I A ( ).,,. [2002 ] I [2003 ] [2004 ] I

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1 I 4 A ( ).. [ ] I [3 ] [4 ] I

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5 5. < y < x x n+ = x n + y n, = ( + ), n =,,..., y n+ x n y n {x n } {y n }.,. x + y x + y x + y = x + y. n x + y, x, y >, xy x + y = (x + y) 4xy (x + y) y y y n x n x x = (x y) (x + y). {x n } {y n },. α, β. n α = β. x n+ = x n + y n, α = α + β,. y n+ = x ny n x n + y n x n+ y n+ = x n + y n x n y n x n + y n = x n y n. x n y n x n y n = x y., {x n } {y n } α α = lim n x n y n = x y.

6 6 lim x n = lim y n = x y. n n < y < x x n+ = x n + y n, y n+ = x n y n, n =,,..., {x n } {y n }. ( ) < b < a a = a + b, b = a b,... a n+ = a n + b n, b n+ = a n+ b n, {a n }, {b n }. lim a n = lim b n = b sinh t. n n t t a = b cosh t.. n= n = π 6. n= n (n + ) n + (n + ) = n + 4n + 4 n (n + ) = n= n(n + ) + 4 (..) n (n + ). n =,,.... n = π 6, (n + ) = π 6 = π , (..). n= n= n= n(n + ) = ( n ) = + n + = 3. n= π 6 + π = n (n + ). n= n (n + ) = ( π π ) = π 6.

7 n S n. S m = S n = ( )n n. ( ) ( ) ( ) m 3 4m, : S S 4 S 6...., S m+ = : ( 3 ) ( 5 7 ) ( ) + 9 4m 4m + S S 3 S S m S m+ S S 4 S m S m+ S 3 S. {S m } {S m+ } ( ). lim S m α lim S m+ β. m m S m+ = S m + 4m + β = α. lim n S n. π/4., ) x arctan x (x x3 3 + x5 xn +... ( )n = ( ) n t n 5 n + t dt

8 8 ( ). x = π 4 S n = ( ) n t n + t dt π 4 S n t n + t dt t n dt = n + n..4?. (). () {a n }. (3) {a n } lim n a n+ a n =. (4) {a n }, {b n } {a n + b n }, {a n b n } {a n }, {b n }. () a < b. x = a + (b a)/. a < x b x = b a b a ( = (b a) ) > x < b., x. x. b a x a =. a < b ( ). () lim n a n = α. ϵ-δ ϵ = N a n α <, n N. a n = (a n α) + α a n α + α + α, n N. M = max{ a,..., a N, + α } n a n M. {a n }. (3). a n = n

9 9., lim a n+ a n = lim n n n + =. {a n }., a n = = , a m + m m. {a n }.. b n = n. lim b n+ b n = lim n + n = lim =. n n n n + + n lim b m b n =. m = 4n n,m. = lim n b 4n b n = lim n 4n n = lim n n =. lim n b n = {b n }. (4). a n = ( ) n, b n = ( ) n.,, a n + b n =, a n b n =. (). () {a n } a n+ a n n. I n = {,,..., n}. A D(A) = lim n n A I n ( A )... () p A = {p, p, 3p,... } D(A). () D(A), D(B) D(A B)..

10 .5?. (). () {a n }. (3) {a n } lim n a n+ a n =. (4) {a n } a n+ a n. n () ( ) x < y. x, y x = x.x x..., y = y.y y...,. ( 3). x < y {x n }, {y n } k x k < y k. z = x.x x... x k y k = y.y y... y k y k.., x < z < y. () ( ) a n =. n. a n lim n a n = (3) ( ) a n = n. a n+ a n = n +. {a n }.,, {a n }.. a n. y = /x x n + x ( ). a n > n+ dx x lim log(n + ) = lim a n =. n n n n = log(n + ).

11 (4) ( ) n, k a n+k a n. a n+k a n a n+k a n+k + a n+k a n a n+k a n+k + a n+k a n+k + a n+k a n a n+k a n+k + a n+k a n+k + + a n+ a n + n+k + + n+k n = ( n + k + + ) k + < = n. n ϵ > < ϵ N m, n N N a m a n < ϵ. {a n }..6 I = [, ] 3 (/3, /3) [ I =, ] [ ] 3 3,., 3 [ I 3 =, ] [ 9 9, 3 ] [ 6 9 9, 7 ] [ ] 8 9 9,. I, I, I 3,.... () I n. () I n L n n. (3) I n ( )?. () I n n. () I n (/3) n L n = ( ) n n = 3 ( ) n (n ). 3 (3) I n. 3. I

12 I 3 = = = = = = = = ,. 3 x = a 3 + a 3 + a a a n ( ). (α, β) ( a α = a ) n + 3 n 3 + n+ 3 + n , ( n+3 a β = a ) n + 3 n 3 + n+ 3 + n , n+3. x α < x < β. ( ) I n. X = n= {a n },,. n= a n 3 n = lim n I n n k= a k 3 k

13 (i) (ii). {a n }, {b n },,? n= a n 3 = b n n 3 n n= 3.7 a, b. ( + an + bn ) n lim n ( e = lim + n (.7.) n n) e ( )., ( lim + a ) n = e a (.7.) n n ( )., ( + a n + b ) n ( = + a n n ( )( n + + n n) a ) n k ( ) k b k n n ( + a n + b ) n n ( + a n) n. lim n M n = (.7.), lim n. M n. k= k= k= n ( ) ( n + a ) n k ( ) k b M k n n n (.7.3) ( + a n + b n ) n = e a n ( ) ( n M n = + a ) n k ( b k + n n ) k+ ( ) n = k + n! (k + )!(n k )! n (n )! k!(n k)! = n ( ) n k

14 4 n ( n M n n k k= = e a b n e a b n ( + a ) n k e a n ) e a ( b n ( + b ) n e a b n n e b = e a + b b. n ) k+ a b n = ne n ( + b n lim M e a + b b n lim n n n ( ) ( n b k= ) n =. k n ) k a > x n = ( + a ) n n e a., ( + a ) n < e a n. ( ) (.7.). (.7.) ( e = lim + x = lim x x) ( + y) y (.7.4) y +. ( lim + a n n + b ) n ( = lim + an + b ) n {( = lim + an + b ) n } an+b an+b n n n n n n. y = an + b, n ( lim + an + b ) n an+b n n n y + (.7.4) ( lim + an + b ) n an+b = e. n n

15 5 ( lim + a ) n {( n n + bn = lim + an + b ) n } an+b an+b n n n.. ( f n (x) = x an+b n, xn = + an + b ) n an+b n = e a lim f n(x) = x a, n lim x n = e n... lim f n(x n ) = e a (.7.5) n ( ) lim f n (x n ). n, log f n (x n ) = an + b log x n n. log lim log x n = log e =. n, an + b lim n n = a lim log f an + b n(x n ) = lim n n n lim f n(x n ) = e a. n log x n = a. (.7.5).. f n (x) nx, x /n f n (x) = nx, /n x /n, x /n

16 6. {x n } x n = /n lim f n(x) =, n lim x n =, n lim f n(x n ) = n. [ (.7.5)..] lim na n = α lim ( + a n ) n = e α. n n (.7.4) (.7.)..8 3 n A n, 3 n B n. A < A < A < < A n < < B n < < B < B < B (.8.) lim (B n A n ) = (.8.) n., {A n }, {B n } ( ). 3 n 3 n A n < B n., 3 n 3 n+ A n < A n+. B n > B n+. (.8.). (.8.), B n A n,.., A n, B n. N = 3 n. N. θ = π N = π N..

17 7 OAB OB =, OA = cos θ, AB = sin θ. OAB = OA AB = sin θ cos θ. N A n = N OAB = N sin θ cos θ = N sin θ = N sin π N., OCD OC =, CD = tan θ, OCD = OC CD = tan θ N B n = N OCD = N tan θ = N tan π N. A n = N π sin N, B n = N tan π N, N = 3 n. (.8.3) B n A n, ABDC N. ( ABDC ) < AC CD < B n A n < AC ( N ). ( N ) ( 3 ). < B n A n < AC ( 3 ) = ( cos θ) ( 3 ). (.8.4) lim cos θ = lim cos π n n 3 = n

18 8 (.8.4) (.8.). A n, B n θ = π 3, π 6, π, π 4,... sin θ, cos θ, tan θ. sin θ =. cos θ, cos θ = + cos θ, sin π 6 =, cos π = 4 sin π 4 = 4 6, cos π 8 4 = (Exel ). n N A n B n

19 ( ) (.8.) (.8.3) B n A n = N tan π N N = N sin π N ( ( cos π N π sin N = N sin π N ) π ) cos N sin θ lim θ θ ( cos π ) π N sin N N cos π N 9 = (.8.5) lim (B n A n ) = lim N sin π ( ( cos π ) π ) cos = π( ) = n n N N N. (.8.5). ( ) 96 (96 = 3 5 ).. A 5, B 5..9.a a a 3... a + a + a 3 + = a n 3 = lim n n n= n k= a k k (.9.). ( a k,,,..., 9.) ( a k (.9.) )., n= a n = n n= b n n {a n } {b n }. n =,,... S n =. a k n k= a k k S S S 3 S n...., a k 9 S n n k= 9 k = n <.

