x) = x n+ + x + b n N,, b R > 0 p(x) = 0 f : (0, ) R f (0, ) x 0 = f : R R f( ) = 0, f() = f () > 0 x x 0 x x x 0 x + x 3x 5 x ± x x + (x ) x 0 x x x 0 x x x 3 4x 3 x 3 x π/ x + (x)
f : [, b] R x 0 [, b] f n x 0 f (x 0 ), f (x 0 ),..., f (n) (x 0 ) T n (f, x 0 ; x) = f(x 0 ) + f (x 0 )(x x 0 ) + f (x 0 ) (x x 0 ) + + f (n) (x 0 ) (x x 0 ) n n! n f (k) (x 0 ) = (x x 0 ) k k! k=0 n f x 0 x 0 = 0 f n f(x) = 0 + (x x 0 ) + + n (x x 0 ) n, k = f (k) (x 0 ), k = 0,,..., n, k! T n (f, x 0 ; x) = f(x) n f x 0 f
f : [, b] R n + [, b] x 0 [, b] T n (f, x 0 ; x) n f x 0 R n (f, x 0 ; x) = f(x) T n (f, x 0 ; x) n f x 0 R n (f, x 0 ; x) = f (n+) (ξ) (x ξ) n (x x 0 ) ξ x 0 x n! R n (f, x 0 ; x) = f (n+) (ξ) (n + )! (x x 0) n+ ξ x 0 x x [, b] ϕ : [, b] R ϕ(t) = R n (f, t; x) = f(x) T n (f, t; x) = f(x) ϕ(t) t [, b] ϕ (t) = f (t) k= n k=0 f (k) (t) (x t) k. k! n [ f (k+) (t) (x t) k f ] (k) (t) (x t)k k! (k )! = f (t) f (t)(x t) + f (t) f (t) (x t) + f (t)(x t)! f (n+) (t) (x t) n + f (n) (t) (x t)n n! (n )! = f (n+) (t) (x t) n. n! ϕ x x 0 ξ x x 0 ϕ(x 0 ) ϕ(x) = ϕ (ξ)(x 0 x). ϕ(x 0 ) = R n (f, x 0 ; x), ϕ(x) = R n (f, x; x) = 0 ( ), ϕ (ξ) = f (n+) (ξ) (x ξ) n, n!
R n (f, x 0 ; x) = f (n+) (ξ) (x ξ) n (x x 0 ). n! f(x) = T n (f, x 0 ; x) + R n (f, x 0 ; x). f n = 0,,,... f(x) = n T n(f, x 0 ; x) = n R n(f, x 0 ; x) = 0, n n k=0 f (k) (x 0 ) (x x 0 ) k = k! n=0 f (n) (x 0 ) (x x 0 ) n n! f(x) = x f (x) = x, f (x) = x, f (x) = x, f (4) (x) = x, f (n) (x) = ( ) n x f (n+) (x) = ( ) n x x R n = 0,,,... f (n) (0) = 0 f (n+) (0) = ( ) n n = 0,,,... T n+ (f, 0; x) = x x3 3! + x5 5! + + ( )n x n+ (n + )! = n k=0 ( ) k x k+. (k + )! n + x x 0 ξ 0 x R n+ (f, 0; x) = ( )n+ ξ x n+, (n + )! R n+ (f, 0; x) = ξ (n + )! x n+ x n+ (n + )!.
