「数列の和としての積分　入門」

Transcription

1 7

2 I = 5. introduction () ( ) g(t) g(t) ( ) II 97 6 ()

3 6. ( ) (Talor ) ( ) ( ) ( ) 8 () 7 9 ( ) ( ) (,, ) () III

4 I

5 5 =. introduction ) r (B.C. ) Newton Leibnitz (7 ) Newton r ) () 8, 8 / 8 8

6 = 6 n n, r S = (r) r = r, = = S =, =, = n, = n,, k = k n,, n = n n = ` ` k k k n n ` n n n n n n n k [ k, k ]. δ k δ k = n f( k ) = ( ) k n Sn S n = f( )( ) + f( )( ) + + f( n )( n n ) n = f( k )δ k = k= n k= ( ) k n n = n n k = k= (n + )(n + ) 6n ;δ k ) S S = lim n S n = lim n (n + )(n + ) 6n = n = 5 = n = = n = ` 5 ` 5 ` = ` ` 8 ` 8 9 CG web site. ) d D

7 = 7 = ` ` n n = ` n n n n n n,, n f( k )δ k < S < k= n f( k )δ k k= n n (k ) < S < n n k= (n )(n ) 6n < S < k= k (n + )(n + ) 6n n n (n )(n ) 6n, (n + )(n + ) 6n S = ( ) = e =, e. ( ) ) a < b, = f() a b a b a = < < < < n < n = b { k } n δ k = k k (k =,,, n), δ k, η k k η k k (k =,,,, n) b a f()d = lim {f(η )( ) + f(η )( ) + + f(η n )( n n } n = lim f(η k )( k k ) k= = lim k= n f(η k )δ k ( ) )

8 = 8 η η = f() n n f() ( ) { k }, f() n k= f(η k)δ k ( ) {η k } ( ) η k f() ( ) f() a b ; f(η k )δ k d δ Leibnitz.) f(ηk ) δ k f() d b a f()d δ k f(η k )δ k () f(η k )δ k d ( ) ( ) ) )

9 = 9. {a n } a k = F (k + ) F (k) F (k) (k =,,, ) n n n a k = {F (k + ) F (k)} = {F (k) F (k + )} k= k= k= = [{F () F ()} + {F () F ()} + {F () F ()} + + {F (n) F (n + )}] = {F () F (n + )} = F (n + ) F () ( ) n k= k(k + ) = = n k= { k k + ( ) + = n + = } ( ) + n n + ( ) ( + + n ) n + F (n) = n a k = k(k + ) = k ( k + = ) ( ) = F (k + ) F (k) k + k n k= k(k + ) = n k= a k = n {F (k + ) F (k)} = F (n + ) F () = n + + k= () k+ k = ( ) k = k k = (k+ k )., n k = k= n ( k+ k ) = k= n ( k k+ ) k= = { ( ) + ( ) + ( ) + + ( n n+ ) } = + n+ = (n ) F (k) = k a k = k = F (k + ) F (k) n n k = {F (k + ) F (k)} = F (n + ) F (n) = (n+ n ) k= k=

10 = S n = a( rn ) r = a(rn ) r r k r k = (r )r k, r=\ r k = rk r rk r F (k) = a rk r ark = F (k + ) F (k). n ar k = k= n {F (k + ) F (k)} = F (n + ) F () = arn r k= 5 ( n k= k ) ar r = a(rn ) r k =,,, (k + ) k = k + k + (k + ) k = k + (k + ) k = + 6 (k + ) (k + ) + (k + ) 6 { k k + k } = k + k + 6 k = k F (k) = k k + k 6, k = F (k + ) F (k) n k = k= n {F (k + ) F (k)} = F (n + ) F () k= (n + ) (n + ) = + n + { = n + { (n + ) (n + ) + } 6 n(n + )(n + ) = 6 } S n = n(a+an)

11 =... ( ) b a f() d def = lim k= ( ) a b F () b a n f(η k )δ k f() = F () def = lim h F ( + h) F () h f() d = lim k= n f(η k )δ k = F (b) F (a) (a = < < < < n = b, k η k k, δ k = k k, = ma{δ k }) δ δ δ δ n a = n n = b η η η η n F () = f(), k =,,,, n F ( k ) F ( k ) = F (η k )( k k ) = f(η k )( k k ) = f(η k )δ k R = F () (η k, [ k, k ] ) Q k =,,,, n F ( ) F ( ) = f(η )δ F ( ) F ( ) = f(η )δ k η k k F ( ) F ( ) = f(η )δ // QR.. + ) F ( n ) F ( n ) = f(η n )δ n n F ( n ) F ( ) = f(η k )δ k k= = a, n = b n f(η k )δ k = F (b) F (a) k= ( )

12 = nx f(η k )δ k k= nx f(η k )δ k k= = f() = f() = f() V k η η η η η η k k η k η k η k, [ k, k ] η k, [ k, k ] ( f() {η k } {η k }.) η k, [ k, k ] n n n f(η k )δ k f(η k )δ k = {f(η k ) f(η k )}δ k k= k= k= [ k, k ] f() V k, V k (k =,,,, n) V f(η k ) f(η k ) V, α + β α + β n n {f(η k ) f(η k )}δ k f(η k ) f(η k ) δ k V k= k= k= k= k= n ( k k ) = V (b a) ( ) V 5) n n n f(η k )δ k f(η k )δ k = {f(η k ) f(η k )}δ k ( ) ( ) ( ) k= b a f() d = lim k= n f(η k )δ k = F (b) F (a) () a k = F (k + ) F (k) = f() = F () = lim h F ( + h) F () h = n a k = F (n + ) F () k= b a f()d def = lim k= n f(η k )δ k = F (b) F (a) a(k) = F (k + )-F (k) F (k), 5) V ( )

13 = 6 f() =, F () = [, ] { k } n = < < < < n = = n. F ( k + ) F ( k ) = ( k + ) ( k ) = ( k ) + k ( ) + ( ) ( ) ( ) ( ) ( ) k k = n n + + = k + k + n n n n F k = F ( k + ) F ( k ) = F (η k ) = f(η k ) k n < η k < k+ n F k = k + k + n = n (η k) k + k +. η k = n η k [ k, k+ ] n n f(η k ) = (η k ) = k= k= n k= k + k + n n = n = = F () F () n ( ) η k [ k, k+ ], [ k, k+ ] η k n I n. ( ) I n n n I = f(η k ) = (η k ) k= n ( ) I n = k= k < η k < k+ ( k n ) < η k < ( k+ n k= n { k } + k + n (η k ) ). k ( ) ( ) + k + k + n < k + k + n n (η k ) < k + k + k n n k + n < k + k + n (η k ) < k + n n k= n k + n n < ( ) I n < k= n n k + n n η k I n ( ) ( ) Comment ( ) ( ) (.)

