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4................................. 4................................. 4 6................................. 6................................. 9.................................................... 3..3.......................... 4..4............................ 5.3............................... 6.3........................ 7 3 9 3............................. 9 3........................... 3................... 3................... 4 4................................. 4....................... 4.. (ctenry)................... 4 4. Newton............................. 6 4.3 e.............................. 7 4.3........................... 7 4.3............................. 8 4.4........................... 8 4.4............................ 9 4.4........................ 3 4.4.3 e..................... 3 4.5............................ 3

4.5............................ 3 4.5............................ 3 4.5.3...................... 3 4.5.4...................... 33 4.6....................... 34 5 35 5............................ 35 5.......................... 37 5............................. 38 5......................... 39 5..3................. 39 5.3.......................... 4 5.4.................... 4 5.4............................. 4 5.4........................ 4 5.4.3 π ( )................. 4 5.5....................... 43 5.5. l Hopitl...................... 44 5.6...................... 46 5.7.............................. 47 5.7....................... 48 5.8........................... 5 5.9 Riemnn zet........................ 5 6 55 6................................. 55 6................................ 55 6.3.............................. 55 6.4...................... 56 6.5........................... 56 6.6............................ 57 7 ε δ 59 7............................. 59 7........................... 6 7.3................................ 6 7.4............................... 6 7.5.............................. 65 7.6................................ 67

3 7.7.............................. 69 7.7.................. 69 7.8 Riemnn zet........................ 74 7.9 n npn.................... 76 7.9. p....................... 76 7.9. > p..................... 77 7.9.3 > p >.................... 77 8 78 8................................. 78 8..................... 84 8....................... 85 8........................... 86 8.3.............................. 88 8.4 Γ............................... 9 8.5 B............................... 9 8.5. e x........................ 9 8.6............................ 96 8.6............................ 97 8.6............................ 98 8.6.3...................... 98 8.6.4...................... 99 8.7............................

4. N R Q C x A x A x A A x A x A, B A B A B B A A B x A x B A B A B A B, A B Λ λ Λ A λ λλ A λ λ Λ A λ λλ A λ Λ N n A n n A n A A A c.

.. 5 ( ) : : : s.t., > s.t.,

6. A x A x A A x.. {, 4, 6, 8,...}. {n: n N} 3. {3n + 5m: n, m Z} 4. {x: x Z} 5. {(x, y): x, y Z, x + y < } (, ), (, ], (, ]. (, ) {x: < x < }, (, ] {x: < x }, (, ] {x: x } A A A B x A x B A B x A x B A\B x A x B

.. 7 A c x A A B x A x B A B A B A B A B A B A B A B x {x} { } λ Λ A λ λ Λ x A λ x {(x, y)} {(x, y): x, y, x + y } x y x λ Λ A λ λ Λ x A λ x {3n + 5m: n, m Z} Z 3. (A B) C (A C) (B C). (A B) C (A C) (B C) 3. (A B) c A c B c 4. (A B) c A c B c 5. (A c ) c A. 4 x (A B) C) x A B x C (x A x B) x C (x A x C) (x B x C) x A C x B C x (A C) (B C)

8.: 4 4 [ n n], (, ) n n ( n, + ) n [, ] R, C, N, Z, Q 5 (Russel prdox). S S c S S S S S S A, B A B {(, b): A, b B} R R R. A {, }, B {, } A B {,,, } A B {(, ), (, ), (, ), (, )}

.. 9. lim sup A n n lim inf n A n n k n n k n A k A k. f : V W V x f(x) W V W A V f(a) {f(x): x A} W B W f (B) {x V : f(x) B} V.3. f(x) x R R R R + R + R R + R +. f(x) + x + x + g(x) x f x < g R\{} 6 f(x) x (, ) [, ). [, 4) (, ] [, ) 7 f(x, y) x + y f (, ), (, ), (, ), (, ) R [, ]. f(r) [, ] f ([, ]) {(x, y): x + y }( )

8 f : (x, y) (r, θ) (, ), (, ), (, ) R. (, ) (/ cos θ, θ) (, π/4) 9 ( ) A, B. y x A. xy B (,, ).. y 3x y x. x y + z x y z ( ) A, B. A xy (, ). B R xy.. R x y z. x + y 4z y x f : V W g : W U (g f)(x) g(f(x)) f g : V U f : V V f(x) x

.. R n f(x) x g(x) x 3 + f g f g. f R + g g [, ) f g R.. f : X Y A, A X B, B Y. f(a A ) f(a ) f(a ). f(a A ) f(a ) f(a ) 3. f (B B ) f (B ) f (B ) 4. f (B B ) f (B ) f (B ).. y f(a A ) x A A s.t. f(x) y x A nd x A y f(x) f(a ) nd y f(x) f(a ) y f(a ) f(a ). y f(a A ) x A A s.t. f(x) y x A or x A y f(x) f(a ) or y f(x) f(a ) y f(a ) f(a ) y f(a ) f(a ) y f(a ) or y f(a ) x A or x A, x A A y f(x) f(a A )

3. x f (B B ) f(x) B B, f(x) B nd f(x) B x f (B ) nd x f (B ) x f (B ) f (B ) x f (B ) f (B ) x f (B ) nd x f (B ) f(x) B nd f(x) B f(x) B B x f (B B ) 4. x f (B B ) f(x) B B, f(x) B or f(x) B f(x) B B x f (B B ) x f (B ) f (B ) f(x) B or f(x) B f(x) B B x f (B B ) f(a A ) f(a ) f(a ) y f(a ) f(a ) y f(a ) x A s.t. y f(x ) y f(a ) x A s.t. y f(x ) x x f(x ) f(x ) x x x x A A y f(x ) f(x ) f(a A ) f(a A ) f(a ) f(a ) f(a A ) f(a ) f(a )

.. 3.. ( ): ( ): f(x) f(x ) x x y f(v ) f ({y}) ( ): f(v ) W 3 sin x, cos x, tn x, e x. sin x: [ π, π ] [, ] cos x: [, π] [, ] tn x: ( π, π ) (, ) e x : R (, ) rcsin x, rccos x, rctn x, log x 4 [, ] [, ] (, ] [, ) (, ] [, ) N Q N Z [, ] (, ] S \{} R 5 f : X Y f : Y X

4 V, W to onto #V #W {,,..., n} to onto n N [, ] X #X..3 b.. b b 3. b, b c c X {y X : y x} x X λ Λ A λ. x, y A λ x y. x A λ y A µ λ µ x y A λ x λ A λ 6 R x y x y Z [, ). #X #X X X. #X #Y #Y #X f : X Y f 3. #X #Y #Y #Z #X #Z f : X Y g : Y Z g f : X Z

.. 5..4 X X X X X A A X {, } N N f : N N N x x x x... yy i i x i j if i j yj i i j nd x i j i j nd x i j y y y y x i x i i x i x i x i x i, Cntor [, ] [, ] [, ] x x x x.. i j yj i xi j i j xi j 9 yy i i y y