n = n+ n = x n+4 (n + 4)! x n+ (n + )! = x (n + 3)(n + 4) R n+(f, 0; x) = 0, n x = n T n+(f, 0; x) = f(x) = x n=0 0 n, ( ) n x n+. (n + )! f (x) = x, f (x) = x, f (x) = x, f (4) (x) = x, x n+ (n + )! f (n) (x) = ( ) n x f (n+) (x) = ( ) n+ x x R n = 0,,,... f (n) (0) = ( ) n f (n+) (0) = 0 n = 0,,,... T n (f, 0; x) = x! + x4 4! + + ( )n x n (n)! = n ( ) k x k. (k)! k=0 n x x 0 ξ 0 x R n (f, 0; x) = ( )n+ ξ x n+, (n + )! R n (f, 0; x) = ξ (n + )! x n+ x n+ (n + )!. n = n+ n = x n+3 (n + 3)! x n+ (n + )! = x (n + )(n + 3) 0 n, x n+ (n + )! R n(f, 0; x) = 0, n
x = n T n(f, 0; x) = ( ) n x n. (n)! f(x) = e x f (n) (x) = e x x R n = n=0 0,,,... f (n) (0) = n = 0,,,... T n (f, 0; x) = + x + x + + xn n! = n n e x k=0 x k k!. x 0 R n (f, 0; x) = e ξ (n + )! xn+, ξ 0 x x > 0 e x 0 < ξ < x R n (f, 0; x) = e ξ (n + )! xn+ < ex x n+ (n + )!. x < 0 x < ξ < 0 e ξ < e 0 = x 0 R n (f, 0; x) = e ξ (n + )! xn+ < xn+ (n + )! < e x x n+ (n + )!. R n (f, 0; x) e x x n+ (n + )!. n = e x x n+ (n + )! n+ n = e x x n+ (n + )! e x x n+ (n + )! = x n + R n(f, 0; x) = 0, n e x = n T n(f, 0; x) = 0 n, n=0 x n n!. f(x) = ( + x), x (, ] f (x) = + x, f (x) = ( + x), f (x) = ( + x), f (4) 6 (x) = 3 ( + x), 4
f (n) (x) = ( )n (n )! x > n =,,... f(0) = 0 ( + x) n f (n) (0) = ( ) n (n )! n =,,... T n (f, 0; x) = x x + x3 3 + + ( )n x n n ( ) k x k =. n k n ( + x) < x x 0 R n (f, 0; x) = ξ 0 x k= ( ) n (n + )( + ξ) n+ xn+, R n(f, 0; x) = 0, n ( + x) = n T n(f, 0; x) = ( ) n x n. n n=0 x 5x + 8, x 3 + 7x +, x 4 x 5 3x 3 + x x + α 0 + α (x ) + α (x ) + + α n (x ) n n T n (f, 0; x) f(x) = + x. f(x) = + x. T 3 (f, 0; x), T 4 (f, 0; x) T 5 (f, 0; x) f(x) = x 4 + x 3 + x + x +,, e e 0 7 n f, g : (, b) R n x 0 (, b) f(x 0 ) = f (x 0 ) = = f (n ) (x 0 ) = 0, g(x 0 ) = g (x 0 ) = = g (n ) (x 0 ) = 0 g (n) (x 0 ) 0 f(x) x x 0 g(x) = f (n) (x 0 ) g (n) (x 0 ).
n f : (, b) R n x 0 (, b) f(x 0 ) = f (x 0 ) = = f (n ) (x 0 ) = 0 f (n) (x 0 ) 0 n f (n) (x 0 ) > 0 f x 0 n f (n) (x 0 ) < 0 f x 0
f : [, b] R [, b] f(x) 0 x [, b] f X OX x = x = b A s A S A s A A S. < b P [, b] x 0, x,..., x n [, b] = x 0 < x < < x n = b.
P [, b] b f : [, b] R [, b] P = {x 0, x,..., x n } [, b] [x i, x i ], i =,,..., n m i = {f(x) : x i x x i }, M i = {f(x) : x i x x i }. f P L(f, P ) L(f, P ) = n m i (x i x i ), i= f P U(f, P ) U(f, P ) = n M i (x i x i ). i= s S f f [, b] m i M i m i M i f P [, b] L(f, P ) U(f, P ), m i M i m i (x i x i ) M i (x i x i ), i =,,..., n. P Q [, b] Q P P Q L(f, P ) L(f, Q), U(f, P ) U(f, Q).