14 = a = < < < < n = b { k } [a, b] n F () = f() F k = f( k ) + ɛ ( F () = lim n n n F k = f( k ) + ɛ k= k= ( ) ɛ E n n F k f( k ) = k= k= n n ɛ k= k= n k= F k = F (b) F (a), E 6) F (b) F (a) = lim k= ) F k = f( k) lim ɛ = k= n f( k ) = n ɛ E b a k= f()d = E(b a) 7) ɛ 8) 7 f() =, F () =,a = < < < < n = b { k }, F ( k+ ) F ( k ) = ( k + ) ( k ) = ( k ) + k ( ) + ( ) n n n {F ( k+ ) F ( k )} = ( k ) + ɛ k (ɛ k = k + ( ) ) k= k= n k= {F ( k+) F ( k )} = F ( n ) F ( ) = F (b) F (a), a k b ɛ k = ( k + )) ( k + ) ( ma( a, b ) + (b a)) k= E = ( ma( a, b ) + (b a)) n {F (b) F (a)} n ( k ) = n n ɛ k ɛ k E = E(b a) k= k= k= k= n lim ( k ) = F (b) F (a). k= b a d = F (b) F (a) 6) k E ( ). 7) [ k, k+ ] f() 8) α α, β lim β = α β β.

15 = 5 = f() = n = 5 n = V V V V V V k (k+) [ k, k+ ] f(), V j = ma{v k } (=\ ) f() f() = ( = ) f() = [, ] k = k (k =,,,, n) n =. n f() < V k = f( k ) f( k+ ) = n k (, ] = k n = n lim V k = lim n k n 5 5 n k + = n ` k + = n 5 n k(k + ), V k (n )V k k = 5 V = f( ) f() = = n V k, V n ( ) f() V f() f () lim V = f () ) f () a b [a, b] n { k } [ k, k+ ] V k, V k V lim V =, + h [ k, k+ ], f( + h) f() = hf (η) (η + h ) f () f () [a, b] M f( + h) f() = h f (η) h M M, + h [ k, k+ ], f( + h) f() M V k M. lim V k = Q.E.D. V k M M M k ( ) M f()

16 = 6 n 6 J n = f(η k ) n =, n =, n = η k (k = k=,,, n) J n F () F () f() = e, [, ] { k } n k = k n (k =,,,, n), = k+ k n n η k (k =,,,, n) [ k, k ] J n = f( k ), I n = f(η k ), I n J n (e ) lim I n J n = k= k= F () = f() F (), n k= f(η k)δ k = F (b) F (a) ( ( )) ( ( ))

17 = 7.. (,, g() = e d = lim k= e d = lim k= n f(η k )δ k = lim n n k= n g(η k )δ k = lim n k= ( ) k n n e k n ) =, (e ) = e, f() = [ ] n = = [ n = e ] = e e sin d f() F () = f() F () f() [a, b] (a < < b) F () = f() F () [a, ] n N, M n N a < n (N + ), n M < (M + ) n N, M, { k } = a, = n (N + ), = n (N + ),, M N = n M, M N+ = [ k, k ] (k =,,,, M N + ) f() m k F n () = M N+ k= m k δ k = M N+ k= m k ( k k ) = M N+ (.) n k= m k (), ) = f() n = f() n a N n N+ M M+ n n n n N a n+ N + n+ M M + n+ n+ n+ ( ) F n () n (f() >.) F n ()

18 = 8 ) F n () (b a) ( [a, b] f() ) lim n F n() 9) M N+ F () = lim F M N+ n() = lim m k δ k = lim m k n n n k= F () F () F () = f() h > ( + h b), [, + h] f() m, M, [ M n, + h] ) n (n =,,, ) k= n f() m n, = f() = M m n h F n ( + h) F n () M h = m = m n h m n F n( + h) F n () h M n N a n N+ M n n n ) + h m F ( + h) F () h M m, M n h h + f() M f() m f(). lim h + F ( + h) F () h = f() h < f(), f() = F () = lim k= n f(η k )δ k = F (b) F (a) (a = < < < < n = b, k η k k, δ k = k k, = ma{δ k }) ( ) {η k } 9)., a n M ( ) a n a n+ (n =,,, ) lim an n ) [, + h] n. ) a n b n (n =,,, ) {a n}, {b n}, lim an lim n n bn

19 = 9 ( ) f() [a, b] a b F () = a f()d = lim k= n f(η k )δ k ( a = < < < < n =, k < η k k, δ k = k k, = ma{δ k }), ( ) {η k } F () F () = f() ).5 () f(), a k = F (k + ) F (k) = f() = F () = lim h F ( + h) F () h S n = n a k = F (n + ) F () k= = lim k= n a k = a n+ = S n+ S n k= F () = def f()d = a n k= n f(η k )δ k = F (b) F (a) f(η k )δ k = F () = lim h F ( + h) F () h = f() ) ( ) f() ( ) {η k } ( ( )) S() Z S() def = f()d a S () = f() ( ) = f() =, t = a, t = S() S () = f() ( ) (i) : F () F () = f(). (ii) F () = f(), ( ) {η k },. (.) (ˆ ˆ ;),,