20 . S S S 3 S n <, n =,,.... {S n } ( ). lim n S n., {a n }, {b n } A n = n k= a k k, B n = n k= b k k, lim A n = lim B n n n. {a n }, {b n } a = b, a = b,..., a k = b k, a k < b k,. ( a k b k {a n } {b n } a k < b k.)., n > k B n A n = A n = A k + n j=k+ B k A k = b k a k k > n j=k+ a j j, B n = B k + b j a j j + (B k A k ) = b j a j n n j=k+ n j=k+ 9 j b k a k k (B n A n ) n j=k+ b j j b j a j j + b k a k k. (.9.) n j=k+ 9 j. k b k a k k k b k a k., a k < b k b k = a k +. (.9.) B n A n = n j=k+ b j a j j + k

21 n lim n n j=k+ a j b j j = k. j k + a j = 9, b j = (? ). {a n }, {b n } a, a,..., a k, a k, 9, 9, 9,... b, b,..., b k, a k +,,,,.... (a k =,,,..., 8.) {a n }, {b n },,..., 9. lim n n j= a j b j j = n a n = 9, b n =. (..).., a.a a a m =, m, n n n n = m..,...., a, a < a +. a = 3., a a a < a +., a, a,..., a n a n+ a n+ ( a n+ + a + + a ) n < a n+ + n n+

22 . {a n } n a k S n = a + k k= < S n < n lim S n =. n.... f(x) = x + 4. () y = f(x). () x = x + 4 (). (3) a n a n+ = f(a n ), a = c, (c > 4). {a n }. (). () x = x + 4 y = f(x) = x + 4 y = x x. α α = + 7. (3) {a n }.

23 3 (i) 4 < c < α c = a < a < < a n < α. (ii) α < c c = a > a > > a n > α. (iii) c = α n a n = α lim n a n = α. (i). c = a < a., 4 < c < a = c + 4 > a < a. c < α. a a = f(c) c = c + 4 c x x 4 = α, β β < < α. c + 4 c = (α c)(c β) c < α >. a < a. n a < a < < a n+., a n+ a n+ = a n+ + 4 a n + 4 = a n+ a n an a n + 4 >. a < a < < a n <...., a n < α., n =. n α a n+ = α a n + 4 = α (a n + 4) α + a n + 4. α (a n + 4) = (α + 4) (a n + 4) = α a n >. a n+ < α. {a n }.. lim n a n = ξ. a n+ = a n + 4 n x + 4 ξ = α {a n } α. ξ = ξ + 4. ( ) {a n } α a n+ α r a n α, n =,,..., (..)

24 4 < r <. r. a n+ α = a n + 4 α + 4 = (a n + 4) (α + 4) an α + 4 = a n α an α + 4 < r < a n+ α = a n α an α + 4. an α + 4 < r n (..). r a n + 4 an α + 4 = α r = /. a n+ α a n α, n =,,.... a n+ α ( ) n a α, n.. () p (p =, 3, 5, 7,... ) ( ). lim N N n= p n. () p, q. lim N N m,n= p n q m. (3)., p, p,..., p s. n. n = p k p k... p ks s, k, k,..., k s =,,,..., n= n (..) p, p,..., p s ( ())., (..)..

25 5 () r r < + r + r + + r n = rn r. n= r n = r.. lim N N n= p = n p = n n= p = p p. () lim (3) () N N m,n= N m,n= p n q = N m p n p n q m = n= p n n= m= N m= q m = q m p p q q. lim N N k,...,k s = p k... p ks s = k = n p k k s = p k s s = p p... p s p s. (..) n = p k p k... p ks s N n, (..). (..) n. n= 5 N n= N+ n > dx x n = p p... p s p s. (..3) = log(n + ), N. n= n =

26 6. (..3) n + = π = π 6.. p: p p

27 7. n =,,... cos nθ cos θ T n (cos θ). cos θ = cos θ. cos θ = T (cos θ), T (x) = x. () n =,,, 3, 4 T n (x). () T n+ (x) T n (x) T n (x). (3) m, n =,,, T m (x)t n (x) dx x. (). (). T (x) = T (x) = x T (x) = x T 3 (x) = 4x 3 3x T 4 (x) = 8x 4 8x + () n. cos(n + )θ = cos nθ cos θ sin nθ sin θ, sin nθ sin θ = (cos(n + )θ cos(n )θ). cos(n + )θ = cos nθ cos θ + (cos(n + )θ cos(n )θ). cos(n + )θ = cos nθ cos θ cos(n )θ.

28 8 cos nθ = T n (cos θ) T n+ (x) = xt n (x) T n (x). (3) x = cos θ. x θ π. T m (x)t n (x) x m = n = m = n m n T m (x)t n (x) x dx = = = π π T (x)t (x) x T m (x)t m (x) x dx = π cos mθ cos nθ cos θ cos mθ cos nθ dθ ( sin θ) dθ (cos(m + n)θ + cos(m n)θ) dθ. dx = dx = π π π ( + )dθ = π. (cos mθ + ) dθ = π. (cos(m + n)θ + cos(m n)θ) dθ =. ( ). cos mθ cos nθ = cos(m + n)θ + cos(m n)θ ( ) T (x) =, T (x) = x T n (x) n ( ).. T n (x) = (, +) n n =,, 3, 4. ( n..). n =,,,.... sin(n + )θ sin θ cos θ n., n =,,, 3, 4 U n (x).

29 9 n = U (x) =. n = sin( + )θ sin θ = sin( + )θ sin θ = sin θ cos θ sin θ = cos θ U (x) = x. n k n k k U k (cos θ)., sin(k + )θ sin θ cos θ sin(n + )θ + sin(n )θ = sin nθ cos θ sin θ sin(n + )θ sin θ + sin(n )θ sin θ = cos θ sin nθ sin θ. sin(n + )θ sin θ = cos θu n (cos θ) U n (cos θ). cos θ n n U n (x) = xu n (x) U n (x). U (x) = 4x, U 3 (x) = 8x 3 4x, U 4 (x) = 6x 4 x +. ( ) U n (x).. + U m (x)u n (x) π x dx =, m = n,. T n (x) U n (x). U n (x) = n x k T n k (x). k=

30 3 n =,,,... P n (x) = n n! d n dx n (x ) n. n = P (x) =.. () P n (x) x n. (.) () P n () =, P n ( ) = ( ) n. (3) xp n(x) P n (x) = np n (x). (4). + P m (x)p n (x) dx = (n + ) δ mn.3 n =,,,... x [, ] T n (x) = cos(n arccos x) T n (x) n., T m (x)t n (x) x. (m, n.) dx () n. cos(n + )θ = cos nθ cos θ sin nθ sin θ, sin nθ sin θ = (cos(n + )θ cos(n )θ). cos(n + )θ = cos nθ cos θ + (cos(n + )θ cos(n )θ). cos(n + )θ = cos nθ cos θ cos(n )θ. θ = arccos x T n+ (x) = xt n (x) T n (x)., T (x) =, T (x) = x. T n (x) n ( ).

31 3 () x = cos θ. x θ π. T m (x)t n (x) x m = n = m = n m n T m (x)t n (x) x dx = = = π π T (x)t (x) x T m (x)t m (x) x dx = π cos mθ cos nθ cos θ cos mθ cos nθ dθ ( sin θ) dθ (cos(m + n)θ + cos(m n)θ) dθ. dx = dx = π π π ( + )dθ = π. (cos mθ + ) dθ = π. (cos(m + n)θ + cos(m n)θ) dθ =. T n (x). T n (x) cos nθ cos θ. n =,,,... x [, ] U n (x) = sin(n + )θ, x = cos θ, sin θ U n (x) n.,. U m (x)u n (x) x dx U n (x)..4. () A cosh x cosh y + B sinh x sinh y = {(A + B) cosh(x + y) + (A B) cosh(x y)} () (3) cosh(x + y + z) = + tanh y tanh z + tanh z tanh x + tanh x tanh y. cosh x cosh y cosh z n sinh(n + )x cosh kx = sinh x. k=

32 3 () ( ) = } {(A + B) ex+y + e (x+y) + (A B) ex y + e (x y) = { A(e x+y + e (x+y) + e x y + e (x y) ) + B(e x+y + e (x+y) e x y e (x y) ) } 4 = { A(e x + e x )(e y + e y ) + B(e x e x )(e y e y ) } 4 = A cosh x cosh y + B sinh x sinh y. (), cosh(x + y) = ex+y + e (x+y) = { e x (e y + e y ) e x y + e x (e y + e y ) e x+y} = { (e x + e x )(e y + e y ) e x y e x+y} = cosh x cosh y ex y + e x+y. e x y + e x+y = { e x (e y e y ) + e x (e y e y ) + e x+y + e x y} = { (e x e x )(e y e y ) + e x+y + e x y} = sinh x sinh y + cosh(x + y). cosh(x + y) = cosh x cosh y + sinh x sinh y cosh(x + y). cosh(x + y) = cosh x cosh y + sinh x sinh y. ( ), sinh(x + y) = sinh x cosh y + cosh x sinh y., cosh(x + y + z) = cosh x cosh(y + z) + sinh x sinh(y + z) = cosh x(cosh y cosh z + sinh y sinh z) + sinh x(sinh y cosh z + cosh y sinh z). cosh x cosh y cosh z.