Q P P = {x 0, x,..., x k, x k,..., x n }, Q = {x 0, x,..., x k, u, x k,..., x n }, = x 0 < x < < x k < u < x k < < x n = b. m = {f(x) : x k x u}, m = {f(x) : u x x k }. L(f, P ) = n m i (x i x i ), i= k n L(f, Q) = m i (x i x i ) + m (u x k ) + m (x k u) + m i (x i x i ). i= i=k+ L(f, P ) L(f, Q) m k (x k x k ) m (u x k ) + m (x k u). {f(x) : x k x x k } {f(x) : x k x u} m k m. m k m. m (u x k ) + m (x k u) m k (u x k ) + m k (x k u) = m k (u x k + x k u) = m k (x k x k ). Q P L(f, P ) L(f, Q) U(f, P ) U(f, Q)
Q P P Q P = P, P,..., P m = Q, P j+ P j L(f, P ) = L(f, P ) L(f, P ) L(f, P m ) = L(f, Q), U(f, P ) = U(f, P ) U(f, P ) U(f, P m ) = U(f, Q). P P [, b] f : [, b] R [, b] L(f, P ) U(f, P ). P P P P = P P P P P L(f, P ) L(f, P ) U(f, P ) U(f, P ) L(f, P ) L(f, P ) U(f, P ) U(f, P ). U(f, P ) {L(f, P ) : P [, b]} U(f, P ) {L(f, P ) : P [, b]} U(f, P ) P [, b], {L(f, P )} {U(f, P ) : P [, b]} {L(f, P )} {U(f, P )}. P [, b] L(f, P ) {L(f, P )} U(f, P ) L(f, P ) {U(f, P )} U(f, P ), {L(f, P )} = {U(f, P )} {L(f, P )} < {U(f, P )} f
f : [, b] R [, b] f [, b] {L(f, P ) : P [, b]} = {U(f, P ) : P [, b]}. f b f b f(x)dx s b f L(f, P ) b f(x)dx U(f, P ) P [, b]. b f(x)dx f : [, b] R f(x) = c c R P = {x 0, x,..., x n } [, b] m i = M i = c, i =,,..., n, L(f, P ) = U(f, P ) = n n n m i (x i x i ) = c(x i x i ) = c (x i x i ) = c(b ) i= i= i= n n M i (x i x i ) = c(x i x i ) = c i= i= i= {L(f, P )} = {U(f, P )} = c(b ), b f(x)dx = b cdx = c(b ). n (x i x i ) = c(b ).
f : [, b] R [, b] f [, b] ε > 0 P [, b] U(f, P ) L(f, P ) < ε. f [, b] {L(f, P )} = {U(f, P )}. ε > 0 ε = ε/ > 0 P [, b] {L(f, P )} ε < L(f, P ) {L(f, P )} L(f, P ) < ε = ε/ P [, b] U(f, P ) < {U(f, P )} + ε U(f, P ) {U(f, P )} < ε = ε/. U(f, P ) {U(f, P )} + {L(f, P )} L(f, P ) < ε/ + ε/ U(f, P ) L(f, P ) < ε. P P P P = P P L(f, P ) L(f, P ) L(f, P ) L(f, P ) U(f, P ) U(f, P ). U(f, P ) L(f, P ) U(f, P ) L(f, P ) < ε,
ε > 0 P [, b] U(f, P ) L(f, P ) < ε. L(f, P ) {L(f, P )} {L(f, P )} L(f, P ) {U(f, P )} U(f, P ), ε > 0 {U(f, P )} {L(f, P )} U(f, P ) L(f, P ) < ε. {U(f, P )} {L(f, P )} < ε, {U(f, P )} = {L(f, P )} f [, b] f : [0, b] R f(x) = x P n = {x 0, x,..., x n } [, b] x 0 = 0, x = b n, x = b n,..., x i = [x i, x i ], i =,,..., n m i = x i, M i = x i. (i )b, x i = ib n n,..., x n = b, L(f, P n ) = n m i (x i x i ) = i= n x i (x i x i ) = i= n i= (i )b n b n = b n n i= (i ) = b n n i=0 i = b (n )n n = n b n
U(f, P n ) = n M i (x i x i ) = i= = b n n i= n x i (x i x i ) = i= i = b n(n + ) n U(f, P n ) L(f, P n ) = n + b n = n + b n. n n n i= b = b n = b n. n b n < ε nε > b, f [0, b] b f(x)dx 0 n b n = L(f, P n) {L(f, P )} = b 0 ib b n n f(x)dx = {U(f, P )} U(f, P n ) = n + b n n b n b 0 f(x)dx n + b n, n b n b n + n b 0 b b f(x)dx = b n, b f(x)dx = b. f(x) = x f(x) = x f : [0, ] R f(x) = { 0, x,, x =.