20 = S n = n a k = a n+ = S n+ S n k= a n S n S n = n k= k k = S k S k (k =,,, ), S n ( ) F () = f() F () a k+ a k = F k F k f()d n k= a k ) b a f()d n k= a k 8 I = sin sin + cos d, J = () t = I = J () I (.) () t =, cos sin + cos d dt = d, t I = = sin sin + cos d = cos t cos t + sin t dt = sin ( t) sin ( t) + cos ( cos sin + cos d = J t) ( dt) () I + J = () I = J sin sin + cos d + cos sin + cos d = sin + cos sin + cos d = d = I = J = ( ) I J sin cos =.( ) F () = f() F () a k+ a k = F k F k f()d n k= a k ( ) ) k, k a k+ a k = F k F k

21 = t =, = sin() dt = d, t = cos(), f(sin )d = = f ( ( )) sin t ( dt) f(cos t)dt = f(cos )d. sin cos = 5 S n = a + a + a + + a n S n = n n {a n } S n n 6 F () = (t ) sin t dt F (). 7 8 n =,,, I = I = n k= + ( ) d k k + (n k + )

22 g() = u g (). ( ) g() = u f(g())g ()d = f(u)du f(u) F (u) F (u) = f(u) df (g()) d = df (u) d = F (u) du d = f(u) du d = f(g())g () f(g())g () F (g()) f(g())g ()d = F (g()) + C = F (u) + C = f(u)du () ( A tpe) g() = u, g(a) = α, g(b) = β β α f(g())g ()d = b a f(u)du α β u a b f(g())g ()d = F (u) + C = F (g()) + C ( ), β α f(g())g ()d = [ ] β F (g()) = F (g(β)) F (g(α)) = F (b) F (a) = α b a f(u)du () g() = u g () = du d. g ()d = du. f(g()) g ()d = f(u) du g ()d du

23 9 ( A tpe) () log d () e e log d () log d = log (log ) d log = u d = du. log d = u du = u + C = (log ) + C du u = log, d ) =. d = du () () log d = (log ) + C e e log [ (log ) ] e d = = (log e ) e (log e) [ = u ] = = udu, log = u d = du. e e u log e log e e e log d = [ u u du = ] = = ( ) (A tpe), f(g())g () g() ( A tpe) () sin cos d () + d () d ) j d (log ) ff = log (log ) = log d

24 () sin cos d = sin (sin ) d sin = u cos d = du. sin cos d = u du = u + C = sin + C () + d = + ( + ) d, + = u ( )d = du., + d = u du = log u + C = log( + ) + C () d = ( ) ( ) d = u d = du. d = u d = du = ( u) + C = ( ) + C ) (, u) (t, ) ) ( B tpe) = g(t), a = g(α), b = g(β) b f()d = β a α f(g(t))g (t)dt a b t α β ( B tpe) () d () + d () = sin θ( θ ), d = cos θ, d = ). 8 < : = 6 6 θ 6 ( sin 6 θ) cos θdθ = [ ( + cos θ)dθ = θ +. sin θ ( d d {f()}n = n{f()} n f () (n=\ ) d d log f() = f () f() R {f()} n f ()d = {f()}n+ + C (n=\ ) n+ d = log f() + C R f () f() (),() ). ] 6 cos θdθ = +

25 5 () = tan θ( < θ < ), d = cos θ, θ. + d = tan θ + cos θ dθ = dθ = [ ] θ = g()g () (.) sin θ = cos θ, + tan θ = cos θ, cos θ = tan θ. a = a sin θ ( θ ) a +, a + = a tan θ ( < θ < ) a = a cos θ ( θ <, < θ ) ) Tpe A Tpe B Tpe B Tpe A a d () 9 () a a d () a a d () () sin d ( + ) + d 5 (.) ) = a cos θ, = a θ θ sin θ sin θ,tan θ = a sin θ ( θ ) = a sin θ ( θ < )

26 6. ( ) () = g(t), a = g(α), b = g(β) b f()d = β a α f(g(t))g (t)dt a b t α β.. g(t) g(t) t α = t t t t k η k µ k t k χ k k k (a) n (b) = f() ( ) b a f()d = lim k= n f(χ k )( k k ) ( ), a = < < < < n < n = b, χ k k χ k k, δ k = k k. i = g(t i ), χ i = g(η i ) g(t), k k = g(t k ) g(t k ) = g (µ k )(t k t k ) µ k (t k < µ k < t k ) b β = t n t = g(t) a f()d = lim k= n f(g(η k ))g (µ k )(t k t k ) f() t = ma{t k t k } t. 5) g(t) f(g(t)) [t k, t k ] η k, t n f(g(η k ))g (µ k )(t k t k ) k= n f(g(η k ))g (η k )(t k t k ) k= ( ) b a f()d = lim k= = lim = t k= β α n f(g(η k ))g (µ k )(t k t k ) n f(g(η k ))g (η k )(t k t k ) f(g(t))g (t)dt 5) ( ).

27 7 k = g (µ k ) t k d f (t)dt ( ) (.) f(ηk ) δ k f() d.. g(t) = f() g(t) α t γ γ t β g(α) = a, g(γ) = c, g(β) = b, a < b [α, γ] l α = t < t < t < < t l = γ g(t ) =, g(t ) =,, g(t l ) = c α = t t t l l (c) (a) n (b) γ α f(g(t))g (t)dt = lim t k= l f(g(η k ))g (η k )(t k t k ) (η k, [t k, t k ] ) t l g(t) γ = t l k k = g (η k )(t k t k ) δ = t m β = t n t lim t k= = g(t) η k [t k, t k ] g(t) l f(g(η k ))g (η k )(t k t k ) = lim k= l f(χ k )( k k ) (χ k [ k, k ] ) α t γ g(t), k k < (k =,,, l) γ α f(g(t))g (t)dt = lim k= l f(χ k ))( k k ) = γ t β g(t) a c f()d ( = c a ) f()d β γ f(g(t))g (t)dt = b f()d = a f()d + b c c a f()d, β α f(g(t))g (t)dt = a ( a f()d + f()d + b f()d c c a a ) = b f()d β α f(g(t))g (t)dt = b f()d a