6. (Cntor ) X X. f : X X X X Y {x X : x f(x)} f(x) Y x X. x Y x Y f(x) Y. x Y x Y f(x) x f(x) x Y ℵ ℵ ℵ i ℵ i+ Gödel) 93.3 f : X Y to onto f : Y X 7 f : R R x, e x, sin x, cos x, tn x 3 rcsin x, rccos x, rctn x 8 rcsin x + rccos x π. X, Y R f f f

.3. 7. x < x < x [x, x ] [x, x ] f(x ) f(x ) to f(x ) < f(x ) y (f(x ), f(x ) x < z < x, x < z < x f(z ) f(z ) y to f y n f() f(y n ) > + ε x n f(y n ) x n x + ε f f(x ) f() to 9 f(x) x 3 3x x +. (, ] (, ] (, ) (, 8] [ 9, 8) ( 9, ).3..3 f f () b f() (f ) (b) f () f (f (b)). :f (x) y x f(y) f (y) dy dx :f ε > h > s.t. h h f () f( + h) f() h b + η f( + h) η f (b + η) f (b) η < ε + h f( + h) f() f () + θε η.4 rcsin x: [, ] [ π, π ] rccos x: [, ] [, π]

8 rctn x: (, ) ( π, π ) ABC(C AB ) x BC ABC rccos x BAC rcsin x rcsin x + rccos x π y rcsin x sin y x cos y dy dx dy dx x y rccos x cos y x sin y dy dx dy dx x y rctn x tn x y dy cos y dx dy dx + x.5 e x : (, ) (, ) log x: (, ) (, ) y log x e y x e y dy dx dy dx x

9 3 3. 3.. xb x (x ) 3. n n! (n ) 4. > log x x (x ) 5. r < lim n n k r n 6. lim x x k e x. x. b x x b. k > b x e x log (x log ) n n! n xk (log ) k k! x b x k! xb k (x ) (log ) k 3. n > n (n ) n 4. x e u (u log x) n ( ) n n n }{{} log x x u e u : log n n (log x x ) x x x x

3 5. n n k r n x f + f n+ n ( ) k n + r n n n ( ) n+ k n r < r < n ( n ) n C r n C 6. 5 x n, r /e n f(x) x k e x f (k) f(k) k k e k < x k e x x k+ e k x (k + )k+ x x 3. 3.. f(n) g(n) lim n f(n) g(n) f(n) o(f(n)) g(n) f(n) O(f(n)) Lndu n k O(n k ), o(n k ) O() o()

3.. 3.. g(x) O(x k ) g(x) x x k o(x k g(x) ) lim x x k 3. x π(x) x dt log t + O( x log x)

4 4. f x f(x + h) f(x) h f (x) df dx (x) f C f C f f(x + h, y) f(x, y) (x, y) lim x h h f f x y f x f x x f f x y x y f y x f y x f y y k C k f y 4.. (cos x) cos(x + h) cos x lim h h lim sin(x + h h/ )sin h h/ sin x

4.. 3 (sin x) sin(x + h) sin x lim h h lim cos(x + h h/ )sin h h/ cos x sin h lim h h h cos h sin h h tn h π π π cos( + b) cos cos b sin sin b cos( b) cos cos b + sin sin b sin( + b) sin cos b + cos sin b sin( b) sin cos b cos sin b cos cos b sin sin b sin cos b cos sin b cos( + b) + cos( b) cos( b) cos( + b) sin( + b) + sin( b) sin( + b) sin( b) e (e x ) e x e h lim e x h h ( e lim + ) n n n

4 4 ( e h lim + h ) n n n ( n ( ) k h e h lim nc k n n) k n ( ) k h lim nc k n n k n ( ) k h h + lim nc k n n k (h) n ( k h n ( ) nc k h n) h k nc k n k k k n ( ) h nc k n k h h k n ( ) nc k n k k ( + n) n h e 4.. (ctenry) s s s + s ρ ρ g 3 T s θ(s) T tn θ(s)) T tn θ(s + ) T tn θ(s + ) T tn θ(s) ρ g T d tn θ(s) ρg ds

4.. 5 C ρg T tn θ(s) Cs θ(s) rctn Cs s (x, y) (, ) x(s) y(s) s s cos θ(t) dt sin θ(t) dt tn θ(t) Ct dθ C cos θ dt x(s) s C cos θ C z(s) dθ cos θ dz z (z sin θ) (log( + z(s)) log( z(s))) C z(s) sin(θ(s)) cos θ(s) + tn θ(s) + C s Cs + C s ( x(s) C log Cs + ) + C s Cs + + C s sinh y ey e y x sinh(log(x + + x )) sinh Cx(s) Cs

6 4 θ y(s) sin θ dθ C cos θ ( ) C cos θ ( + C s C ) C ( + sinh (Cx(s)) ) (cosh(cx(s)) ) C y (cosh x ) C e x cosh x ex + e x x ( + x + + x + x ) + x + y C ( + C x + ) C x + 4. Newton f(x) Mthemtic FindRoot[f,{x,x }] x x f y f (x )(x x ) + f(x ) x x {x n } f(x) x x n+ x n f(x n) f (x n ) x x f(x ) f (x )

4.3. e 7 f(x ) (f (x ) ) T x x f(x) f (x) T T x f (x) f(x)f (x) f (x) f(x)f (x) f (x) f C f (x) > δ > f(x) T < 4.3 e 4.3. n + r n + ( + n) n n r n n k n+c k (n + ) k nc k n k ( + ) n+ ( + n n + n) nc k n k r n(n + ) > [ (n + ) (n + k) n (n k) (n + ) (n + ) n n n ( k Jensen : f i n+c k (n + ) k nc k n k n n n λ i f(x i ) f( λ i x i ) (λ i, λ i ) i f(x) log x x x n, x n b n n b ((n ) + b) n i ] k! > ) + (n + ) n+

8 4 ( ) n n n ( n + n ) n ( ) n n n ( ) n n + Jensen ) n n + n n + n ( ) 4.3. nc k n k n (n ) n n ( + n) n + (n k + ) n ( + n) n k e x n k k! k! k 3 n k k! e k! k 4.4 e x n k x k k! + eθx n! xn n+ n n +

4.4. 9 4.4. sin x, cos x e iθ cos θ + i sin θ cos x x + x4 4! + sin x x x3 3! + x5 5! + R n (x) n! x n sin x n 4k (sin x) (n) cos x n 4k + sin x n 4k + cos x n 4k + 3 cos x n 4k (cos x) (n) sin x n 4k + cos x n 4k + sin x n 4k + 3 (sin x) cos x (cos x) sin x x > < c < x cos c sin x x cos c sin x x cos cos x + x x sin x (cos t) dt t dt x x x x x3 3! cos t dt ( t ) dt x sin t dt