33 33 (3). n cosh kx = n (e kx + e kx ) k= = { } e x (e nx ) + e x (e nx ) e x e x = { } e x (e nx ) + e x (e nx ) e x e x e x e x { = e x (e nx ) e x (e nx ) } (e x e x ) { = e (n+)x e (n+)x (e x e x ) } 4 sinh x sinh(n + )x sinh x = 4 sinh x sinh(n + )x = sinh x. k= sinh x + sinh y = sinh x + y cosh x + cosh y = cosh x + y cosh x y cosh x y sinh x sinh y = cosh x + y cosh x cosh y = sinh x + y sinh x y sinh x y sinh(x + y + z) sinh x sinh y sinh z = + coth y coth z + coth z coth x + coth x coth y.5 ( ). () arcsin x x = arccos ( x ) + x + x x () arctan x = arcsin = arccos + x + x (x ) () θ = arcsin x + x sin θ = cos θ = sin θ = x + x. ( ) ( ) x x =. (.5.) + x + x

34 34, x sin θ. θ π/ ( ). cos θ. (.5.) cos θ = x + x θ = arccos x + x. () θ = arctan x x x = tan θ, θ < π., cos θ sin θ, x + x = + tan θ = cos θ cos θ = + x. sin θ = cos θ = x x. θ = arcsin x = arccos. x + x..6 α = arctan 5. tan 4α = arctan 5 arctan 39 = π 4..

35 35 tan α = /5,. tan 4α = tan α = tan α tan α = 5. tan α tan α = 5 / 9 6 = 9 = + 9. ( tan 4α π ) = tan 4α tan π tan 4α tan π 4 p.98.. = / = 39. arctan 39 = 4α π 4 = 4 arctan 5 π 4. π 4 = 5 arctan 7 + arctan 3 79 = arctan + arctan 5 + arctan 8 arctan x + arctan y + arctan z = x + y + z = xyz..7 () arctan x. () arctan p + arctan q = π 4 (p, q). (). () α = arctan p, β = arctan q α + β = π/4. = tan π 4 = tan(α + β) = tan α + tan β tan α tan β. tan α = /p, tan β = /q q + p = pq. = p + q p q = q + p pq. (p )(q ) =. (p, q) = (, 3), (3, ).

36 36.8 x π 4. arcsin( sin x) + arcsin( cos x) = π, ( sin x) + ( cos x) = sin x + cos x = ( cos x) + cos x =. AB = sin x, AC = cos x, BC =. sin C = AB BC = sin x, sin B = AC BC = cos x. arcsin( sin x) + arcsin( cos x) = C + B = π. < x < π/4 x =, π/4.. f(x) x. sin x cos x f (x) = sin x + cos x cos x cos x = cos x + sin x cos x cos x cos x sin x cos x = + =. cos x cos x sin x (, π/4). f(x). f(x) [, π/4] f(x) = f() = arcsin + arcsin = π.

37 37.9., sinh x + sinh y = sinh x + y cosh x y. sinh x sinh y, cosh x + cosh y, cosh x cosh y. sinh x + y cosh x y.. = e(x+y)/ e (x+y)/ e (x y)/ + e (x y)/ = { e (x+y)/ e (x y)/ + e (x+y)/ e (x y)/ e (x+y)/ e (x y)/ e (x+y)/ e (x y)/} = ex + e y e y e x = sinh x + sinh y. sinh x sinh y = cosh x + y cosh x + cosh y = cosh x + y cosh x cosh y = sinh x + y sinh x y cosh x y sinh x y cosh x = cosh x, cosh 3x = 4 cosh 3 x 3 cosh x, sinh x = sinh x cosh x, sinh 3x = 4 sinh 3 x + 3 sinh x, tanh x = tanh x + tanh x, tanh 3x = 3 tanh x + tanh3 x + 3 tanh. x. < x < arcsin x ( + x ) + arcsin + x ( + x ) x.

38 38 < x < x, + x, ( + x ). ( x) + ( + x) = ( + x ) 3 3. sin α = x ( + x ), sin β = + x ( + x ) α = arcsin x ( + x ), β = arcsin + x ( + x ). arcsin x ( + x ) + arcsin + x ( + x ) = α + β = π. ( ) f(x) = arcsin x ( + x ) + arcsin + x ( + x ) f (x) =. f(x) (, ). x =... π 4 = arctan + arctan 5 + arctan 8

39 39 tan tan(α + β + γ) = = = = tan α + tan(β + γ) tan α tan(β + γ) tan α + tan β+tan γ tan α tan β tan γ tan β+tan γ tan β tan γ tan α( tan β tan γ) + tan β + tan γ tan β tan γ tan α(tan β + tan γ) tan α + tan β + tan γ tan α tan β tan γ (tan β tan γ + tan γ tan α + tan α tan β), α = arctan, β = arctan 5, γ = arctan 8, tan(α + β + γ) = ( ) =. 5 α + β + γ = π/4. (p, q). arctan + arctan p + arctan q = π 4., arctan p + arctan q + arctan r = π 4 (p, q, r). ( ) p q r. p q r > 3 arctan p arctan p + arctan q + arctan r = π 4 p tan π = p 3 p =,, 3. [ : (, 4, 3), (, 5, 8), (3, 3, 7)]

40

41 [, ] f(x) [, ] x = f(x) x. f(x) {f(x) ; x }. () f(x) [, ], f(α) =, f(β) = α, β f(x) [, ]., α < β. y = f(x) y = x f(x). g(x) = x.. f(α) =, g(α) > ; f(β) =, g(β) < ; F (x) = f(x) g(x) F (α) <, F (β) >. F (c) = c α < c < β. c x. α > β ( ).

42 R? { x n x x sin f(x) = g(x) = x x x n x < x = h(x) = { e x x > x () x C - ( ). n f (n ) (x) = n!x, x ; = n!x, x <. f (n ) (x) = n! x. x R x =. f (n ) (x) x =. f(x) C n - C n -. () x C - ( ). x =., sin θ lim x x sin x lim x =. x lim x g(x) = g() g(x). x =. g(x) g() lim = lim sin x x x x.. x. x =, n =,,..., πn lim x = sin πn =. n, n =,,,..., πn + π/ lim (πn sin + π ) = n x lim x sin x. g(x) x =. g(x) C - C -. (3) x C - ( ). x =. x < h (n) (x) =. x > h(x) = e x, h (x) = x e x, h (x) = ( x + x ) e 3 4 x,...

43 . h (n) (x) = p n(x). p n (x)., n =. n n ( h (n+) (x) = np n(x) + p n(x) x n+ x + p ) n(x) e nxp n (x) + x p n x n x x = n(x) + p n (x) e x n+ x. p n+ (x) = nxp n (x) + x p n(x) + p n (x) n +. ( ) p n (x) lim x + h(n) (x) = lim x + x n e x = lim p n y n e y =. y y lim x h (n) (x) = h(x)., h(x) C -. x n e x R? sin x x x n log x x > f(x) = x g(x) = x = x = ( x) n log x x < f(x) f (x) f (x). x 3 sin f(x) = x x x =. x =. x = : f(x) f() = x sin x x sin θ f(x) f() x x x. f(x) x = f () =. x, f (x) = 3x sin x + x3 cos x ( x ) = 3x sin x x cos x. lim x f (x) = f (x) x = R.

44 44 3, x f (x) = 6x sin x + 3x cos ( x ) cos x x + x sin ( x ) x = 6x sin x 3 cos x cos x x sin x = 6x sin x 4 cos x x sin x. x 3. x = (π/ + nπ) n x sin ( ) ( ) π π x = + nπ sin + nπ = π + nπ n. f(x)., f (). f (x) f () = 3x sin x x cos x [, ] x. 3.4 f(x) = sin x x. () y = f(x). () f(x) x =. (3) y = f (x) ( ). () x = ( () ). () x., x > f(x) = sin x x f (x) = cos x, x >.

45 45, x < f(x) = sin x + x f (x) = cos x +, x <., x =. f(h) f() D + f() = lim h + h f(h) f() D f() = lim h h sin h h = lim h + h sin h + h = lim h h sin h = lim h + h =. sin h = lim h h + =. D + f() = D f() =. f () =. f(x) x { cos x, x f (x) = cos x +, x. (3) f (x) x = ( ). π π π π π π 3.5 f(x) = x log x. () x > y = f(x). () lim x log x f(x) x f() x +. n (3) n a,..., a n a k a k = k= n f(a k ) log n (4) (3) a,..., a n? k=

46 46 3 (). f (x) = log x + x x = log x +. f (e ) = x = e f (x). (), y = e t (.) lim x log x = lim x + y + y log y = lim log y y + y. log y lim y + y t = lim t + e = t lim x log x =. x + f(x) x = f() = f(x) [, + ). (3) n a,..., a n a k f(a k ) =., a k. a k >., f(x) (, ). f(p x + p x ) p f(x ) + p f(x ), x, x (, ), p, p, p + p =.. p,..., p n, p + + p n = p,..., p n f(p x + + p n x n ) p f(x ) + + p n f(x n ), x,..., x n (, ), ( p.7)., p = = p n = /n ( ) x + + x n f n n (f(x ) + + f(x n )). (3.5.)