28 8 Comment = g(t) t k = t k+ t k > k > ( g (t k ) > k > g (t k ) < k < ) g (t) < a c t α γ, γ α f(g(t))g (t)dt = c a f()d t α β a c a b t n f(g(t k ))g (t k )δt k = k= } {{ } α t β l m f(g(t k ))g (t k )δt k + f(g(t k ))g (t k )δt k + k= } {{ } α t γ l m f( k )δ k + k= } {{ } δ k n f( k )δ k k=m } {{ } a b k=l k=l } {{ } γ t δ n + f( k )δ k f( k )δ k } {{ } δ k k=m } {{ } δ k n f(g(t k ))g (t k )δt k k=m } {{ } δ t β. δt k β α f(g(t))g (t)dt = b a f()d, t α β a b (g(t), ) a b, a b,. ) g(t)

29 9. ( ) = g(t) δt k = t k+ t k > k+ k >. k+ = g(t k+ ), k = g(t k ) ) f() > f()δ k = f()( k+ k ) = k, = k+, = f(), = δs δi = f()δ k = f()( k+ k ) { { f() > (i) k+ k >, (iv) f() < δi = δs. k+ k < { { f() > (ii) k+ k <, (iii) f() < δi = δs k+ k > j (i) f() > k+ k > j (ii) f() > k+ k < j (iii) f() < k+ k > j (iv) f() < k+ k < k k+ k+ k = f() = f() k k+ k+ k = f() = f() { f() > (i) k+ k > f( k ) = f(g(t k ))( k = g(t k )). δ f (t k )δt k δ k : δt k = g (t k ) : = f(g(t))g (t), : g (t k ), g (t k ) : ( ) f( k )δ k f(g(t k ))g (t k )δt k δt, δ, = b f()d = β a α f(g(t))g (t)dt f() < g (t) < = f(g(t))g (t) = f() = f(g(t)) g (t) f(g(t k ))g (t)δt k f( k )δ k f(g(t k ))δt k k t k+ t k t k+ t k t k+ g (t k ) g (t k ) t

30 I = d, J = e et dt = t d = dt, t I = J h (h > ) t δt h δt = ( + h) = h + h h e t δt e h = e δ I = J = e = e t e e t + h ( + h) δ δt I = d, J = e et dt = t d = dt, t I = J h (h > ) t δt δt h. e t δt e h = e δ. e t δt = e δ. δt, δ I,J I = J < < δ = h >, δt = h < e t δt < e δ <. ( ) I = e, =, =, = ( ) J = e t, =, =, = ( ) =. = e = e t e δ + h t ( + h) δt = h

31 I = ( )e d = t ( )d = dt, t J = e t dt I = J = h (h > ) t δt ( )d = dt, δt = { ( + h) ( + h) } ( ) ( )h ( )e δ = ( )e h e ( )h = et δt I J I = J < < <, δt = ( ) < ( )e δ < et δt <. ( ) ( ) < < >, δt = ( ) > ( )e δ > et δt >. ( ) = et, =, t =, t = S, I = ( S) + } {{ } }{{} S = = ( )e ( )e = e + h + h g() g( + h) g() e t ( )e ( )h ( )h t t t = t = + h + h g() g( + h) g( + h) g() < < < <

32 CG δi = ( )e δ, δj = et δt, δs δi = δj = δs δi = δj = δs ( )e δ et δt t t t t t t

33 Tpe B.Tpe A 5 I = d = sin θ ( θ ) d = cos θdθ, θ sin θ cos θ dθ = cos θ dθ I = J θ δθ (δθ > ) J = δ cos θdθ = d, δ cos θδθ < θ < δ >, cos θ >, δi = δ cos θ cos θdθ δj = cos θδθ I I = J =. cos θ = cos θ θ δθ cos θ sin θ + δ θ θ + δθ θ δ = cos θδθ Comment = sin θ θ θ = tan θ θ I = sin d, J = sin tdt = f() = sin = sin t I = J I = + d = g(θ) = tan θ = f() = + θ = f(g(θ)) = f(g(θ))g (θ)

34 . d = sin θ = θ θ = g(t) g (t) g(α) = a, g(β) = b = sin θ ( θ ) d = cos θdθ.. θ 6, 6 θ = sin θ θ 5 6 d = = = 6 6 ( sin θ) cos θdθ [ sin θ + θ cos 6 θdθ = ] 6 = (cos θ + )dθ + d = = = = ( sin 6 θ) cos θdθ = 5 cos 6 θdθ + ( cos θ)dθ (cos θ + )dθ + [ sin θ + θ ] 5 6 [ + sin θ θ cos θ cos θdθ ( cos θ )dθ ] 5 6 = + ( 5 ) ( ) = +, cos θ = cos θ = { cos θ ( θ ) cos θ ( θ 5 6 ) θ 6, θ f(g(θ))g (θ)dθ = cos θ cos θdθ 5 6 f(g(θ))g (θ)dθ = 5 6 cos θ cos θdθ 6 f(g(θ))g (θ)dθ = θ 6

35 5 = g(θ) = sin θ g (θ) = cos θ = g(θ) = tan θ g (t) = cos θ = ±, ±, ± 5, : I = + d = tan θ ( < θ < ) d = cos θ dθ. θ I = + d = + tan θ cos θ dθ = dθ = g ( 5 ) = θ 5 I = 5 + tan θ 5 cos θ dθ = dθ = 5,! = tan θ = dθ = θ k θ k > d = k k > ( g(θ k ) = tan θ k = k ) I = I = 5 f(g(θ))g (θ)dθ = f(g(θ))g (θ)dθ = 5 dθ = dθ = [ ] θ [ θ ] 5 = > = > I +I = =\ g (θ) dθ = θ k θ k > d = k k < I I 6) = + = tan θ 5 θ 6) tan θ θ = R +tan θ cos θdθ, R R +tan θ cos θdθ = lim α R α +tan θ cos θdθ = lim α α dθ = lim α `α = g(t)?