P = {x 0, x,..., x n } [, b] x j < < x j, < j < n m i = M i = 0 i =,,..., n, i j, m j = 0, M j =. j L(f, P ) = m i (x i x i ) + m j (x j x j ) + i= j U(f, P ) = M i (x i x i ) + M j (x j x j ) + i= n i=j+ n i=j+ U(f, P ) L(f, P ) = x j x j. m i (x i x i ) = 0 M i (x i x i ) = x j x j. ε > 0 U(f, P ) L(f, P ) < ε P x j < < x j, < j < n x j x j < ε f [0, ] f(x)dx 0 0 = L(f, P ) {L(f, P )} = f(x)dx = {U(f, P ) U(f, P ) = x 0 j x j < ε 0 0 f(x)dx < ε ε > 0 0 f(x)dx = 0 x = [0, ] f : [, b] R [, b] f [, b]
f [, b] f() f(x) f(b) x [, b], f [, b] f() = f(b) f f() < f(b) ε > 0 n + x i = + i b, i = 0,,..., n, n x i x i b x 0 = x n = n b P n = {x 0, x,..., x n } [x i, x i ], i =,,..., n m i = f(x i ), M i = f(x i ). L(f, P n ) = U(f, P n ) = n m i (x i x i ) = i= n i= n M i (x i x i ) = i= f(x i ) b n n i= f(x i ) b n = b n (f(x 0) + f(x ) + + f(x n )) = b n (f(x ) + f(x ) + + f(x n )). U(f, P n ) L(f, P n ) = b n (f(x n) f(x 0 )) = (b )(f(b) f()). n n (b )(f(b) f()) n < ε nε > (b )(f(b) f()), f [, b] f [, b] f : [, b] R [, b] f [, b]
f [, b] [, b] ε > 0 f [, b] [, b] ε = ε > 0 δ > 0 b x, x [, b] x x < δ f(x) f(x ) < ε = ε b. δ = δ(ε ) = δ(ε) ε x, x P = {x 0, x,..., x n } x i x i < δ, i =,,..., n f [x i, x i ], i =,,..., n u i, v i [x i, x i ] m i = f(v i ), M i = f(u i ). U(f, P ) L(f, P ) = < u i v i x i x i < δ, M i m i = f(u i ) f(v i ) < n M i (x i x i ) i= n i= ε b. n m i (x i x i ) = i= ε b (x i x i ) = ε b n i= n (M i m i )(x i x i ) i= (x i x i ) = ε (b ) = ε, b f [, b]
f : [, b] R [, b] c [, b] f [, b] f [, c] [c, b] b f(x)dx = c f(x)dx + b c f(x)dx. f, g : [, b] R [, b] f + g [, b] b (f(x) + g(x))dx = b f(x)dx + b g(x)dx. f : [, b] R [, b] c [, b] cf [, b] b cf(x)dx = c b f(x)dx. c > 0 P = {x 0, x,..., x n } [, b] [x i, x i ], i =,,..., n m i = {(cf)(x) : x i x x i }, M i = {(cf)(x) : x i x x i } m i = {f(x) : x i x x i }, M i = {f(x) : x i x x i }. m i = cm i M i = cm i, i =,,..., n, L(cf, P ) = cl(f, P ) U(cf, P ) = cu(f, P ), {L(cf, P )} = c {L(f, P )} {U(cf, P )} = c {U(f, P )}. f [, b] {L(f, P )} = b f(x)dx = {U(f, P )},
{L(cf, P )} = c {L(f, P )} = c b cf [, b] b cf(x)dx = c f(x)dx = c {U(f, P )} = {U(cf, P )}, b f(x)dx. c < 0 m i = cm i M i = cm i, i =,,..., n, c > 0 c = 0 b cf(x)dx = b 0 f(x)dx = 0 = 0 b f(x)dx = c b f(x)dx. f : [, b] R [, b] m f(x) M x [, b] m(b ) b f(x)dx M(b ). P = {x 0, x,..., x n } [, b] [x i, x i ], i =,,..., n m m i, M i M, m(b ) n m i (x i x i ) i= n M i (x i x i ) M(b ) i= m(b ) L(f, P ) U(f, P ) M(b ). f [, b] m(b ) {L(f, P )} = b f(x)dx = {U(f, P )} M(b ),
f : [, b] R [, b] f(x) 0 x [, b] b f(x)dx 0. f, g : [, b] R [, b] f(x) g(x) x [, b] b f(x)dx b g(x)dx. m = 0 f, g [, b] f g (f g)(x) 0 x [, b] b (f g)(x)dx 0 b (f(x) g(x))dx 0 b f(x)dx b b g(x)dx. f(x)dx b g(x)dx 0 f : [, b] [m, M] [, b] ϕ : [m, M] R [m, M] ϕ f : [, b] R [, b] f : [, b] R [, b] f [, b] b b f(x)dx f(x) dx. ϕ(x) = x f f(x) f(x) f(x) x [, b],
b ( f(x) )dx b f(x)dx b b f(x) dx b f(x) dx b f(x)dx f(x) dx. b f(x)dx b f(x) dx f, g : [, b] R [, b] f fg [, b] f [, b] ϕ(x) = x f fg fg = (f + g) (f g), 4 f : [, b] R [, b] F : [, b] R F (x) = x f(t)dt f f : [, b] R [, b] F f [, b] f [, b] f [, b] M > 0 f(x) M x [, b].