3 4 cos x x x sin t dt x + x4 4! (t t3 3! ) dt x x3 3! sin x x x x cos x + x4 4! x xn n! 4.4. cos x eix + e ix, sin x eix e ix i cosh x ex + e x, sinh x ex e x, tnh x ex e x e x + e x cosh x sinh x 4.4.3 e. e e p q e < 3 (+ 6 )6.5 > 5 q > e e + + + + q! + eθ (q + )! ( < θ < ) q! q!e eθ q+ < eθ q + < e q + < q q + <

4.5. 3 4.5 Newton 3.. F m 3. 4.5. 3 3 3 6 N 6N x v dx dt v, mdv dt F F mg ( ) ( ) d x v dt v g (F kx) ( ) d dt x v ( ) ( ) x k v (x, v) 4.5. (x(t), y(t)) r(cos ωt, sin ωt) (v x (t), v y (t)) rω( sin ωt, cos ωt) ( x (t), y (t)) rω cos ωt, sin ωt)

3 4 3 3 5 5 5 5 5 5 5 5 5 3 5 5 5 3 4.: F m mrω F m v r ( ) G mm r v m v r GM g 9.8m/s r F mg G mm r GM r g v rg 64 9.8 799.6 7.996km/s 4.5.3 F U r

4.5. 33 U U r GmM r U r G mm r dr G mm r.km/s v mv G mm r GM r gr v 64 9.8 4.5.4 t F (t) v(t) F (t)v(t) dt m(t)v(t) dt m dv v(t) dt dt v t dv dv dt dt mv dv mv U mv F U x F dv m v dx mv dv dt dt dx m d x dt m

34 4 4.6 F f F (x) f(x) x f(t) dt f(x) dx F (b) F () x x e t dt x b π e x dx X b f f(x) dx

35 5 5. 5. ( ) f [, b]. < c c > h f (c) f(x + h) f(x) + f (c)h 3. < θ < f(x + h) f(x) + f (x + θh), f(b) f() + f ( + θ(b )) 5. (Rolle) f [, b] C f() f(b) f (c) < c < b. f f (, b) < c > f(x ) f(c) x c f x c x, x c f (c) f (c) f (c) 5.3 ( ) F, G [, b] C < c < b F (b) F () G(b) G() F (c) G (c)

36 5. Q(x) F (x) F () F (b) F () (G(x) G()) G(b) G() Q() Q(b) < c < b Q (c) Q (c) F (c) F (b) F () G(b) G() G (c) 5.4 ( ) f x C n f(x) n k R n (x) f (k) () (x ) k + R n (x) k! R n (x) f (n) (c) (x ) n n! x c f(x) n k r n f (n) (c) n!. n F (x) f(b) f(x) G(x) (b x) n k f (k) () (x ) k + r n (x ) n k! f (k) n (x) (b x) k f(b) k! k f (k) (x) (b x) k, k! < c < b n F (x) k n k F (b) F () G(b) G() F (c) G (c) f (k+) k! n (b x) k f (k) (x) + (b x)k (k )! k f (k+) n (x) (b x) k + k! k f n) (x) (b x)n (n )! f (k+) (x) (b x) k k!

5.. 37 n F (b) F () F () f(b) k f (k) () (b ) k k! (G(b) G()) F (c) G (c) (b ) n f ( n)(c)(b c) n (n )!n(b c) n (b ) n f (n) (c) n! 5. ( ). R n (x) x f(x) f() f (n) (t) (n )! (x t)n dt x R n (x) [f (n ) (x t)n (t) (n )! f (t) dt R (x) ] x + x f (n ) () (n )! (x )n + R n (x) f (n ) (x t)n (t) dt (n )! 5. e x n k x k k! + eθx n! xn n+ n n +

38 5 5.. sin x, cos x e iθ cos θ + i sin θ cos x x + x4 4! + sin x x x3 3! + x5 5! + R n (x) n! x n sin x n 4k (sin x) (n) cos x n 4k + sin x n 4k + cos x n 4k + 3 cos x n 4k (cos x) (n) sin x n 4k + cos x n 4k + sin x n 4k + 3 (sin x) cos x (cos x) sin x x > < c < x cos c sin x x cos c sin x x cos cos x + x x sin x (cos t) dt t dt x x x x x3 3! cos t dt ( t ) dt x sin t dt

5.. 39 cos x x x sin t dt x + x4 4! (t t3 3! ) dt x x3 3! sin x x x x cos x + x4 4! x xn n! 5.. cos x eix + e ix, sin x eix e ix i cosh x ex + e x, sinh x ex e x, tnh x ex e x e x + e x cosh x sinh x 5..3 log( + x) x x + x3 3 (log( + x)) (n) n (n )! ( ) ( + x) n (log( x)) (n) (n )! ( x) n log( x) (x + x + x3 3 + + xn n xn + ( )n n + ( )n n( + θx) n xn ) n( θx) n xn

4 5 x x (log( x)) x + x + x + + x n + xn x R n (x) x x < t > R n (x) x t n t dt t n dt x < log + 3 log + 3 ) log( x) (x + x + x3 3 + log( + x) x x + x3 3 n n x n n n xn ( ) log + x x x n+ n + x < +x x ( log 3 + 3 3 3 + 5 3 5 + ) 7 3 7 + log n 3 3 +/7 4 4 log 3 log log /7 9 3 8 8 log 3 3 log + +/7 log /7 5 5 4 4 log 5 3 + log 3 + +/49 log /49 49 7 48 48 log 7 log + log 3 + +/97 log /97 n

5.3. 4 x n+ log(n + ) log n j ( ) j+ j + n + n + + R n log l l+ R n 3 ( j + n + j ( n + j ) j+ ) j+ 3 n(n + )(n + ) < n 3 n l l+ 3l l+ < l ( ) 3 n + (/(n + ) 5.3 d dx ( + x) ( + x) n ( + x) C k x k + R n (x) k R n (x) C n ( + θx) n x n x ( ) ( n + ) ( + t) n (x t) n dt (n )! x + x + x 8 + x3 6 5x4 8 + x + x + 3x 8 5x3 6 + 35x4 8 + x n C n ( + t) n (x t) n dt

4 5 5.4 5.4. dy. rctn x y x tn y cos y dx (rctn x) cos y +x. rccos x y x cos y sin y dy dx rccos x [, π] (rccos x) sin y x 3. rcsin x y x sin y cos y dy dx rccos x [ π, π ] (rcsin x) cos y x rccos x + rcsin x π 5.4. (rctn x) + x + ( x ) + ( x ) + + ( x ) n + ( x ) n + x rctn x x x3 3 + x5 xn + + ( )n 5 n + x ( t ) n + t dt rccos x π x x3 6 3x5 4 rcsin x x + x3 6 + 3x5 4 + 5x7 + 5.4.3 π ( ) rctn rctn π 4 4 rctn 5 π 4 rctn 39