36 6. f() > S = b a f()d f() < S = b { f()} d, a, a b f() g(), f() g(), = f(), = g(), = a, = b S f() g() = f() = g() S = b a def = lim {f() g()} d k= n {f(η k ) g(η k )} ( k k ) ( a = < < < < n = b, k < η k < k, = ma{ k k }) a Q b = f() = g() S ( ) {f() g()} (Q) ) (a) k k n (b) = f() = g() f(),g() f() g() = f() = f() = g() = α { f(α) = g(α) f (α) = g (α) ) = g() f() g() = ( α) h()(h() ) f() g() = = α. α ) (?) :Q δ k Qδ k ( δ k = k k ) ) f() g() = F ().(i) F (α) = F (α) = F () ( α) F () ( α) Q(), a + b F () = ( α) Q() + a + b F () = ( α)q() + ( α) Q () + a ( )

37 7 6 C : = f() = + a + b l : = g() = m + n = = f(), = g(), =, = S a, b, m, n C : = + a + b = f() = g() = f() = g() = f() g() = + a + b (m + n) = ( ) l : = m + n S = = {f() g()} d [ ( ) ( ) = ] = 6 Comment = ( ) = ( ) ( a, b, m, n ) f(), g() = α f() = g() ) { f(α) = g(α) = f() = g() = α f (α) = g (α) C : = f() = + a + b l : = g() = p + q + r =, = Q = f() = g() S a, b, p, q, r. < f() = sin, g() = a cos = f() = g() a = f(), = g() S a. ( ) F (α) = F (α) = j aα + b = a = a = b = F () ( α) F () = ( α) h() ( ), F (α) = F (α) = ). Q.E.D.

38 8. a C : j = (t) = (t) b C C : { = (t) = (t) d d dt dt t t (t = f () ) = f() = (t) (t) >, a < b = a, = b, C S 7 S = b a d = β α d dt, (α) = a, (β) = b ( ) dt () { = θ sin θ ( θ ) = cos θ S ( k, k ) ( k, k ) (, ) (, ) ( n, n) (, ) = k k = θ sin θ, d = ( cos θ)dθ. θ θ k θ k θ Comment = θ sin θ S = = = [ = d = ( cos θ) ( cos θ)dθ ( cos θ + cos θ ) dθ ( cos θ + θ sin θ + sin θ + cos θ θ [, ] = θ < θ < θ < < θ n = n θ k, k, k, k k = (η k )(θ k θ k ) (θ k < η k < θ k ) S = lim k= n k ( k k ) = lim θ k= n (θ k ) (η k )(θ k θ k ) = ] = d dθ dθ ) dθ

39 9 θ δθ (δθ > ) δ ( cos θ)dθ = d, δ ( cos θ)δθ δ >, cos θ, δi = δ ( cos θ) ( cos θ)dθ δj = ( cos θ) δθ δ S = lim δi = lim θ δj = ( cos θ) dθ cos θ ( cos θ) = ( cos θ) + δ δ = ( cos θ)δθ θ sin θ θ δθ θ δ ( cos θ) δθ, ( δθ ) t t t t t t

40 d dt (, ) ( d dt >, d dt <. ) 8 C : { = t + = t + t + C S t= 5 t = 5 t = S S = f () 5 t = ( t ) d dt = t, d dt = t + t d dt d ( dt ) ( ) + + ( ) + ) ( ( 5 9 ) 5 C t C = f (), t C = f (), S, S S = 5 f ()d, S = f ()d = f () = f () (, ), 5 t S 5 = f () (, ), t S = S S S = S S = Comment d dt dt d dt dt = d dt dt = ( t + t + ) tdt = 9 t δt (δt > ) δ dt = d, δ tδt < t <, δ <,, δi = δ ( t +t+) tδt δj = ( t +t+) tδt ( ) ( ) < t <, δ >,, δi δj ( ), δi t S, δi t S lim δ δi = ( t +t+) tdt S

41 t = t + t + t( t + t + ) = t( t + t + ) g(t + δt) g(t) 5 g(t + δt) g(t) δ = tδt δ = tδt = g(t) = t + t + δt t t t + δt t( t + t + ) t δi = δ δj = ( t + t + )tδt ( ) (t) >, t (, ) C : = (t), = (t) S t = α a b t = β S = β α d dt, (α) = a, (β) = b, α < β ( ) dt ( S).

42 9 C : { =sin θ cos θ =sin θ + cos θ + ( θ ) S d dθ = cos θ + sin θ = ( sin θ + ), d dθ = cos θ ( sin θ = sin θ θ 6 6 d dθ d dθ «««! «! + C θ, S = = = = = f ()d d dθ dθ d dθ dθ + d dθ dθ ) θ = f (), θ 7 = f () f ()d + d dθ dθ + d dθ dθ f ()d d dθ dθ d dθ dθ (sin θ + cos θ + )(sin θ + cos θ)dθ θ =, θ = 7 = f () θ = 6 { = sin θ + cos θ + ( } + ) sin θ cos θ + sin θ + cos θ dθ ( cos θ = + + cos θ + + ) sin θ + sin θ + cos θ dθ [ + = θ + sin θ + ] cos θ cos θ + sin θ = ( + ) S θ = = f () θ = C ( ) C C : = f(t), = g(t) (α t β) t α β (f(t), g(t)) C α t γ d d dt, γ t δ dt, δ t β d dt, C : S CG web site. t = α, β t = γ S t = δ a b c

43 ? = g(t) t δt(δt > ) δ, δi = δ. = g(t) d = g (t)dt. δt δ g (t)δt δi = δ g (t)δt (i) α < t < γ. g (t) > δ >. δi (ii) γ < t < δ. g (t) < δ <. δi ( ) (iii) δ < t < β. g (t) > (i) δi δi = δ t = α t = β t = γ t = γ t = δ δi = δ t = δ k k n m k k n t = α t. t k t k. t n = γ t n = γ t n+. t k t k t = f(t). t m = δ t = f(t) t m = δ. t l = β t = f(t) t = α t = α t = γ t = γ S t = δ t = δ S S

44 S, S, S l g (t k )δt k = k= n m g (t k )δt k + g (t k )δt k + k= k= k=n n m δ k + δ k + k=n S + ( S ) + S l δ k k=m l g (t k )δt k k=m δt = J = β α g (t)dt = β α d dt dt = S + ( S ) + S = S δi = δ t = α, β t = α. t k t k. X X + h δi = δ ( ) t i t i. β t = f(t) (, ) C (, ) [X, X + h] (h > ), t (, ) g (t)δt = δ > δj = g (t)δt = δ > δj ( ) g (t)δt = δ < δj = g (t)δt = δ < δj ( ) ( ) t k t t k t i t t i δj C = X, = X + h δs t α β δj δt J J = β α d dt dt = β α g (t) dt = lim g (t) δt = lim δj = lim δs = S t t t

45 5 δi = δ ( ) t = α, β t = α X δi = δ X + h t k t k t i t i β = f(t) t (, ) C C S [X, X + h] (h > ) t (, ), g (t)δt = δ < δj = g (t)δt = δ < δj ( ) ( ) g (t)δt = δ > δj = g (t)δt = δ > δj ( ). ( ). t k t t k t i t t i δj, C = X, = X + h δs ( ) t α β δj δt J J = β α d dt dt = β α g (t) dt = lim g (t) δt = lim t S = J = β α t d dt dt δj = lim ( δs) = S t, > < (.)