x 0 [, b] F x 0 ε > 0 δ > 0 F (x) F (x 0 ) = x x0 x x [, b] x x 0 < δ F (x) F (x 0 ) < ε. x f(t)dt x0 f(t)dt = f(t) dt, x 0 < x < x 0 + δ, x 0 f(t) dt, x 0 δ < x < x 0 x f(t)dt, x 0 < x < x 0 + δ, x 0 f(t)dt, x 0 δ < x < x 0 x x0 x x 0 Mdt, x 0 < x < x 0 + δ, x0 M(x x 0 ), x 0 < x < x 0 + δ, = = M x x 0 < Mδ, M(x 0 x), x 0 δ < x < x 0 x Mdt, x 0 δ < x < x 0 Mδ ε δ ε M F f F (x) F (x 0 ) M x x 0 M > 0. F M > 0 f : [, b] R [, b] M > 0 f(x ) f(x ) M x x x, x [, b]. f [, b] f [, b] f : [, b] R [, b] f x 0 [, b] F f x 0 F (x 0 ) = f(x 0 ).
f x 0 ε > 0 δ > 0 t [, b] t x 0 < δ f(t) f(x 0 ) < ε. F (x) F (x 0 ) x x 0 x x 0 F (x) F (x 0 ) f(x 0 ) = x x 0 = = ( ) F (x) F (x0 ) = f(x 0 ) f(x 0 ) = 0. x x 0 x x 0 x f(t)dt x 0 f(t)dt f(x 0 ) x x 0 x x 0 f(t)dt f(x 0 )(x x 0 ) x x 0, x 0 < x, x 0 f(t)dt + f(x x 0)(x 0 x), x < x 0 x x 0 x x 0 (f(t) f(x 0 ))dt x x 0, x 0 < x, x0 x (f(t) f(x 0))dt, x < x 0, x 0 x F (x) F (x 0 ) f(x 0 ) x x 0 = x = x x 0 f(t)dt x x 0 f(x 0 )dt, x x 0 x 0 < x, x0 x x 0 x 0)dt, x x 0 x < x 0 x 0 (f(t) f(x 0 ))dt x 0 x x x 0 f(t) f(x 0 ) dt x x 0, x 0 < x, x0 x f(t) f(x 0) dt, x < x 0. x 0 x x x 0, x 0 < x, (f(t) f(x 0))dt, x < x 0 x 0 x x (x 0 δ, x 0 + δ) {x 0 } x 0 t x x t x 0 t (x 0 δ, x 0 + δ) F (x) F (x 0 ) f(x 0 ) x x 0 < x x 0 εdt x x 0, x 0 < x, x0 x εdt x 0 x, x < x 0 = ε(x x 0 ) x x 0, x 0 < x, ε(x 0 x) x 0 x, x < x 0 = ε.
ε > 0 δ > 0 x [, b] 0 < x x 0 < δ F (x) F (x 0 ) f(x 0 ) x x 0 < ε, F x 0 F (x 0 ) = f(x 0 ) f : [, b] R [, b] F f [, b] F (x) = f(x) x [, b]. f : [, b] R [, b] ξ (, b) b f(x)dx = f(ξ)(b ). F (x) = x f(t)dt [, b] f [, b] F [, b] F () = 0 ξ (, b) b f(x)dx = F (b) F () = F (ξ)(b ) = f(ξ)(b ). f : [, b] R [, b] G : [, b] R G (x) = f(x) x [, b] b f(x)dx = G(b) G(). F f F (x) = f(x) = G (x) F (x) = G (x) x [, b], F (x) = G(x) + c c R.