47 47 ( x + + x n n ) log ( ) x + + x n n n (f(x ) + + f(x n )). x k = a k a k = ( ) ( ) log n n n (f(a ) + + f(a n )). (f(a ) + + f(a n )) log n.. (4) f(x) (3.5.) x = = x n. (3) a = = a n = n. ( ), x log x x, x, (3.5.). (x =, () log =.) g(x) = x log x x + g (x) = log x + x = log x. x x = g (x) = x = g (x). g() =. (3.5.). n, n a,..., a n a k a k =. x = na k (3.5.) k= na k log(na k ) na k (3.5.3) na k log n + na k log a k na k a k log n + a k log a k a k n. n a k = k k= log n + n a k log a k =. k= n a k log a k log n. (3.5.4) k= (3.5.3) k. na k (3.5.) x na k =. (3.5.4) a k = /n, k =,,..., n,.

48 ().,. () f(x) (a, b)., x (a, b) f (x) = f(x). (). () (a, b) x < y. f [x, y] (x, y) f (y) f (x) = f (z)(y x) z (x, y). f (z) = f (y) f (x) =. f (x) (a, b). A : f (x) = A, a < x < b., f(x) f(y) f(x) = f (z)(y x) z (x, y) f (z) = A f(y) f(x) = A(y x). f(x) = Ax + (f(y) Ay). x, y (a, b) y x. B = f(y) Ay f(x) = Ax + B.

49 () y = sinh x cosh x () y = x x (3) y = log(cos x) (4) y = arcsin x x () (sinh x) = cosh x, (cosh x) = sinh x. y = (sinh x) cosh x + sinh x(cosh x) = cosh x + sinh x = cosh x. (,.) () y = e log x x (3) ( ) log x y = e log x x x y = (cos x) cos x (4) (arcsin x) = x. = log x x x x = ( log x)x x. = cos x( sin x) cos x = sin x cos x = tan x.. y = (arcsin x) x arcsin x x = x x arcsin x x x. 4.. n 3 () lim n n! n () lim n (log n) (3) lim x x cos x x () n 3. n 3 n! = n n n (n 3)(n 4)... n n n = ( ) ( ) n. (n 3)(n 4)... n n

50 5 4 () log n = x n = e x. n x lim n n = lim (log n) x e x/ =. x e x. e x/. (3), x cos x x { cos x x > = cos x x <. x cos x lim x + x = lim cos x =, lim x + x cos x x x = lim ( cos x) =, x. 4.3? ( ). () (a, b) f(x) lim. x a+ f(x), lim f(x) x b () f(x) (a, b). f(x) f(x). (3) f(x), g(x) f(x) f(g(x)) g(x). (4) f(x), g(x) (a, b) f (x) = g (x) x (a, b). lim f(x) = lim g(x) f(x) x a+ x a+ g(x). (). f(x) = (b x)(x a), a < x < b, (a, b). lim f(x) = lim f(x) = +. x a+ x b (). (, ) f(x) = x. f(x) = x = x f(x) = x x =.

51 (3). f(x) = x, g(x) = x. f(x) f(g(x)) = g(x) = x = x. g(x) = x x =. (4). (a, b) f(x) g(x). c, f(x) = g(x) + c. lim f(x) = lim g(x) x a+ x a+ c =. f(x) g(x). ( ) lim f(x) = lim g(x),. x a+ x a+ =. f(x) = (b x)(x a), g(x) = +, a < x < b. (b x)(x a) f(x). f(x) = x + () y = f(x). (). F (x) = (3) y = F (x). { x x x < x. f(t)dt, x. (4) F (x) = f(x).. () (), x < x F (x) = F (x) = x = x f(t)dt = f(t)dt = ) ( x x x ( t ) dt + ( t ) dt = [ t ] t=x t t= ( + ) = x x. x (t + ) dt = ( ) x + + x.

52 5 4 (3) x x F (x) = x + x x, x. (4) F (x) < x < F (x) = ( x x ) = x. < x < ( x F (x) = + x ) = x +.

53 53 x =., F (x) F () D + F () = lim x + x F (x) F () D F () = lim x x = lim x + x + x x x = lim x x x =, =. F (x) = f(x), < x <, < x <. x = ± F (x). f(±). x =, F (x). f(x) x = F (). 4.5 C - f(x) f(c + h) + f(c h) f(c) lim h h = f (c)., f (c). /.. f(c + h) + f(c h) f(c) lim h h = lim h f (c + h) f (c h) h = lim h f (c + h) + f (c h) = f (c). f(c + h) = f(c) + f (c)h + f (c + θ h) f(c h) = f(c) f (c)h + f (c θ h) < θ, θ <. f(c + h) + f(c h) = f(c) + (f (c + θ h) + f (c θ h))h. h, h, f(c + h) + f(c h) f(c) h. h = (f (c + θ h) + f (c θ h)).

54 54 4. f(c + h) = f(c) + f (c)h + f (c) f(c h) = f(c) f (c)h + f (c) < θ, θ <. h + f (c + θ h) 6 h f (c θ h) 6 f(c + h) f(c h) = f (c)h + h3 6 (f (c + θ h) + f (c θ h)). (4.5.) f(c + h) f(c h) = 4f (c)h + 8h3 6 (f (c + θ 3 h) + f (c θ 4 h)). (4.5.) (4.5.) (4.5.) (f(c + h) f(c h)) (f(c + h) f(c h)) h 3, h 3, = h3 6 (f (c + θ h) + f (c θ h)) 8h3 6 (f (c + θ 3 h) + f (c θ 4 h)). h {(f(c + h) f(c h)) (f(c + h) f(c h))} h3 lim h = 3 (f (c + θ h) + f (c θ h)) 4 3 (f (c + θ 3 h) + f (c θ 4 h)). {(f(c + h) f(c h)) (f(c + h) f(c h))} h3 = 3 f (c) 4 3 f (c) = f (c). f (c) = lim {(f(c + h) f(c h)) (f(c + h) f(c h))}. h h3 f f(c + h) f(c + h) + f(c h) f(c h) (c) = lim. h h 3

55 f(x) = arctan x + arctan x. () f (x). () y = f(x) (f(x) ). () (arctan x) = x +. ( f (x) = (arctan x) + arctan ) = x x + + ( ) ( x ) =. + x f (x) =. () f(x) (, ), (, + ). f(x)., x > f(x) = arctan x + arctan x = arctan = π. x < f(x) = arctan x + arctan x = arctan( ) = π. f(x) = arcsin x + arccos x. 4.7 f(x) (, + ) lim x f (x) = a. () f(x). () h > lim x {f(x + h) f(x)}.

56 56 4 (), lim x f (x) = a. f (x) = a + x f(x) = ax x. f (x) = a + g(x) lim x g(x) =. () f(x + h) f(x) = f (x + θh)h < θ = θ(x, h) <. ϵ > M > f (x) a < ϵ, x > M. x > M x + θh > M f (x + θh) a < ϵ f(x + h) f(x) ah = f (x + θh)h ah < ϵh, x > M. lim {f(x + h) f(x)} = ah. x 4.8 P (x) =, P n (x) = n n! d n dx n (x ) n, n =,,...,.. () P n (x) n x n. () P n (x) = (, +) n. () (x ) n = x n +... P n (x) = n n! d n dx n (xn +... ) = n n! {(n)(n )... (n + )xn +... } P n (x) n x n (n)! (n)(n )... (n + ) = n n! n n! n! = (n)! n (n!).

57 57 : () n (n)! = (n)(n )(n )... 3 = (n)(n )... (n )(n 3)... 3 = n n! (n )!! (n)! n (n!) = n n! (n )!! = n (n!) (n )!!. n! f(x) = (x ) n = (x + ) n (x ) n. k =,,..., n f (k) (x) =, k (, ). k =. f() = f( ) = Rolle (, ) f (x) = x ( ). k =. k k. < x < < x k < f (k) (x ) = f (k) (x ) = = f (k) (x k ) = (4.8.). k + n k +. k ( ) f (k) (x) = dk k dx (x + k )n (x ) n = {(x + ) n } (j) {(x ) n } (k j) j j= k n f (k) ( ) = f (k) () =. (4.8.) Rolle (, x ), (x, x ),..., (x k, ) f (k+) (x) = x. f (k+) (x) = k + (, ).., f (n) (x) =, n (, )., P n (x) f (n) (x) n P n (x) = n (, ). n =,,... d n dx n e x = ( ) n H n (x)e x H n (x). H (x) =. () H n (x) n. () H n+ (x) = xh n (x) H n(x). (3) H n (x) α < β H n (x). (4) H n (x) = n. ( ) H n (x).