46 6 C > < ( = f() = g() ) t = α, β X X + h δi = δ ( ) δi = δ ( ) t = α. t k t k. t i t i. β t = f(t) (t) < C <, (, ) C. [X, X + h] (h > ) = X C,. =, =. δs C = X, = X + h, δ g (t)δt >, δ g (t)δt < δt = t k t k, δt = t i t i δt δt δj δj + δj = g (t k )δt + g (t i )δt δ + δ = h + ( h) = ( ) h ( ) δj + δj = δs δj t J J = β α d β dt dt = g (t) dt = lim g (t) δt = lim δj = lim δs = S α t t t ( ) δs > < ) ) < δj = g (t)δt = δ < δj ( ) ( ) δj = g (t)δt = δ > δj ( ) δt δt δj C = X, = X + h δs

47 7 t = α, β t = α, β δs a S δs b X X + h α α δt δt δt δt β β t t (, ) C [X, X + h] (h > ) C = X,,,,,,,. [X, X + h] t δt, δt, δt, δt, C S C = X, = X + h δs δs a, δs b, δ g (t)δt >, δ g (t)δt < δt δt δj = g (t)δt δj + δj δ + ( δ) = ( ) δ = h δs a δt δt δj δj + δj δ + ( δ) = ( ) δ = h δs b, δt, δt, δt, δt δj δs a + δs b = δs δj t J J = β α d β dt dt = g (t) dt = lim g (t) δt = lim δj = lim δs = S α t t t C. )

48 8 ( ) C ( ) C C : = f(t), = g(t) (α t β) t α β (f(t), g(t)) C, C S t = α, β S S = β α d dt dt ( ) C ( ) C C : = f(t), = g(t) (α t β) t α β (f(t), g(t)) C, C S t = α, β S β S = d α dt dt C 8, 9 C 5) C t β t γ ((t), (t) = (h(t), ) C (h(t) h(β) = f(β), h(γ) = f(α) ) β t γ S = γ α d β dt dt = d γ α dt dt + d β β dt dt = d γ α dt dt + d β β dt dt = d α dt dt 8 < = f(t) C : = g(t) : α t β 8 < = f(t) C : = g(t) : α t γ t = α t = β t = α, γ {z } t = β C : = l : = + t (t=\ ) () t () =, = S { = sin θ + cos θ 5 C : ( θ ) = sin θ + cos θ 5) = f(), = g(), = a, = b S

49 9. (r, θ), r θ (, ) r θ (, ) = r cos θ, = r sin θ, r = + S C, C : r = f(θ) C θ, α θ β S α β S = β α {r(θ)} dθ r θ r θ S r S = r θ S = δθ k S k α θ β, α = θ < θ < θ < < θ n = β [α, β] n θ θ k θ θ k S k θ k η k θ k S k = {r(η k)} δθ k (δθ k = θ k θ k ) η k (θ k < η k < θ k ) ( ) S = n S k = lim k= n θ k= {r(η k)} δθ k = lim n θ k= β {r(η k)} (θ k θ k ) = α {r(θ)} dθ ()

50 5 { = e θ cos θ C : ( θ ) C = e θ S sin θ j = e θ cos θ C : = e θ sin θ ( θ ) (, ) S θ { = r cos θ = r sin θ r(θ) = e θ S = r dθ = (eθ ) dθ = [ eθ dθ = eθ] = (e ) C S = = = = d dθ dθ = e θ (sin θ sin θ cos θ)dθ = eθ dθ [e θ] [ e θ sin θ e θ sin θ(e θ cos θ) dθ = e θ (cos θ + sin θ)dθ ] = (e ) θ cos θ sin θ e dθ e θ sin θ(e θ cos θ e θ sin θ)dθ, (.) 6 C : { = θ(sin θ + cos θ) = θ(sin θ cos θ) ( θ ) = S

51 5. S() : = a, = b ( a b) V V = b a S()d S() a b V = lim k= n S(η k )( k k ) = lim k= n S(η k )δ = b a S()d S(η k )δ ( a = < < < < n = b, k < η k k, δ k = k k, = ma{δ k }) a k η k k b = f() = a, = b (a < b) f() = f() S() = f() = {f()} V a S() b V = b {f()} d = b a a ( = f()) d = f() f()

52 5 = g() = a, = b (a < b) g() S() d S() = g() = {g()} S() = g() V V = d {g()} d = d c c ( = g()) d c, ( ) = sin ( ) = + Q Q R QR (, ) QR R z z t R Q = + = sin S() = QR V = S()d = Q d = ( + sin ) d { = ( + ) ( + ) sin + sin } z d { ( + ) = ( + ) + } ( = + ) ). ) Z Z Z ( + ) sin d = ( + )( cos ) d = ( + )( cos ) ( + ) ( cos )d = ( + ) cos + sin + C Z ( + ) d = ( + ) + C, Z Z sin d = cos d = sin + C

53 5 r V + = r = ± r. : r ( r r) r V = r = r r d = r r (r )d (r )d = [r ] r = r r r = r r = f(), = g(),., : + ( ) V ( ) +( ) = = ± + ( ) ( ) ) V = = = 8 ( + ) d ( ) d { ( + ) ( ) } d d = 8 = ( ) = + = + ( ) = ) ( ), R = sin θ.k..