5.5. 43 α rctn 5 tn α tn α tn α /5 /5 5 tn 4α tn α tn α 9 tn(4α π tn 4α ) 4 + tn 4α 39 5.5 k > ( + c n k )n, n ( + n + c n k )n. log( + x) n log( + c n k ) n + θc/n k c n k n log( + n + c ( n k ) n n + c n k ( + θ(/n + c/n k ) ( n + c ) n k ) ( + c n k )n, ( + n + c n k )n e 3 ( e t/n + e t/n ) n, ( e t/ n + e t/ n ) n n. ( e t/n + e t/n ) n ( ( + t t + eθt/n n n + t n + e θ ) n θt/n t ( + e 4n + e θ t/n t 4n t/n t n ) ) n

44 5 ( ) n e t/ n + e t/ n e t / ( ( + t + t n n + t 3 eθt/n 3!n + t + t 3/ n n + e θ ( + t n + t 3 eθt/n + t/n t 3 ) n e θ 3!n3/ 3!n 3/ t/n t 3 3!n 3/ ) ) n 4. K(x) e x sin x x x 3 lim x x e x sin x ( + x + x + ec 3! x3 )(x x3 3! x + x + 3 x3 + x 4 K(x) sin c x 4 ) 4! 5.5. l Hopitl 5. (Cuchy ) f, g [, b] (, b) ξ (, b) f (ξ)(g(b) g()) g (ξ)(f(b) f()). h(x) (f(x) f()(g(b) g()) (f(b) f())(g(x) f()) h() h(b) Rolle ξ (, b) h (ξ) f (ξ)(g(b) g()) (f(b) f())g (ξ)

5.5. 45 5.5 (l Hopitl) f, g [, b] (, b) lim x f (x) g (x) f(x) f() lim x g(x) g() lim f (x) x g (x) 5. lim x f(x) lim x g(x) lim x f (x) g (x). f(x) lim x g(x) lim f (x) x g (x) x ( x > ) g (x) g(x) g() x > Cuchy ξ (, b) g(x) g() f (ξ)(g(x) g()) (f(x) f())g (ξ) f (ξ) f(x) f() g (ξ) g(x) g() x ξ x f g C n f (k) () g (k) () ( k n ) g (n) () f(x) f() + n! f (n) (ξ)(x ) n g(x) g() + n! g(n) (ξ )(x ) n x g (n) (ξ ) f(x) f() g(x) g() f (n) (ξ) g (n) (ξ ) f 5.6 (l Hopitl ) lim x g(x) lim (x) x g (x) f(x) lim x g(x) lim f (x) x g (x). x > N lim x f (x) g (x) c ε > N f (x) g (x) c < ε

46 5 M > N x > M mx{ f(n), g(n) } g(x) x > M Cuchy ξ (N, M) ε f(x) f(n) g(x) g(n) f (ξ) g c + θε, ( < θ < ) (ξ) f(x) f(x) f(n) g(x) g(x) g(n) f(n) g(x) g(n) f(x) f(n) g(x) g(x) g(n) f(n) g(x) + g(n) g(x) f(x) f(n) g(x) g(n) ε + ( c + )ε ( c + )ε f(x) g(x) f (ξ) g (ξ) ( c + )ε N x, ξ 5 sin x lim x x lim x lim x lim n lim x cos x x log x x n nα ( R) ( ) x n e x e /x lim x x n ( ) 5.6 f C r f(x + h, y + k) r i i! i i f ic j x j y i j (x, y) hi k i j + R n (x, y) j

5.7. 47 θ R n (x, y) r! r r f rc j x j y r j (x + θh, y + θk) hj k r j j. g(t) f(x + th, y + tk) g(t) r i t g (i) (t) g (i) () i! t i + g(r) (θt) t r r! i i f ic j x j y i j (x + th, y + tk) hj k i j j r i ( h i! x + k ) i f(x, y) + ( h y r! x + k ) r f(x + θh, y + θk) y 5.7 R n (x) x f(x) n f (n) () (x ) n n! 5. x x < x x. n f (n) () n! ε > n n, m n m k x k < ε kn n x n < ε n x n n x n x x n < ε x x n

48 5 m n x n ε < x/x kn n k kx k Cuchy r (r ) k kx k x < r x > r r 5. x < r 5.7. 5.7. lim n+ n n r. lim sup n n n r.. ε > n s.t. n n n+ n < r ε n+ x n+ n x n n+ n x x r ε n x n n x n n+x n+ n x n < n x n x r ε ( < n x n x r ε ) n n

5.7. 49 x x < r ε x < r ε nn n x n ε > n s.t. n n n+ n > r + ε n+ x n+ n x n > x r + ε ( x r ε ) n n nn n x n x > r ε > x > r + ε ( ) n n n x n > n x n x r + ε. x ε ε > n s.t. n n k sup k < k n r ε n n < r ε, nx n < ε > n s.t. n n {k n } k sup k > k n r + ε nk x n k > ( ) nk x r + ε ( ) n x r ε x > r ε x > r + ε

5 5 5.8 5.8 f(x) n nx n r x < r f (x) n n x n f(x) dx n n n n + xn+. ˆx ( ˆx < r) < ρ < n nˆx n nˆx n < M M x < ρ ˆx n n x n nˆx n n xˆx n ˆx < M ˆx nρn n nρn < ε > n s.t. nn nρ n < ε x < ρ ˆx n n x n n n n x k n n nx n < ε ˆx, ρ x < r x, x x, x < ρ ˆx m > n m k kx k m k kx k n k kx k n k kx k x x x x m kn+ k k x k k (x k x + θ k (x x ), θ k < ) m kn+ n x k x k x x k x, x ( x < r ) k kx k k kx k x x N k x k x k kn+ k + kx k kn+ kx k x x x x N x ε x x N k k kx k n n x n n

5.9. Riemnn zet 5 f ( )F f F (x) f(x) f(x) b n x n n nb n x n n n x n n n (n + )b n+ F (x) n n n xn n n n + xn+ 5.3. f(x) n (x ) n b n (x ) n n r f (n) () n! n n b n n b n 5.9 Riemnn zet Riemnn zet ζ(z) n n z

5 5 z ζ() + + 3 + + + + n + + ( 3 + 4 + + + n n + n ) ( + + n + + + ) n z < n z > n z z < + z + + ( n ) z ( + z + 3 z ) + + + z + + n ( n ) z + ( z ) + + ( z ) n z ( ) ( n ) z + + ( n+ ) z ζ() π sin x x nπ n, ±, ±,...) 6 sin x n Z(x nπ) (x nπ)(x + nπ) sin x x n N(x (nπ) ) sin x x ( ( x )) nπ n N