58 f(x). x x, f(x) = + e /x x =. () f(x) x =? () f(x) x =? (3) a, b. lim {f(x) (ax + b)} =. x + () x C -. lim x + e/x = +, lim x e/x =,. lim f(x) = lim x + x +, f(x) x =. () x = = f(), lim f(x) = lim + e/x x x f( + h) f() lim h + h = lim h + f( + h) f() lim h h h + e /h h x = = f(), + e/x = lim =. h + + e/h = lim =. h + e/h f(x) x =. y = f(x) f (x) = ( + e/x ) x( x e /x ( + e /x ) = + e/x + x e/x ( + e /x ) >, x,,.,. lim f + e /x + x (x) = lim e/x = x ± x ± ( + e /x )

59 : 4.9 (3), { } x = lim (ax + b) x + + e/x x ( + e /x )(ax + b) = lim x + + e /x = lim x + ( a ae /x )x ( + e /x )b + e /x. x = t., lim t + (4.9.),. ( a ae t ) = lim ( + t et )b t + ( + e t a ae t = lim t + + e t t = lim t + + e t + e = t, lim e t t + t ) ( + e t )b ( a et + a ( + e t )b t t ( = lim a + a ) b t + t a =, a b =, =, lim t + ( + e t ) =, ). (4.9.) a =, b = 4.

60 6 4 ( ) lim {f(x) (ax + b)} = x +. f(x) (ax + b) lim x + x a = f(x) lim x + x = x = a I f(x), (i) x a (ii) lim x a f (x) f(x) x = a. [ ].. () n f(x) = f (a) = lim x a f (x) { e /x, x > x f (n) (x) = p n(x) x n e /x, x >, n p n (x). () lim f (n) (x) = lim f (n) (x) = x + x. (3) f(x) x =. 4. α = arctan 5. (). () arctan x x x3 3 + x5 5 tan 4α = + 9, 4 arctan 5 arctan 39 = π 4 π 3.

61 6 tan α = /5, tan 4α = tan α = tan α tan α = 5. tan α tan α = 5 / 9 6 = 9 = + 9. ( tan 4α π ) = tan 4α tan π tan 4α tan π 4 (), 4 arctan ( ( = / = 39. arctan 39 = 4α π 4 = 4 arctan 5 π 4. arctan π 3.4. ( ) 5 ) 3 ) 5 ) + = ( 5 ( ) 3 = π 4( ) = ( ) 5 = π 4 = 5 arctan 7 + arctan 3 79 = arctan + arctan 5 + arctan 8 arctan x + arctan y + arctan z = x + y + z = xyz. 4. a >. x f n (x) = x n a x, n =,,...,. () x > f n(x) = x. () n =,, 3 y = f n (x). (3) () x x n. lim x n, n lim f n (x n ). n

62 6 4 () f n (x) = x n a x > log f n (x) = n log x x log a. f n(x) f n (x) = n x n x log a log a =. x, x > f n(x) = x =. x = n log a f n(x). n log a f n(x) () a = e. a x y.. x e x. xe x x, x e x x, x 3 e x x 3, xe x x e x x 3 e x (3) lim x n n = lim n n log a =., f n (x n ) = x n na x n = ( ) n n a n log a. log a

63 63 b = a n log a log b = n log a = n log a b = e n. f n (x n ) = x n na x n = ( ) n n. e log a lim f n(x n ) =. n 4... () f(x), x = a. () [, ] f n (x) f(x) = lim f n (x) n. x n lim x n = a lim f n (x n ) = f(a) x x. (3) lim x f(x) = n lim x e nx f(x) =. () f(x) = x.. x = +,. x =., y = x x =. () f n (x) = x n. f(x) = lim n f n (x) = {, x <,, x =. x n = n, f(a) = f() = a = lim n x n =. ( lim f n(x n ) = lim n = e n n n). lim f n(x n ) f(a). n

64 64 4 (3) x e nx (n: ).. n f(x) = e x. lim x e nx f(x) = lim e x nx = x () f(x). [, ] f n (x) f(x) = lim f n (x). n,? 4.3 n =,,,... H n (x) = ( ) n e x dn dx n e x (4.3.). () H n (x) n. () xh n (x) H (x), H (x),.... (3) H n(x) H (x), H (x),.... (). H (x) =, H (x) = x, H (x) = 4x, H 3 (x) = 8x 3 x,. H n (x) = (x) n + P n (x), P n (x) n (P (x) = P (x) = ).. n =. n n.. e x dn ( ) n e x dx n e x = (x) n + P n (x) x. ( ) n dn+ e x = d dxn+ dx ((x)n + P n (x))e x = (n(x) n + P n (x))e x x((x) n + P n (x))e x = ( (x) n+ + n(x) n + P n (x) xp n (x))e x.

65 65 dn+ H n+ (x) = ( ) n+ e x e x dxn+ = (x) n+ n(x) n P n (x) + xp n (x). n(x) n P n (x) + xp n (x), n. P n (x) H n+ (x) = (x) n+ + P n (x) n +. H n (x) n n. () H n (x) (4.3.) H n (x) = ( ) n e x dn ( xe x). dxn ( ( ) d n n ( xe x) = )( x) dn n d e x + dxn dxn dx ( x) d n e x dxn = x dn d n e x (n ) e x. dxn dxn { H n (x) = ( ) n x e x = x( ) n e x dn dx dn e x dxn n e x = xh n (x) (n )H n (x). (n )e x (n )( ) n e x dn e x dxn } dn e x dxn xh n (x) = H n+(x) + nh n (x). (4.3.) n = xh (x) = H (x). H (x) = (4.3.). (3) (4.3.) x { H n(x) = ( ) n xe x dn dx n e x = x( ) n dn x e dx n e x + e x dn+ dx ( ) n+ e x n+ e x } dn+ e x dxn+.

66 66 4 () H n(x) = xh n (x) H n+ (x). (4.3.3) H n(x) = {H n+ (x) + nh n (x)} H n+ (x) = nh n (x). H n(x) = nh n (x), n =,,,.... ( ) (4.3.3). H (x), H (x),.... (4.3.3), H n (x). ( ) H n (x) n.. H n (x) (x) n. 4.4 x >. x x3 3 + x5 5 x7 7 < arctan x < x x3 3 + x5 5. ( ),., f(x) = arctan x ) (x x3 3 + x5 5 x7 7 f(x) > (x > ).. f (x) = + x ( x + x 4 x 6 ) = x8 + x > x. f(x). x >, f(x) > f() =.,. g(x) = x x3 3 + x5 5 arctan x g (x) = x + x 4 + x = x6 + x > x. g(x). x >, g(x) > g() =. < x π x x3 x3 < sin x < x 3! 3! + x5 5!.,.

67 f(x) [, ] (, ) f() = f() =. () g(x) = f(x)e x. g (c) = c < c <. () α f (c) = αf(c) c < c <. () f(x), e x [, ] (, ) g(x). g() = g() =. g (c) = c < c <. () g(x) = f(x)e αx () g (c) = c < c <. g (x) = f (x)e αx αf(x)e x = (f (x) αf(x))e x g (c) =, f (c) αf(c) =. f(x) (, ) lim f(x) = lim f(x) = x x. α f (c) = α c < c <. [ : y = f(x) (x, y ) y = α(x x ) + y.] 4.6 () y = arcsin x. () (arcsin x). (3) arcsin x x = arcsin x = a n x n n= (A) a n n. (4) (A) 5 f(x) = a n x n (5 ) n=. arcsin f() 3.

68 68 4 () y = arcsin x y = sin x π/ x π/. y = x. () y = arcsin x x = sin y. x = y cos y. (arcsin x) = y = cos y. π/ y π/ cos y cos y = sin y = x. (arcsin x) = x. (3) (arcsin x) = x = ( x ) / = x arcsin x = ( ) / ( x ) n = n n= ( / )( ) n xn+ n n +. n= ( ) / ( ) n x n. n. ( ) / = ( ) ( ) ( n n! )... ( ) (n ) n= = ( )n n n! (n ) = ( )n (n )!!. (n)!!

69 69 arcsin x = (4) (3) n= f(x) = ( ) n (n )!! (n)!! n= (n )!! (n)!! ( ) n xn+ n + = x n+ n= (n )!! (n)!! n + = x + x x5. f() = + 9 = x n+ n +. arcsin f() = π f() =.57.4 = () [, ] f(x) f() = f() f (c) = c < c <. () [, ] f(x). (). f(x) = x. f() = f() =. x / f(x) ±. x = / f(x). f (c) = c < c <. () f(x) = { x x. : = x < x < < x n = [, ]. ξ,..., ξ n. n S( ; {ξ k }) = f(ξ k )(x k x k ) k=.,.,... f(x).

70 () {a n } lim n a n+ a n =.. () x sin x x = ( ).. (3). x log x lim x x (4) y = arcsin x. (5). π/3 dx cos x () a n = n lim a n+ a n = lim n n n + =. lim a n = n n= n = : a n = log n, a n = n. () f(x) = sin x f (x) = cos x, f (x) = sin x, f (3) (x) = cos x, f (4) (x) = sin x. n =,,,... f (4n) (x) = sin x, f (4n+) (x) = cos x, f (4n+) (x) = sin x, f (4n+3) (x) = cos x. f (4n) () =, f (4n+) () =, f (4n+) () =, f (4n+3) () =.