54 5 = f() = g() a b f() g() = f(), = g(), = a, = b V = f() f()(a b) V f()(a b) V = g() V = = b a b a {f()} d b a {g()} d [ {f()} {g()} ] d a b = f() = g() a b = g() = f() S() f() g(), S() S() = {f()} {g()} = [ {f()} {g()} ] V = b a S()d = b a [ {f()} {g()} ] d, V = b a [f() g()] d () = f() = g() (f() g().) <

55 55 = = + V = = + = = ( ) = +. = + = =, = = + V = ( + ) d + [ 5 = 5 + ] ( + ) d + [ ( + ) = = ] ( ) d [ ] = + 7 = sin = cos

56 56. 5 ( ) { = θ sin θ ( θ ) = cos θ V V = d = θ sin θ d = ( cos θ)dθ. = cos θ. θ. (, ) C V = = d = ( cos θ) dθ = ( cos θ) ( cos θ)dθ ( cos θ + cos θ cos θ)dθ cos θdθ = ( sin θ)(sin θ) dθ, sin θ = u, cos θdθ = du, cos θdθ =. ( = cos θ ) V = { cos θ + } [ ( + cos θ) dθ = θ sin θ + ( θ + sin θ ) ] θ u. = 5 Comment θ δθ (δθ > ) δ ( cos θ)dθ = d, cos θ δ ( cos θ)δθ δ >, θ δθ δv, + δ δ = ( cos θ)δθ θ sin θ δv δ = ( cos θ) ( cos θ)δθ δθ δθ V = lim δv = ( cos θ) dθ θ θ = θ sin θ

57 57 6 ( = cos ) = cos ( ), V = cos V = = cos = cos (cos cos ) cos = cos d = sin d,. V =, V = = Comment sin d = (sin ) d = d d = ( sin )d = ( cos ) d = [ sin ] [ ] ( cos ) sin d = V = ( ) sin d ( cos )d = cos d = t, = cos t V = d. = cos t = t δ(δ > ) δ, δv, d = sin d, δ sin δ δ <, δv δ ( sin )δ = sin δ δ V = lim δv = lim sin δ = sin d

58 58 7 ( ) C : { =sin θ cos θ =sin θ + cos θ + ( θ ) V ( ) C C θ, 7 θ = f (), θ 7 = f () V = = = = = = {f ()} d d dθ dθ d dθ dθ + d dθ dθ 7 7 {f ()} d + d dθ dθ + d dθ dθ + 7 (sin θ + cos θ + ) (sin θ + cos θ)dθ 7 d dθ dθ d dθ dθ {f ()} d θ =, θ = 7 = f () θ = 6 { sin θ + sin θ cos θ + cos θ + 6 sin θ + 6 } cos θ (sin θ + cos θ)dθ θ = = f () θ = (, ) sin θdθ = sin θdθ = cos θdθ = cos θdθ =. sin θ cos θdθ = ( cos θ) cos θdθ =, sin θ cos θdθ = ( sin θ) sin θdθ =. V = ( 6 sin θ + 6 ) cos θ dθ = { ( cos θ) + } ( + cos θ) dθ = 6( + ) Comment ( ),

59 59. t δt (δt > ) V δv δ = (cos θ + sin θ)δθ > V (δv > ), δ = (cos θ + sin θ)δθ < V (δv < ), δv > δv < V. z z z z z

60 6 t = α, β δj = h t = α, β X X + h δj ( ) h t = α. t k t k X X + h δj ( ) ( h) ( ) t = α. t k t k δj = ( ) ( h) ( ).. t i t i t i t i.. β t = f(t) β t = f(t) C = f(t), = g(t) (α t β) ( ) t α β (f(t), g(t)) C. (, ) [X, X + h] (h > ) C V, = X, = X + h δv, t (, ) g (t)δt = δ > δv = δ = g (t)δt g (t)δt = δ < δv = ( δ) = g (t)δt. δj = g (t)δt,,,, δt = t k t k, δt = t i t i δt δt δj δj = δj + δj = ( ) g (t k )δt + ( ) g (t i )δt ( ) h + ( ) ( h) = ( )h = δv J = β α g (t) dt = β d α dt dt, β J = g (t) dt = lim g (t) δt = lim δj = lim δv = V α t δt t C < () ) (*) C ) ( ) C < ( ) ( ) <.

61 6 ( ) C ( ) C C : = f(t), = g(t) (α t β) t α β (f(t), g(t)) C C, C V t = α, β β V = d α dt dt ( ) C ( ) C C : = f(t), = g(t) (α t β) t α β (f(t), g(t)) C C, C V t = α, β β V = d α dt dt C = f() C C V β V = d α dt dt t = α, γ t = α, γ t = β {z } 8 = f() = sin ( ) V 9 C : r = a( + cos θ) ( θ ) t = β

62 6. 8 () z (cos θ, sin θ, ), Q(cos ( ) ( ) θ +, sin θ +, ) θ Q, z = z = V z. z. z z. z. z Q A(,, ) z = Q S θ z θ t R Q -t z = t SR = ( t) S + t SQ ( t) + t z = t ( t ) Q z = t Q R R z t R : RQ = R z z : Q z R z = t : ( t) ( z, z ) z = t z S(, t) = t cos θ sin θ + t sin θ cos θ = ( t) cos θ t sin θ ( t) sin θ + t cos θ t t z = t S(t) S(t) = SR = { ( t) + t } V V = S(t)dt = { ( t) + t } dt = [ (t ) + t ] =

63 6, 9 ( ) z A z A(,, ),B(,, ),C(,, ) ABC z V C B z = t ( t ) z, AB Q, AC R, QR H, z = t S(t) AQ : QB = (A Q z ) : (Q B z ) = ( t) : t z A Q = t A + ( t) 6 t B = t + t = t ( t) + t t ( R = Q, R = A = C = ) R = t A + ( t) C = t + t = ( t) + t t 6 t t Q t B R C QR H (i) Q = t t H QR H R S(t) = (R H ) R H = HR S(t) = HR = (R ) = H Q R

64 6 (ii) Q = t t H QR Q R S(t) = (R Q ) R = H + HR, Q = H + HQ R Q = (H + HR ) (H + HQ ) = HR HQ { ( ) } t S(t) = {(R ) (Q ) } = H Q ( ) = t t R V = S(t)dt + S(t)dt = dt + ( ) t t dt = Comment (ii) S(t) = (HR HQ ) Q,R z Q,R S(t) Q R = t (. Q Q R R, A,B,C z A,B,C ABC A B C z = k ABC z V A B C z V V = ( ) ( ) = = = V z A A z A C B C B C B

65 65 A,B,C z A,B,C ABC A B C z = k ABC z V A B C z V ( z =, z =.),.