5.9. Riemnn zet 53 sin x x x sin x x ( ( x )) nπ n N sin x x x3 6 + x 3 ζ() + + 3 + π ζ() π 6 zet ϕ(s) ϕ(s) s + 3 s ( + s + ) ( 3 s + s + 4 s + ) 6 s + ζ(s) s ζ(s) ( s) ζ(s) ϕ() ζ(), ϕ( ) 3ζ( ), ϕ( ) 7ζ( ) f(x) + x + x + x f( ) d dx f( ) ( + x + 3x + ) x ( x) x 4 d df (x dx dx )( ) ( + 4x + 9x + ) x + x ( x) 3 x d d f( ) ϕ(), dxf( ) ϕ( ), dx (x f)( ) ϕ( ) ζ(), ζ( ), ζ( ) ζ( n) Riemnn ζ(z) Re z > Re z z Re z ζ(s) p p : prime s

54 5 π(x) x π(x) x log x

55 6 6. 6. 6. : f x ε > δ > s.t. x < δ f(x) f() < ε : f (x, y) (, b) ε > δ > s.t. d((x, y), (, b)) < δ f(x, y) f(, b) < ε f I I f : X Y 6 solution x ε > δ > s.t. x x < δ f(x) f(x ) f (x ) x x < ε f(x) f(x ) < (ε + f (x ) ) x x δ ε+ f (x ) ε x x < δ f(x) f(x ) < ε 6.3 X, Y O Y f (O) G Y f (G)

56 6 x X f(x) U f (U) x V 7 x 6.4 6. f [, b] c, d [, b] s.t. f(c) mx x [,b] f(x) f(d) min x [,b] f(x). sup x [,b] f(x) M ε > x(ε) s.t. f(x(ε)) M ε x n x(/n) {x n } f( lim n x n) lim n f(x n) lim n (M n ) M f {f(x) > n: x b} x n [, b] {x n } f(x n ) 6.5 6. ( ) f [, b] f() < f(b) f() y f(b) x b s.t. f(x) y. c, d [, b] f(c) min f(x), f(d) mx f(x) c < d c > d sup inf x sup{c < x < d: f(x) < y} x > x f(x) y f f(x ) lim x x f(x) y

6.6. 57 {x n } s.t. x n x f(x n ) < y f f(x n ) f(x ) f(x ) y f ε > δ > x x < δ f(x) f(x ) < ε n n n x x n < δ f(x n ) f(x ) < ε f(x n ) < y f(x ) < y + ε ε f(x ) y f(x ) y 6.6 6. : f I ε > δ > s.t. x x < δ (x, x I) f(x) f() < ε : f I ε > δ > s.t. d((x, y), (x, y )) < δ, ((x, y), (x, y ) I) f(x, y) f(, b) < ε I ε > δ > x x < δ f(x) f(x ) < ε 6.3 [, b]. ε > c [, b] δ c > x c < δ f(x) f(c) < ε {c δ c, c + δ c } [, b] [, b] compct c,... c k δ min δ ci x x < δ f(x) f(x ) < ε. ε δ > x, x s.t. x x < δ f(x) f(x ) ε

58 6 δ n x, x x n, x n [, b] {x n } lim n x n x lim n x n x f(x n ) f(x n) ε. ε > x I x δ(ε, x ) > s.t. x x < δ(ε, x ) f(x) f(x ) < ε U x (x δ(ε, x ), x +δ(ε, x )) x U x I I x,..., x n n k U(x k) I δ min k n δ(ε, x k ) δ > x x < δ x i, x j x x i < δ x x j < δ f(x) f(x i ) < ε, f(x ) f(x j ) < ε U xi U xj x f(x i ) f(x ) ε f(x j ) f(x ) < ε f(x) f(x ) < 4ε

59 7 ε δ 7. A mx A. b A b. A A sup A. b A b. ε > b A s.t. b > ε A b A {,,...} sup A sup n n 8 9 3 A {,,...} sup A lim n n 3 A B sup A sup B sup(a B) sup A, inf(a B) inf A, sup(a B) mx{sup A, sup B} sup λ Λ A λ sup λ Λ (sup A λ ) sup(a B) sup A inf(a B) inf A 3. mx( n + b n ) mx n + mx b n

6 7 ε δ. mx( n ) min n 3. mx( n b n ) mx n min b n 33. sup( n + b n ) sup n + sup b n. sup( n b n ) sup n inf b n 34. mx x [,] (f(x) + g(x)) mx f(x) + mx g(x). mx x [,] ( f(x)) min x [,] f(x) 3. mx x [,] (f(x) g(x)) mx x [,] f(x) min x [,] g(x) 35. sup( n + b n ) sup n + sup b n. sup( n ) inf n 3. sup( n b n ) sup n inf b n 36. sup x [,] (f(x) + g(x)) sup f(x) + sup g(x). sup x [,] ( f(x)) inf x [,] f(x) 3. sup x [,] (f(x) g(x)) sup x [,] f(x) inf x [,] g(x) 37 n e /n ( n )n n sin π n cos nπ 4 b n sin n 3 π (lim n b n b) sin n 38 n lim sup( n b n ) (lim n ) (lim sup b n ) 39 n > lim sup n lim n f(a) f (A)

7.. 6 7. lim sup n lim n sup n k n k inf sup k k n lim inf n n lim n inf k n k sup inf k n k A n { n, n+,...} n 7. lim sup n lim inf n lim n lim n lim sup n lim inf n.. lim sup n lim inf n. lim n 7. ( ) f : [, b] R c inf x [,b] f(x) d sup x [,b] f(x) c < y < y f(x) x [, b] 7.3 lim n n ε > n s.t. n n n < ε 4. r n. nr n 3. r n n 4. n!r n 4 lim n n lim n b n b n + b n lim + b n, lim nb n b n

6 7 ε δ 7.4 ε > n s.t. n, m n n m < ε I I I n n I n R A B A B A b B < b. mx A min B. mx A min B I I I 3 I n I n [ n, b n ] { n } b {b n } lim n n b lim n b n n n n n I n ( ε, + ε) I n { n } N n N n I n ( ε, + ε) lim n n A B (A B R, A B ) x A y B I [x, y] x+y A I [ x+y x+y, y] B I [x, x+y ] n I n {} ε > ε A, + ε B A mx A

7.4. 63 min B B min B mx A A {x: n s.t. x n }, B {x: n s.t. x > n } Dedekind mx A min B ε > ( ε, ] n n lim n n Cuchy {I n } I n { n, n } Cuchy m n m I n m n I n I n { n } n n n I n I n I n Cuchy { n } Cuchy n s.t. m, n n n m < n n n n n I [ n, n + ] I n n n I 4, 8,... n < n < n n n n I n ( ε, +ε) n n n < ε lim n n 7. { n } Cuchy nk lim n n. Cuchy ε > k s.t. l k nl < ε n s.t. m, n n n m < ε

64 7 ε δ n mx{n k, n } n n n n nk + nk < ε 7.3 Cuchy. ) n n n n ε > n n n n < ε n > m n s n s m s n n km+ n k n k n km+ n < ε Cuchy ) {s n } Cuchy {s nk } m, n n k s n s m < k n k > n k s nk s nk+ < k k s nk s nk k (s nk s nk+ ) k k N (s nk s nk+ ) s n s nn+ k {s nn } {s n }