71 7 f(x) = sin x { f (4n) () sin x = x 4n + f (4n+) () (4n)! (4n + )! x4n+ + f (4n+) () (4n + )! x4n+ + f } (4n+3) () (4n + 3)! x4n+3 = = n= { n= m= (4n + )! x4n+ + ( ) m (m + )! xm+. (3). x sin x = x log x x log x + x lim = lim x x x x log } (4n + 3)! x4n+3 m= ( ) m (m + )! xm+. log x + 3 = lim x x (log ) = lim /x x x (log ) =. 3 (4) y = arcsin x y = sin x π/ x π/. y = arcsin x x = sin y. x = y cos y. (arcsin x) = y = cos y. π/ y π/ cos y cos y = sin y = x. (arcsin x) = x. (5) sin x = dx + t cos x = t + t dt = ( = + t + ) t t = tan x t t, cos x = + t + t, dx = + t dt. t dt dt = log + t log t = log + t t

72 7 4 t = tan x x π/3 t tan π 6 = / 3. π/3 [ dx cos x = log ( ) + t t ] / 3 π/3 dx π/3 cos x = s = sin x ds = cos x dx π/3 log( + 3) = log 3/ dx cos x = ds s = = [ log + s s = log + / / 3 = log. = log( 3 + ). 3 cos x π/3 cos x dx = 3/ ] 3/ cos x sin x dx. ( + s + s = log + 3/ 3/ ) ds = log = log( + 3) = log( + 3) = log = log( ) = log( 3 + ) log 4.8 α = arctan 5 tan α, tan 4α., 4 arctan 5 arctan 39 = π 4. tan α = 5 tan α = tan 4α = ( tan 4α π ) = 4 tan α tan α = /5 (/5) = 5. tan α tan α = 5/6 (5/) = 9. tan 4α tan(π/4) + tan 4α tan(π/4) = = 39.

73 73. 4α π 4 = arctan 39. arctan x π.

74

75 x =. ( n..) () y = sinh x cosh x () y = x + x (3) y = arccos x (4) y = (+x) /3 ( x) /3 () e x = y = sinh x cosh x = 4 (ex e x ). n= x n n! ( y = (x) n 4 n! n= ) ( x) n. n! n= x (), y = 4 n= (x)n+ (n + )! = 4 n (n + )! xn+. n= y = x + x = + x = x. x = x n n= y = x = x n. (3) (arccos x) = = ( x x ) /, (arccos x) = n= ( ) / ( x ) n. n n=

76 76 5 arccos = π/ arccos x = π ( ) / ( ) n n n + xn+. n= ( ) / ( /)( / ) ( / n + ) = n n! ( )( ) ( n + ) = n n! = ( )n (n )(n 3) 3 = n n! x x arccos x = π n= = ( )( 3) ( n + ) n n! ( ) n (n)! n n!(n)(n ) = ( )n (n)!. n (n!) (n)! n (n!) x 4n+ n +. (4) y = ( x ) /3. ( ) /3 ( ) /3 y = ( x ) /3 = ( x ) n = ( ) n x n. n n n= f (n) (). n= 5. x = ( ). f(x) = x g(x) = log(x ) h(x) = cos πx f(x) f(x) = a n (x ) n n=. a n = f (n) () n!. (). ( ) f(x) = x = ((x ) + ) = (x ) n. n n=

77 77 (), g(x) = log(x ) = log x = log( + (x ))., x + x = ( x) n. n= ( ) n x n+ log( + x) = n + n= ( ) n = x n. n n= (3), ( ) n g(x) = (x ) n. n n= cos πx = cos(π(x ) + π) = cos π(x )., cos x : cos x = n= ( ) n (n)! xn. ( ) n π n h(x) = (x ) n = (n)! n= ( ) n+ π n (x ) n. (n)! n= 5.3 n R n (x). () R n (x) = ( ) n x arctan x = x x3 3 + x5 xn + + ( )n 5 n + R n(x) t n dt. + t () x R n(x) 4 n+ n. (3) arctan + arctan 3 = π 4 π 4. arctan x = x x3 3 + x5 xn + + ( )n 5 n + R n(x)

78 78 5 x + x = x + x ( ) n x n + R n(x). R n(x) = + x ( x + x ( ) n x n ) = + x ( x ) n ( x ) = ( )n x n + x. R n (x) R n () = ( ) n x t n + t dt. R n (x) x = R n () =. () () x x R n (x) = t n ( )n + t dt x t n + t dt. x, x x R n (x) = t n ( )n + t dt x t n + t dt. t x / (3). R n (x) R n (x) t n + t tn x t n dt = x n+ n + n+ (n + ) n+ n = 4 n+ n. S n (x) = x x3 3 + x5 xn + + ( )n 5 n π = 4 arctan + 4 arctan 3 = 4S n ( ) ( ) ( ) ( ) + 4R n + 4S n + 4R n 3 3

79 79 () 4R n ( ) ( ) + 4R n n+ n = n n. < 5, π = 4S n ( ) + 4Sn ( 3) 4. n n < 5 (n ) log + log n > 5 n log =.3. n = , n = n = 8., A n = 4S n ( ) + 4S n( 3 ) :, n 6 4. n R n (x). () R n (x) = ( ) n x () ( ) n= n xn A = A = A 3 = A 4 = A 5 = A 6 = A 7 = A 8 = A 9 = log( + x) = x x + x3 xn + + ( )n 3 n + R n(x) (5.3.) n t n dt. + t < x log( + x). (3) = ( ) n = log. n n= (4) x x >, ( ) n= n xn n.

80 a. (x a) n n. () cosh x + sinh x (a = ) () arctan x x (a = ) (3) x (a = ) () cosh x + sinh x = e x = e e (x ). e x cosh x + sinh x = e ((x )) n e n = (x ) n. n! n! (), arctan x x n= (arctan x) = + x = ( x ) n = arctan x = n= n=. arctan x = ( ) n x x n + (x)n+ = x n= n= n= ( ) n x n. n= ( ) n n + xn+. ( ) n n+ n + (3). x = (x + ) = { (x + )} / = n= x n+ = n= ( ) / ( ) / ( (x + )) n = ( ) n (x + ) n. n n n= ( ) n n+ n + (x + ) n n =, n ( ) ( ) ( ) ( ) ( ) / ( ) n... n + = ( ) n n n! ( ) ( ) ( + ) +... ( ) + n = n! (n 3) (n)! = = n n! ( (n)) n n! = (n)! n n!n!. x = n= (n)! n n!n! (x + )n. x n.

81 8 5.5 f(x) x = a (x a) n c n. (.) () log(x + ), a = () sinh x x, a = (3) (x + ) cos x, a = π (4) 5x 4 x 3 + x x +, a = (), g(x) = log(x + ). g (x) = (log(x + )) = x + = (x + ). g (n+) (x) = ( (x + ) ) (n) = ( )( )... ( n)(x + ) (n+) = ( ) n n!(x + ) (n+). log(x + ) = n= g (n) () n! x n = + n= ( ) n n! (n + )! xn+ = ( ) n x n. n n= x x log(x ( ) n + ) = x n. n n= (),. e x = n= sinh x = (ex e x ) = x n n! x n ( x) n. n! n x n ( x) n = n = m + n= x n ( x) n = x m+. sinh x x = x sinh x = m= m= x m+ (m + )!. x (m+) (m + )! = x 4m (m + )!. m=

82 8 5 (3) (x π) n y = x π y. ( f (n) (π).) (x + ) cos x = ((y + π) + ) cos (y + π) = (y + π + ) cos y. cos z. y = x π cos z = (x + ) cos x = (y + π + ) (x + ) cos x = = = m= m= m= ( 4) m (m)! m= ( ) m (m)! zm. m= ( ) m (m)! (y)m+ + ( 4) m (m)! y m+ + (x π) m+ + ( ) m (m)! (y)m m= ( ) m (π + ) (m)! (π + )( 4) m y m. (m)! m= (4) f(x) = 5x 4 x 3 + x x +. (y) m (π + )( 4) m (x π) m. (m)! m= f (x) = x 3 6x + x f (x) = 6x x + f (3) (x) = x f (4) (x) =. f( ) = 5x 4 x 3 + x x + = f ( ) = 9 f ( ) = 74 f (3) ( ) = 3 f (4) ( ) =. f(x) = 4 n= f (n) ( ) n! (x + ) n = + ( 9)(x + ) + 74! (x + ) + 3 (x + ) 3 + (x + ) 4 3! 4! = 9(x + ) + 37(x + ) (x + ) 3 + 5(x + ) 4.

83 f(x) x = a. (.) () sin x, a = () cos x x, a = (3) x, a = (4) 6x 5 3x 4 x +, a = e. ( p.84.) () e x e = n k= k! + e θ (n + )! (5.6.) < θ <. () e e = q. p, q. () p e θ n = p p +. (3) e < 3 eθ p +. () (e x ) = e x (e x ) (n) x= =. e x e x = n k= k! xk + eθx (n + )! xk+ < θ = θ(x) <. x = (5.6.). () (5.6.) e = q p, n = p p!. e θ p + = q p p! q p = p k= k! + eθ (p + )!. p p! p k! = q(p )! p(p )... (p k + ). k= k=

84 84 5 (3) e < 3, < θ < < e θ < e < 3., p < eθ p + eθ < 3 e θ, p +. eθ = p +, e θ = p =. e = q p = q e. (5.6.) e > + + n k= k! > e < 3 < e < 3. e. 5.7 () = n n= = n n= () S n S n = ( )n n. = n ( ) k k= k S < S 4 < < S n < < S n+ < < S 3 < S (3) () lim n S n. () : n= n = , {}} n { n + + n n n n =.