0 = F () = G() + c c = G(), b f(x)dx = F (b) = G(b) + c = G(b) G(). G G (x) = f(x) x [, b] f F f f b f(x)dx f G : [, b] R [, b] G [, b] b G (x)dx = G(b) G(). P = {x 0, x,..., x n } [, b] [x i, x i ], i =,,..., n ξ i (x i, x i ) G(x i ) G(x i ) = G (ξ i )(x i x i ). m i = {G (x) : x i x x i }, M i = {G (x) : x i x x i }, m i G (ξ i ) M i, L(G, P ) = n m i (x i x i ) i= n G (ξ i )(x i x i ) i= n M i (x i x i ) = U(G, P ), i= n L(G, P ) (G(x i ) G(x i )) U(G, P ) i= L(G, P ) G(b) G() U(G, P ) {L(G, P )} G(b) G() {U(G, P )}
G [, b] {L(G, P )} = b b G (x)dx G(b) G() G (x)dx = {U(G, P )}, b G (x)dx, b f(x)dx f G G (x) = f(x) x [, b] x n dx = xn+, n N {0}, n, n + dx = x, x e x dx = e x, xdx = x, xdx = x, dx = x, x dx = x, x dx = x, x
dx = x. + x f, g : [, b] R [, b] f g [, b] b f(x)g (x)dx = f(b)g(b) f()g() fg [, b] b (fg) (x) = f (x)g(x) + f(x)g (x). f (x)g(x)dx. f g fg [, b] (fg) b f(x)g (x)dx = = b b ((fg) (x) f (x)g(x))dx (fg) (x)dx 0 xe x dx = e (e e 0 ) = e e + = π 0 π 0 x xdx = π 0 b xdx = π 0 = 0 = e xdx = e 0 xdx = f (x)g(x)dx = f(b)g(b) f()g() x(e x ) dx = e 0 e 0 x( x) dx = π π + 0 0 π e (x) xdx = e e e 0 b (x) e x dx = e 0 x( x) dx = e f (x)g(x)dx, 0 e x dx = (x) ( x)dx = e x x dx =
e e e e dx = e (e ) = e e + = e x x dx = x x dx e e x x dx = x x dx = e x( x) dx = ( e) ( ) e ( x) xdx = x x dx = e x xdx = (e x ) xdx = e x x e x ( x) dx = e x x e x xdx = e x x (e x ) xdx = e x x e x x+ e x ( x) dx = e x ( x x) e x xdx e x xdx = e x ( x x) e x xdx = ex ( x x). g : [, b] R [, b] g [, b] f : g([, b]) R [, b] b f(g(x))g (x)dx = g(b) g() f(u)du. F f g(b) g() f(u)du = F (g(b)) F (g()). F f (F g) (x) = (F g)(x) g (x) = (f g)(x) g (x) x [, b], F g (f g) g b f(g(x))g (x)dx = (F g)(b) (F g)() = F (g(b)) F (g()) = g(b) g() f(u)du. u = g(x) du = g (x)dx u x
b g() g(b) g() > g(b) g(b) g() f(u)du = g() g(b) f(u)du u u g(x) π u = x du = xdx π 0 4 x xdx = π 4 0 xdx = π 0 π 4 0 π 4 0 u 4 du = xdx 0 π x x dx = 4 0 u = x du = xdx π 4 0 π 4 xdx = 0 u du = u du = e x x dx u = x du = dx x 0 4 x xdx u 4 du = 5 5 05 5 = 5. x x dx. du = u = = = e e x x dx = u du = du = ( ) = ( ). u 0 x + x dx u = + x du = xdx xdx = du 0 x + + x dx = du +0 u = x + x + dx u du = ( ) =.
x + x + dx = (x + x + ) + dx = (x + ) + dx. u = x + du = dx (x + ) + dx = du = u = (x + ). u + e x + e dx x u = e x du = e x dx dx = du e = du x u e x u + e dx = du x + u u = u u( + u) du. e x + e x dx = u u( + u) = u + u, ( u ) du = + u u du = u ( + u) = e x ( + e x ) + u du f : [, b) R b R b = + f x [, x] < x < b f(t)dt x b f [, b) b f(t)dt = x b x f(t)dt. ± b f(t)dt ± f : (, b] R R = f [x, b] < x < b b f(t)dt = x + b x f(t)dt,