7.5. 65 7.5 { n (t)} n t 7.4 [, b] f n f f. f n ε > n s.t. n n f n (x) f(x) < ε f(x) f(x ) f(x) f n (x) + f n (x) f n (x ) + f n (x ) f(x ) ε f n δ > s.t. x x < δ f n (x) f n (x ) < ε f(x) f(x ) 3ε f [, b] C[, b] f sup f(x) x [,b] 7.5 C[, b]. {f n } Cuchy ε > n s.t. m, n n f n f m < ε x [, b] f n (x) f m (x) < ε {f n (x)} Cuchy f(x) n n f n (x) f(x) < ε f n f f n f f C[, b] C[, b] 7.6 f n [, b] f lim n f n (x) dx f(x) dx

66 7 ε δ. ε > n s.t. n n sup f n (x) f(x) < ε f n dx f(x) dx f n (x) f(x) dx ε(b ) 7.7 [, b] f n f f n g f g. (f n f m )(x) (f n f m )(x ) f n f m x x ε > n s.t. n n f n g < ε m, n n f m f m < ε f n (x) f n (x ) f m(x) f m (x ) x x x x < ε m f n (x) f n (x ) f(x) f(x ) x x x x < ε n δ > f n (x) f n (x ) f x x n(x ) < ε g(x ) f(x) f(x ) x x g(x ) f n(x ) + f n(x ) f n(x) f n (x ) x x + f n (x) f n (x ) f m(x) f m (x ) x x x x < 4ε

7.6. 67 7.6 n n s N + + N ε > n s.t.n n s n s < ε ε > n s.t. m, n n s n s m < ε > n s.t. n n n < ε s n + + + n s n+ + + + n+ < s n s n+ n+ n r n ( r < ) s n rn r r n r 7.8 n n lim n n. lim n n δ > n > δ n {n k } n m n k m kn k > δ 7.9 ( ) n n n b n n b n n n 7. () n n n n

68 7 ε δ. b k n k b k ε > n s.t. m, n n m kn+ b k < ε m kn+ k m kn+ k m kn+ b k < ε n k k 7. + 3 4 n n k n k n k. n k k ε > n s.t. m, n n m kn+ k < ε kn+ k ε b + b + n n n n b k (b k n ) k n m, n n m kn+ b k kn + k ε n k b k n n n b k k b k k + k k k k kn + k kn + k ε

7.7. 69 7.7 7. { mn } lim n mn m lim m m lim m mn b n lim n lim mn lim m lim mn m n. ε > n s.t. n n mn m < ε m s.t. m m m < ε n n m m mn mn m + m < ε n n m b n b n mn + mn < 3ε mn n m mn n m m n mn n mn n m m nm mn 7.7.

7 7 ε δ 5 ( ) X X X i + + X i + 5% 5% n S n X + + X n S n r k k r k + (k r) n k n+r n k p r (n) nc k n n + r p r (n) n + r p (n) q n q n n n p n n p n n p q n 4 q 4 4 q p 4 q 4 q n n P (t) t n p n Q(t) n t n q n n

7.7. 7 p q q n Q(t) t m mc m m (7.) m ( + x) C m x m m C m ( ) ( m + ) m! ( /)( 3/) ( / m + ) m! m 3 (m ) ( ) m m! ( ) m (m)! m m!m! ( ) m mc m m /C m (7.) Q(t) t m ( ) m /C m m m m /C m ( t ) m m t P (t) Q(t) n m n n q n p n + n m p m q n m n p m q n m m p q m n

7 7 ε δ Q(t) t n q n + + + n n t n n m m nm m n + Q(t)P (t) p m q n m t n p m q n m t n+m p m q n P (t) Q(t) t (7.) n P () P () (7.) P (t) t P (t) ( t ) / ( t) t( t ) / P (t) n p nt n P (t) n np nt n P () n np n np n P () n

7.7. 73 7. ( ) n! n n e n πn b b n n! n q n n C n/ n n n e n πn ((n/) n/ e n/ πn) n πn p n t t n p n / C n/ ( ) n/ ) ( ) ( ( n + ) ( ) n/ (n/)! ( n + ) / C n/ ( ) n/ n ( )n/ nc n/ n ( ) n/ n n C n/ n n πn π n 3/ n n p n

74 7 ε δ 7.8 Riemnn zet Riemnn zet ζ(z) n n z z ζ() + + 3 + z < + + + n + + ( 3 + 4 + + + n n + n ) ( + + n + + + ) n n z > n z z < + z + + ( n ) z ( + z + 3 z ) + + + z + + n ( n ) z + ( z ) + + ( z ) n z ( ) ( n ) z + + ( n+ ) z ζ() π sin x x nπ n, ±, ±,...) 6 sin x n Z(x nπ)

7.8. Riemnn zet 75 (x nπ)(x + nπ) sin x x n N(x (nπ) ) sin x x ( ( x )) nπ n N sin x x sin x x n N x ( ( x )) nπ sin x x x3 6 + x 3 ζ() + + 3 + π ζ() π 6 zet ϕ(s) ϕ(s) s + 3 s ( + s + ) ( 3 s + s + 4 s + ) 6 s + ζ(s) s ζ(s) ( s) ζ(s) ϕ() ζ(), ϕ( ) 3ζ( ), ϕ( ) 7ζ( ) f(x) + x + x + x f( ) d dx f( ) ( + x + 3x + ) x ( x) x 4 d df (x dx dx )( ) ( + 4x + 9x + ) x + x ( x) 3 x d d f( ) ϕ(), dxf( ) ϕ( ), dx (x f)( ) ϕ( ) ζ(), ζ( ), ζ( ) ζ( n)

76 7 ε δ Riemnn ζ(z) Re z > Re z z Re z ζ(s) p : prime p s π(x) x π(x) x log x 7.9 n npn n tn p n s n + + n s ε > n s.t. n n s n s < ε 7.9. p lim n n ε > n n n s.t. n ε ε ε ε n n s n s < ε s n s < ε s n s < ε n > ε s n s s n + n s > ε + ε > ε lim n np n

7.9. n npn 77 7.9. > p p > > p > np n n np n (n + )p n+ p n (n( p ) p ) p p npn > (n + )p n+

78 8 8. f [, b] [, b] : < < < n b S (f) S (f) n sup f(x)( i i ) x [ i, i n sup f(x)( i i ) x [ i, i i i 8. 4. S (f + g) S (f) + S (g) S (f + g) S (f) + S (g). S (αf) αs (f) S (αf) αs (f) 3. f g S (f) S (g), S (f) S (g) 4. S (f) S ( f) 8. S S S S