85 85 n k= k = + + ( ) ( ) ( ) ( + 8 n {}}{ = + n. n n ) n= n n = n= n= n = +, n= n = n= n = +. () S n < S n. S n = S n + n n = S n + n(n ) > S n. S n+ < S n. S n < S n+.. S n+ = S n + n + > S n (3) {S n }, {S n+ }. lim S n = α, n lim S n+ = β n. S n < α β < S n+. β α < S n+ S n = n + n α = β. lim n S n = α., ϵ > N, N S n α < ϵ, n N ; S n+ α = S n+ β < ϵ, n N. N = max{n, N + } lim n S n = α. S n α < ϵ, n N.

86 c y = cx y..,. y (a, b), a,. b = ca c = b/a y = (b/a )x. ( ) c y = cx y. y, y c = x c c. y = cx (a, ca ) y = cx ca. y = f(x) ca = f(a), f (a) = ca. c c. f (a) = ca = a ca = a f(a). a. a x f(x)f (x) = x (f(x)) f(x) = x + C. x + y = C C:.

87 87 c 4x + y = c.,. 5.9 arctan x π 3.., x (arctan x) = + x = ( x ) n = arctan x = n= n= ( ) n x n. n= ( ) n n + xn+. tan θ = x. tan π/6 = / 3 π 6 = ( ) ( ) n n+ 3 = ( ) n n + n= 3 n + 3. n n= π = 3 ( ) n n + 3. (5.9.) n n= π.

88 88 5. (5.9.) : ( ) n n + 3 = n n= ( = 3 ) ( ) ( = 4k + 3 ) k 4k k+ k= k= >. S n = n ( 3 4k + π : 3 k 4k + 3 S < S < S < π. S = , S = π > 3.3. ( ) ) 3 k+ n 3 4 S n x =. x x sin πx log x (). x = + (x ) = / + (x )/ = ( x ) n = n= ( ) n (x ) n = n= (). n= x = x x + = + (x ) 3 + (x ) ( ) n n+ (x )n. = + (x ) /3 + (x )/3 = ( (x )) n 3 n= n= } ( = {( ) n ( )n (x ) n = ( ) n n= 3 n+ n= 3 n+ ( x 3 ) (x ) n. ) n

89 89 (3), sin πx = sin(π(x ) + π) = sin π(x ). sin x. (4), sin πx = n= ( ) n (n + )! (π(x ))n+ = log x = log{ + (x )} = log log( + x) n= ( ) n π n+ (x ) n+. (n + )! ( + x ) ( = log + log + x ). ( ) n log x = log + n n= ( ) n x = log + ( ) n (x ) n. n n n= 5. (). () (). n ( ) k xk+ k + k= n + t = ( t) k + ( t)n + t k= = log( + x) + ( )n+ (3) (). x t n + t dt, x > () : n ( t) k = ( t)n ( t). k=. () () x >. x n dt + t = k= x ( t) k dt + x ( t) n + t dt

90 9 5. (3) () x = n log( + x) = ( ) k xk+ k + + n ( ) k k + k= k= = log + ( )n+ x ( t) n + t dt. t n + t dt. t n ( )n+ + t dt t n + t dt n ( ) k = log. k + k= t n dt = n + n.

91 x + () x arctan x dx () dx (3) x 3 dx e x () x arctan x dx = x x arctan x = x arctan x + x dx ( + x = x arctan x (x arctan x) = (x arctan x + arctan x x) ) dx (). x + x 3 = x + (x )(x + x + ) = a x + bx + c x + x +. x x = a =. bx + c x + x 3 = x x x + x +. (x + x + ) = x + x + x + x 3 = x = x x + x + x + + x + x + x + x + x + + (x + ) x 3 dx = log(x ) log(x + x + ) + x = log x + x + + arctan x arctan x + / 3/

92 9 6. (3) e x = t e x dx = dt. dx e x = dx x + A = A arctan x A dt t(t ) = ( t + ) dt t = log t + log(t ) = x + log(e x ). = log t + log(t ) = log t t ( = log ) = log( e x ) t. 6. ( ). () x 3 (x + )(x + )(x + 3) () (x a)(b x) (3) + a cos x ( < a < ) (), 3 x x 3 (x + )(x + )(x + 3) = (x + )(x + )(x + 3) 6x x 6 (x + )(x + )(x + 3) = + A x + +. B x + + C x + 3 A + B + C = 6, 5A + 4B + 3C =, 6A + 3B + C = 6. A =, B = 8, C = 7. x 3 (x + )(x + )(x + 3) = / x x + 7/ x + 3. x 3 (x + )(x + )(x + 3) dx = x log x log x + 7 log x + 3.

93 (). a < b [a, b], b < a [b, a]. a < b. a > b (x a)(b x) = a, b. (x b)(a x) ( ). a < b b x >. = b x (x a)(b x) b x x a.. x = at + b t +, dx = (x a)(b x) t + (b a)t t = b x x a (b a)t b x = t +, (a b)t t (t + ) (a b)t dx = (t + ) dt dt = dt t + = arctan t. 93 ( ) arctan dx b x = arctan (x a)(b x) x a. (6..) arctan x + arctan x = π arctan ( ) b x π x a x a x a = arctan = π + arctan b x b x.. ( ), dx (x a)(b x) = arctan (x a)(b x) = x + (a + b)x ab = = (a b) 4 (a b) 4 { ( x a + b x a b x ) ( x a b a + b a b ) }

94 94 6 a < b dx = (x a)(b x) b a dx ( x ). a+b a b a b dx = (x a)(b x) y = x a b dy = a b dx ( dy ( ) = arcsin y a + b ). y a+b a b a b dx (x a)(b x) = arcsin ( x a b a + b ) = arcsin a b arcsin ( ) dx x a b = arcsin. (x a)(b x) b a ( ) arcsin ( ) x a b arcsin b a arcsin x + arccos x = π = π ( ) x a b arccos b a. ( ) dx x a b = arccos (x a)(b x) b a ( x a b a b.... (3). ). t = tan x. + t = cos x = + cos x cos x = t + t.

95 95, dt = cos x dx dx = dt + t. dx + a cos x = +t + t a dt = +t ( + a) + ( a)t dt = + a dt + a +a t. dx + a cos x = + a s = a + a t + a + s a ds = arctan s. a ( ) dx + a cos x = a arctan a + a tan x. () : ( ) b x x a b arctan x a arcsin b a. ( ). dx = log( x a + x b) (x a)(x b) dx (x a) + b = log(x a + (x a) + b ) a x dx = a x x + a log x a + a x x a 3 x dx = ( x ) 3/ 3 3 arcsin a 6.3 ( ). sin x () () sin log x (3) + sin x + cos x x x +

96 96 6 () t = tan x sin x = t + t cos x = t + t dx = + t dt. sin x + sin x + cos x dx = = = t +t + t + t + t dt +t +t t + t + t + t t ( + t)( + t ) dt. + t dt t ( + t)( + t ) = + t + + t + t = + t + ( + t ) + + t + t sin x + sin x + cos x dx = log + t + log + t + arctan t. = log + tan x + log cos x + x = log + tan x log cos x + x ( = log + tan x ) cos x + x = log cos x + sin x + x. sin x + sin x + cos x dx = log x cos + sin x + x. () sin log x dx = x sin log x x cos log x x dx = x sin log x cos log x dx = x sin log x x cos log x + x( sin log x) x dx = x sin log x x cos log x sin log x dx.

97 97 sin log x dx = x sin log x x cos log x. sin log x dx = x (sin log x cos log x). (3) x + x = t, x = t, dx = t dt. t t x + = x + t = t t + t = + t t dx x x + = t ( dt = t +t t t dt = + t + ) dt t t t = log + t + log t = log t. + t dx x x + = log + x x + x +. x ( () lim n n + + n ) n ( ) () lim n n + n + n + n + + n + n [, ] : < /n < /n < < n/n = [(k )/n, k/n] k/n f(x)dx = lim n n n k= ( k f n)..

98 98 6 () () ( lim n n + + n ) n ( = lim n n ) + n n n n dx [ ] = + x = log + x = log. lim n lim n n = ( ) n + n + n + n + + n + n n + n + n n dx + x = [ + x ] = () π sin nx sin x dx () log( + x) dx (3) x(arctan x) dx ()., I n = π sin nx sin x dx I =, I = π (6.5.). I = π sin x sin x dx = π sin x cos x sin x dx = π cos x dx =. (6.5.)

99 99, n 3. sin nx sin x sin(n )x cos x + cos(n )x sin x = sin x sin(n )x = cos x + cos(n )x sin x sin(n )x cos x + cos(n )x sin x = cos x + cos(n )x sin x sin(n )x = cos x + cos(n )x cos x + cos(n )x sin x sin(n )x = ( sin x) + cos(n )x cos x + cos(n )x sin x sin(n )x = sin(n )x sin x + cos(n )x cos x + cos(n )x sin x sin(n )x = + cos(n )x. sin x I n = π ( sin(n )x sin x (6.5.), (6.5.) I n = ) + cos(n )x dx = I n, n 3. { n π n. () t = x t = x, dx = tdt. I = log( + x) dx = log( + t) tdt = [ ] t log( + t) t ( I = log dt = log t + ) dt + t + t [ t ] ( ) = log t + log + t = log + log =. (3) I = x(arctan x) dx [ x ] = (arctan x) = π 3 = π 3 x + x arctan x dx arctan x dx + x arctan x + x dx arctan x dx. + x t + t dt.