8.. 79. sup( n +b n ) sup n +sup b n inf( n +b n ) inf n +inf b n f(x) dx inf S (f) f(x) dx sup S (f) inf sup 8. δ n S (f) n sup f(x) δ S (f) S (f) S (f) + n sup f(x) δ S (f) S (f). Mδ sup f(x)( i i ) Mδ i x i i b δ > s.t. δ S (f) f(x) dx < ε S (f) f(x) dx < ε

8 8. S (f) ε n n sup f(x) δ < ε δ δ 8. S (f) n sup f(x) δ S (f) S (f) dx + f(x) ε 43.. (f + g) dx f dx + g(x) dx (f + g) dx f(x) dx + αf(x) dx α f(x) dx αf(x) dx α f(x) dx g(x) dx 3. f g 4. f(x) dx g(x) dx,. () S (f) S ( f) f(x) dx S (f + g) S (f) + S (g) g(x) dx sup(a B) sup A + sup B (f + g) dx (f + g) dx S (f) + S (g) ε > f g S f S g f dx + ε g, dx + ε

8.. 8 f g (f + g) dx f dx + g dx + ε ε (),(3) (4) sup A inf( A) 8. f f(x) dx { n } x, x [, b] n n n x, x n 8. { n } n : n < < n N n b {ξ n i } ( i ξ n i n i ). f(x) dx lim n i N n f(ξi n )( n i n i ) δ n δ 8.3 ε δ S n S n ε N n S n (f) f(ξi n )( n i n i ) S n (f) i 8., n,... lim n S n (f) lim S n n (f). n S n (f) inf S (f) sup S (f) S n (f)

8 8 8. [, b]. f [.b] ε > δ > x x < δ f(x) f(x ) < ε δ n S (f) S (f) ( sup f(x) inf f(x))( i i ) i i x i i x i n ε ( i i ) i ε(b ) f(x) dx f(x) dx < S S < ε(b ) 8.3. f [, b] n n S n (f) n n f( i ), i S n (f) n f( i ) n i S n (f) S n (f) 8.4. (f ± g) dx f dx ±. αf dx α f dx 3. f g f dx g dx 4. b f dx f(x) dx. g dx ( ) f(x) f(x) f(x)

8.. 83 8.5. f, g fg. f inf x b f(x) > f(x). f(x)g(x) f(x )g(x ) (f(x) f(x ))g(x) + f(x )(g(x) g(x ) M ( f(x) f(x ) + g(x) g(x ) ) M ((sup f(x) inf f(x)) + (sup g(x) inf g(x))) S (fg) S (fg) n (sup fg(x) inf fg(x)) ( i i ) i M n ((sup f(x) inf f(x)) + (sup g(x) inf g(x))) ( i i ) i ( M f dx f dx + g dx δ inf f(x) f(x) f(x ) fx) f(x ) f(x)f(x ) δ f(x) f(x ) g dx ) 8.6 < b < c [, c] c f(x) dx f(x) dx + c b f(x) dx. [, c] b δ sup f(x)δ 44. x, x. e x 3. sin x, cos x

84 8 8.. 8. ( ) f f(x) dx (b )f(ξ) ( ξ [, b]) 8.3 () f ϕ f(x)ϕ(x) dx f(ξ) ϕ(x) dx ( ξ [, b]). inf f(x) x [,b] ϕ(x) dx inf f(x) x [,b] f(x)ϕ(x) dx sup f(x) x [,b] f(x)ϕ(x) dx ϕ(x) dx sup f(x) x [,b] ϕ(x) dx 8.4 ( ) ϕ f. f(x)ϕ(x) dx ϕ() ξ f(x) dx + ϕ(b) ξ ϕ ϕ() f(x) dx f(x)ϕ(x) dx ϕ(b) f(x) dx f(x) dx ϕ() ξ f(x) dx + ϕ(b) f(x) dx ξ ξ < ξ ξ ϕ() f(x) dx + ϕ(b) f(x) dx ϕ() ξ f(x) dx ϕ(b) ξ ξ ξ ξ ϕ() f(x) dx ϕ(b) f(x) dx ξ ξ ξ (ϕ() ϕ(b)) f(x) dx ξ f(x) dx f M ξ ξ f(x) dx M(ξ ξ)

8.. 85 8.. 8.5 () f [, b]. d dx x f(t) dt f(x) [, x] ε > δ > s.t. x y < δ f(x) f(y) < ε F (x) x h > h < δ F (x + h) F (x) f(x) h f(t) dt h h ε x+h x x+h x (f(t) f(x) dt f(t) f(x) dt 8. f [, b] f (x) dx f(b) f(). d dx x F (x) f (t) dt f (x) x f (t) dt F (x) f (x) F (x) f(x) + C

86 8 8.4 F (x) F. n log(x + ) n 3. I n (x + ) n dx ( ) x I n (n )(x + ) + n 3 n n I n n > rctn x n f(x, y), g(x, y) f(sin x, cos x) g(sin x, cos x) x rctn r r tn x tn x r r cos x r + r r sin x r

8.. 87 dx dr + r r 3 () x +, (b) x, (c) x () t rctn x, (b) t rccos x, (c) t rcsin x () (b) (c) x + cos t, x tn t, x cos t, dx dt cos t dx dt sin t cos t dx dt cos t 3 8. ( + x 4 dx / x/ ) x / + x/ + x + x + dx x + ( / x/ ) (x /) + / + / + x/ (x + dx /) + / y x / / x/ (x /) + / dx /4 y/ y + / 4 rctn y 4 log(y + /) 4 rctn (x dy ) 4 log(x x + ) + x 4 dx 4 rctn (x ) + 4 rctn (x + ) + 4 log x x + x + x + + x 4 dx π 4 4 (π + π 4 ) + 4 (π π 4 )

88 8 R R +z 4 z e πi/4, e 3πi/4 lim z (i±)/ C z i z 4 + 4 e 3πi/4, 4 e 9πi/4 i 4, i 4 dz i πi ( + z4 4 + i 4 ) π R R z 4 + dz + x dx z Re iθ π R 4 e 4iθ + Reiθ dθ R R 4 π R π + x 4 dx 4 8.3 8.5 f [, b) (b ) < c n s.t. m, n n g(x) dx g(x) dx < ε n f(x) dx

8.3. 89 8. tn x dx log cos x ( π, π ) 8.3 e x dx e x e x (x > ) N lim e x [ dx lim e x ] N N N e 8.4 log x dx log x dx x log x x 8.5 π I I π/ π/ π/ π I π log log(sin x) dx π log π log(sin x) dx + log(sin x) dx + ( ) sin x log dx π/ π/ log(sin x) dx π log log(sin x) dx log(cos x) dx x l Hopitl x log(sin x) log(sin x) dx x x log(sin x) dx

9 8 8.4 Γ Γ(x) t x e t dt (x > ) t t x e t/ t t x e t e t/ for suffciently lrge t < x < t t x e t t x t x dt. Γ() [ ] x tx x. x > Γ(x) (x )Γ(x ) Γ(x) t x e t dt [ t x ( e t ) ] (x )t x ( e t ) dt (x )Γ(x ) 8.6. Γ(n) (n )!. ( ) Γ Γ t / e t dt e x dt ( ) π e x dx π (t x, dt xdx t / dx)

8.5. B 9 8.5 B B(p, q) x p ( x) q dx (p, q > ) < p < x < q < < x < x p ( x) q x p q x ( p q ) < q < x p dx p [xp ] p < q < x 8.5. e x 8.6 π e x dx. e (x +y ) x +y N ( N e (x +y ) dxdy e dx) x x +y N x r cos θ, y r sin θ ( ) ( ) x x r θ cos θ r sin θ sin θ r cos θ y r y θ r x +y N e (x +y ) dxdy π π/ N N rdr r dθ e e t dt π 4 ( e N ) e (x +y ) dxdy (t r, dt rdr)

9 8 e (x +y ) dxdy π 4 : ( ) e x(+y) dy dx I e x ( ( x e x x dx I ) e xy dy dx ) e t dt dx ( y t) e s ds I ( x s) ( ) e x(+y) dx dy + y dy [rctn y] π 8.6 () f [, b] [c, d] C. d dy f(x, y) dx f (x, y) dx y Fubini G(y) g(y) f(x, y) dx f f(x, y) dx y d c g(y) dy d c f (x, y) dydx y (f(x, d) f(x, c)) dx G(d) G(c) d c G (c) g(c)

8.5. B 93 8.7 : π e x dx. f(t) g(t) ( t ) e x dx e (+x )t dx + x f (t) g (t) t t u tx g (t) e x dx e t ( t( + x ))e (+x )t dx + x t te (+x )t dx e t u du f (t) f(t) + g(t) dx f() + g() + + x π 4 g(t) e (+x )t + x e t e t dx e t (t ) lim t f(t) π 4 8.8 B(p, q) Γ(p)Γ(q) Γ(p + q)

94 8. N N Γ(p)Γ(q) lim e x y x p y q dxdy N c c c lim e x y x p y q dxdy D D (u x + y, v x x + y ) ( M lim M M lim M lim b Γ(p + q) B(p, q) e u u p+q du lim b e u (uv) p (u uv) q u dv x uv y u uv ( ) ( ) x u u v v u v u y u v v ) v p ( v) q dv u u du 8.9 (. B ( ). Γ π, ) π. ( ( π B ), Γ ( Γ() )) ( ( Γ ) ). ( B ), dx x( x) x x t x +t dx t (+t ) dt ( B, ) + t + t dt + t [ rctn t ] t π t ( + t ) dt

8.5. B 95. ( B, ) dx /4 (/ x) dt /4 t /4 dt t [ rcsin t ] π ( π ) π (t x) [ ] rccos t ( π) π 45 π/ sin m x cos n x dx B ( m +, n + ). π/ sin m x cos n x dx t m/ ( t) n/ dt t( t) B ( m + t (m )/ ( t) (n )/ dt, n + ) (sin x t) 46 n. I n (x ) x x dx x x x x I n () n I (x ) x x x n dx n x x n x dx x

96 8 I n (x ) c n ( x ) n/ c I n (x ) x x I n (x + x ) dx c n x x ( x x ) (n )/ dx π/ c n ( x ) (n )/ ( sin θ) (n )/ x cos θ dθ (x x sin θ) π/ π/ c n ( x ) n/ cos n θ dθ π/ ( c n ( x ) n/ B, n + ) ( c n c n B, n + ) ( c n B, n ) ( B, n + ) ( c B, 3 ( B ) ), 4 ( B, n + ) Γ(/)Γ(3/) Γ(/)Γ(4/) Γ(/)Γ((n + )/) Γ(4/) Γ(5/) Γ((n + )/) π (n )/ Γ(3/) Γ((n + )/) π n/ Γ((n + )/) 8.6 Newton 3.. F m 3.

8.6. 97 3 3 5 5 5 5 5 5 5 5 5 3 5 5 5 3 8.: 8.6. 3 3 3 6 N 6N x v dx dt v, mdv dt F F mg ( ) ( ) d x v dt v g (F kx) ( ) d dt x v ( ) ( ) x k v (x, v)

98 8 8.6. (x(t), y(t)) r(cos ωt, sin ωt) (v x (t), v y (t)) rω( sin ωt, cos ωt) ( x (t), y (t)) rω cos ωt, sin ωt) F m mrω F m v r ( ) G mm r v m v r GM g 9.8m/s r F mg G mm r GM r g v rg 64 9.8 799.6 7.996km/s 8.6.3 F U r U U r GmM r

8.6. 99 U r G mm r dr G mm r.km/s v mv G mm r GM r gr v 64 9.8 8.6.4 t F (t) v(t) F (t)v(t) dt m(t)v(t) dt m dv v(t) dt dt v t dv dv dt dt mv dv mv U mv F U x F dv m v dx mv dv dt dt dx m d x dt m

8 8.7 C (x(t), y(t)) (t [, b]) x (t) + y (t) dt (x(t), y(t)) (x(t + h), y(t + h)) (x(t + h) x(t)) + (y(t + h) y(t)) x (t + θ h) + y (t + θ h) h ( θ, θ ) t x y f(x) (, f()) (b, f(b)) y (t) dy dt f (x) + f (x) dx 47 Asteroid x cos 3 θ, y sin 3 θ π 6 6 ( 3 cos θ sin θ) + (3 sin θ cos θ) dθ 3 4 π/ sin θ dθ 6 [ ] π/ cos θ π/ cos θ sin θ dθ 48 Crdioid r ( + cos θ) x cos θ( + cos θ), y sin θ( + cos θ) ( ) dx + dt ( ) dy ( + cos θ) dt π ( + cos θ) dθ π θ cos dθ 8

8.7. 49 f n f [, b] C (f n (x) f(x) f n(x) f (x)) f n () f n (b) f() f(b). f n f + f n (x) dx + f (x) dx 8.7 ( ) ABC AB BC AB,BC AC M,N,P AM,MP,PN,NB AB BC C AC x(u, v) Φ Φ(u, v) y(u, v) L(t) (u(t), v(t)) ( t ) z(u, v) dφ dt dt dφ dt Φ du u dt + Φ v dv dt ( Φ u, Φ v ) ( ) du dt dv dt ( E F ) F G dφ dt dt (( Φ u, Φ ( u Φ u, Φ v ( du dt, dv dt ) ( du dt, dv dt ) E ( du dt ) ( Φ ) ( Φ u, Φ v v, Φ v )) ) ( ) Φ u ( Φ Φ u, Φ v ) v ) ( E F F G ) ( du dt dv dt dt ) + F du dv dt dt + G E, F, G ( ) du dt dt dv dt ( ) dv dt dt

8 Φ 3 u Φ(u, v) v Φ u, Φ v ( E F ) ( ) F G