100 6 x = tan θ, I = = π/4 I = I = arctan x dx arctan x = θ, dx = cos θ dθ. [ θ tan θ θ [ cos θ dθ = ] π/4 = π 4 + [ log cos θ π/4 ] π/4 ] π/4 θ tan θ sin θ cos θ dθ = π 4 + log = π 4 log I = [ (arctan x) ] arctan x dx + x π/4 tan θ dθ π arctan x dx = + x 6 I. I = π 3. ( I = π 3 I + I = π π 3 4 ) log + π 3 = π 6 π 4 + log. ( ) I. I = arctan x dx = = π 4 [ log( + x ) [ ] x arctan x ] = π 4 log. x + x dx 6.6. π/3 sin x + cos x + sin x + cos x dx

101 t = tan x sin x = t t, cos x = + t + t, dx = + t dt. sin x + cos x t + sin x + cos x dx = + t + ( (t + )(t + ) dt = t + ) dt t + = arctan t log x + = x log tan x. + π/3 sin x + cos x [ + sin x + cos x dx = x log tan x ] π/3 + = π ( ) 3 log f(x). () f (x). (). f(x) = arctan x + x + arctan + x x (3) lim f(x), x xf(x) dx. lim f( x), lim x + f(x) x + () (arctan x) = + x. f (x) = = ( x ) +x ( x +x ) + + ( +x ) x ( +x x ) + = (+x)+( x) (+x) ( x +x ) + + ( x) + ( + x) + ( + x) + ( x) =. ( x)+(+x) ( x) ( +x x) + (), lim f(x) = x ( lim arctan x x + + arctan x + ) x x = lim x (arctan( ) + arctan( )) = π.

102 6, ( lim f( x) = lim arctan + x ) x + arctan = lim x + x + x + x arctan t = π t. ( lim arctan x x + + x + arctan + x ) = arctan + arctan = π x. (3) f (x) = f(x) (, ), (, ), (, ). π/ < x <, f(x) = π/ < x <, π/ < x < +. f(x) [, ]. xf(x) dx = xf(x) dx : 6.7 xf(x) dx = π π ( ) = π. ( ) f (x) = [ x ] xf(x) dx = f(x) x [ x f ] (x) dx = f(x). (.) 6.8. I = sin(log x) dx () I I. () I.

103 () log x (, ] x =. lim x + log(x) = log x [, ]. I [, ] I = lim ϵ + ϵ sin(log x) dx.. () sin(log x) dx. 3 sin(log x) dx = x sin(log x) = x sin(log x) = x sin(log x) x{sin(log x)} dx x cos(log x) x dx cos(log x) dx (6.8.) cos(log x) dx = x cos(log x) = x cos(log x) = x cos(log x) + x{cos(log x)} dx x{ sin(log x)} x dx sin(log x) dx (6.8.) ( ) sin(log x) dx = x sin(log x) x cos(log x) + sin(log x) dx = x sin(log x) x cos(log x) sin(log x) dx. sin(log x) dx = x {sin(log x) cos(log x)}. I = lim ϵ + ϵ [ ] x sin(log x) dx = lim {sin(log x) cos(log x)} ϵ + ϵ = lim ϵ {sin(log ϵ) cos(log ϵ)} (6.8.) ϵ + ϵ {sin(log ϵ) cos(log ϵ)} ϵ { sin(log ϵ) + cos(log ϵ) } ϵ

104 4 6, (6.8.), lim ϵ + ϵ {sin(log ϵ) cos(log ϵ)} =. I =. ( ) t = log x. x = e t, dx = e t dt I = lim ϵ + ϵ sin(log x) dx = lim (sin t)e t dt = ϵ + log ϵ lim L L e t sin t dt. e t sin t dt = e t sin t e t cos t dt ( ) = e t sin t e t cos t e t ( sin t) dt = e t sin t e t cos t e t sin t dt. I = e t sin t dt = et (sin t cos t). [ ] e t lim (sin t cos t) = L L () π sin nx sin x dx () log( + x) dx (3) x(arctan x) dx ().,. I = π sin x sin x dx = I n = π sin nx sin x dx I =, I = π (6.9.) π sin x cos x sin x dx = π cos x dx =. (6.9.)

105 5, n 3. sin nx sin x sin(n )x cos x + cos(n )x sin x = sin x sin(n )x = cos x + cos(n )x sin x sin(n )x cos x + cos(n )x sin x = cos x + cos(n )x sin x sin(n )x = cos x + cos(n )x cos x + cos(n )x sin x sin(n )x = ( sin x) + cos(n )x cos x + cos(n )x sin x sin(n )x = sin(n )x sin x + cos(n )x cos x + cos(n )x sin x sin(n )x = + cos(n )x. sin x I n = π ( sin(n )x sin x (6.9.), (6.9.) I n = ) + cos(n )x dx = I n, n 3. { n π n. () t = x t = x, dx = tdt. I = log( + x) dx = log( + t) tdt = [ ] t log( + t) t ( I = log dt = log t + ) dt + t + t [ t ] ( ) = log t + log + t = log + log =. (3) I = x(arctan x) dx [ x ] = (arctan x) = π 3 = π 3 x + x arctan x dx arctan x dx + x arctan x + x dx arctan x dx. + x t + t dt.

106 6 6 x = tan θ, I = = π/4 I = arctan x dx arctan x = θ, dx = cos θ dθ. [ θ tan θ θ [ cos θ dθ = ] π/4 = π 4 + [ log cos θ π/4 ] π/4 ] π/4 θ tan θ sin θ cos θ dθ = π 4 + log = π 4 log I = [ ] I = (arctan x) arctan x dx + x π/4 tan θ dθ π arctan x dx = + x 6 I. I = π 3. ( I = π 3 I + I = π π 3 4 ) log + π 3 = π 6 π 4 + log. ( ) I. [ I = arctan x dx = x arctan x = π 4 [ log( + x ) ] ] = π 4 log. x + x dx 6.. I = sin(log x) dx () I I. () I.

107 () log x (, ] x =. lim x + log(x) = log x [, ]. I [, ] I = lim ϵ + ϵ sin(log x) dx.. () sin(log x) dx. 7 sin(log x) dx = x sin(log x) = x sin(log x) = x sin(log x) x{sin(log x)} dx x cos(log x) x dx cos(log x) dx (6..) cos(log x) dx = x cos(log x) = x cos(log x) = x cos(log x) + x{cos(log x)} dx x{ sin(log x)} x dx sin(log x) dx (6..) ( ) sin(log x) dx = x sin(log x) x cos(log x) + sin(log x) dx = x sin(log x) x cos(log x) sin(log x) dx. sin(log x) dx = x {sin(log x) cos(log x)}. I = lim ϵ + ϵ [ ] x sin(log x) dx = lim {sin(log x) cos(log x)} ϵ + ϵ = lim ϵ {sin(log ϵ) cos(log ϵ)} (6..) ϵ + ϵ {sin(log ϵ) cos(log ϵ)} ϵ { sin(log ϵ) + cos(log ϵ) } ϵ

108 8 6, (6..), lim ϵ + ϵ {sin(log ϵ) cos(log ϵ)} =. I =. ( ) t = log x. x = e t, dx = e t dt I = lim ϵ + ϵ sin(log x) dx = lim (sin t)e t dt = ϵ + log ϵ lim L e t sin t dt = e t sin t e t cos t dt ( ) = e t sin t e t cos t e t ( sin t) dt = e t sin t e t cos t e t sin t dt. L e t sin t dt. I = e t sin t dt = et (sin t cos t). [ ] e t lim (sin t cos t) = L L. 6. t t π/ (cos 3 t, sin 3 t).. x = cos 3 t, y = sin 3 t π/ L = (x ) + (y ) dt. (x ) + (y ) = (3 cos t( sin t)) + (3 sin t cos t) = 9 cos 4 t sin t + 9 cos t sin 4 t = 9 cos t sin t(cos + sin t) = 9 cos t sin t. L = π/ 3 cos t sin t dt = 3 π/ sin t dt = 3 [ ] π/ cos t = 3.

109 9 6.. / / x 4 + x 3 + 3x + x (x + )(x + ) dx. x 4 + x 3 + 3x + x (x + )(x + ) = x + x (x + )(x + ), x (x + )(x + ) = A x + + Bx + C x +. x + x = x = A A = x + x=. Bx + C x + = x (x + )(x + ) / x + = (x ) x 4 + x 3 + 3x + x (x + )(x + ) = x + (x + )(x + ) = x + + x x + (x ) x + x +. / /. 3. / / x 4 + x 3 + 3x + x (x + )(x + ) dx = / / dx x + / dx / x +., / / x + dx = [ ] / log(x + ) = ( log 3 / log ) = log 3.,, / / / x + dx = [ arctan x ] / = arctan. / x 4 + x 3 + 3x + x (x + )(x + ) dx = log 3 arctan.