66 第章 66 体積 例 (立体図形の回転体) の直円錐を 頂点を原点に固定して 平面上を図のようにすべることなく転がす 半径 高さ 円錐の中心軸が一周して元の位置に戻るとき 円錐が通過する領域の体積 V を求めよ 類題 東京電気大) z. z z z z.5. z 原則は 廻してから切るのではなく 切ってから廻せ です 円錐を 平面 z = t で切った断面を考えま す (下の図は 交線が見やすいように底面を抜いてあります ) 図からもわかりますが 平面と円錐の 交線は双曲線になります (数 C の範囲)

67 67 z z Q R S M Q S R M Q t θ M z = t z Q, R S, RS M z = t S(t) S(t) = (R Q ), R R = (,, t) R = 9 + ( ) =. R = R = t z., tan θ = Q = = t S(t) = { ( t ) ( ) } t = ( ) t t t V = S(t)dt = ( t ) dt = 8 Comment z, z = t, QS ( QS QS S(t) ) QS t ( ).

68 68... ( ) z A(,, ), D D : = D z E V z z = t D R D Q H A Q R Q H Q z = t ( < t < ) Q,,Q =, Q, z = t z R(,, t), R Q H, S(t) S(t) S(t) = (RQ RH ) = HQ = R S(t) D = D, D z E z = t E E, E E V = ( ) = = H = A AH = t. S(t) = H = ( t ) V = S(t)dt = ( t )dt = [ ] ( t )dt = t t = Comment (6 ) D Q Q Q Q Q Q D Q D D D Q

69 69 ( ) z A(,, ), D D : = D z E V z = Q z = A Q Q R Q R Q z = t u = H Q = M = D,Q,R,H, S(t),Q =, Q H Q, {( S(t) = (RQ R ) = RQ + QQ ) ( R + )} = (RQ R ) D = D, D z E E E =, z = u u z D (, ), V = (u ) + z = u = ± z + RQ R = ( + t ) ( t ) = t S(t)dt = ( RQ R ) dt = t dt = = ( ) Q M HQ H = (HQ + H)(HQ H) = HM Q = t ( ) ( ) Comment Q = Q = D H HQ = Q, ) ). V = =

70 7 ( ) z A(,, ), D D : z = + D z E V z z R Q H Q R Q R Q z = t A H Q D C,Q,R,H,,Q, S(t) S(t) S(t) = (RQ RH ) = HQ = R S(t),D = D, D z E E E z 5,Q R = AH Q = Q z, C D, C 5) D, C z V = ( ) = = H = A AH = ( t ) = t. V = Comment S(t)dt = ( t )dt = ( t )dt = [t t] = D z ABCD A(,, ), B(,, ), C(,, ), D(,, ) z E V 5).

71 7 () z D D : ( ) + + z, D z E V z z z = t t R(,, t) Q t A H Q R T T T Q H A S(t) H Q Q = t z = t ( < t < ) D z = t,q,r,t R Q H, S(t) S(t) { S(t) = (R RT ) = + ( t ) ( t ) } = t V = S(t)dt = t dt = = Comment QT =, Q T Q T z E E E z Q,T Q T z E E E, (.) z ABCD A(,, ), B(,, ), C(,, ), D(,, ) z E V

72 7. S(t) S(t) () (,?) 5 ( ) ( ) V.5 z. z.5. z = k z = k( k ) z k k k (,, k) k + z, = + z, = z = k C z + z C (,, z) C : + z ( ) C z + z C : + z ( ) D (,, z) { + z + z z = k, () { { + k + k k k k k

73 7 z = k ( k ) S(k) S(k) = ( k ) = ( k ) V = S(k)dk = ( k )dk = 8 ( k )dk = 6 C z = k k.) C z = k k ) (.) + (5 ) z

74 7 6 ( ) z ) D, A (,, ) B ( ),, D E V z z = t : z = + B z t + R R Q B S Q S A A t t D +. : z = +. E (,, z) { + z + = t ( t ) = t { + t z t + { t t z t + = t S(t) S(t) = S Q = ( t t + ) V = S(t)dt = t ( t + ) dt = t t dt + t dt t = u, tdt = du, t t dt = t dt t dt = + V = ( ) 8 t u ( u ) [ du = ] u = + = = 8 Y Y = t t

75 75 S(t) = S Q = ( t t + ) () z B S t z : z = + R S A Q t A B t t + Comment (), = t CG

76 76 7 ( ) z, z + + ( ),, +,, z V ( 98 ) = k = k, z = k = k z z = ( k) + k + k + S(k) k k + (, k, ) = k( < k < ) = k ( ) z ( k) + k + k +, k k + ( k) + k + k + = f() = k z = f(k) f(k) = k + >, f(k + ) = k > k k + f() >. = k S(k) S(k) = k+ k { ( k) + k + k + } d = {(k + ) + ( k)} = k + V = S(k)dk = k + dk = 7 = k ( = k )

77 77 5 = e =, e (7 ) [, ] n s n n n s n = f( k )δ k = e k n n = { + e n n + e n + + e n n k= k= } = n (e n ) n e n h = n n, h +. S Comment S = lim n s n = lim n n e e n = lim h + e e h h = e ( ) e d = [ ] e = e S n = n k= f( k)δ k = n k= e k n n S n = n(a+an) ( ) ( ) { (k + ) k = k + (k + ) k = (k + ) (k + ) k k = k f(k) = k k (k =,,, ), k = f(k + ) f(k) n k = k= n n {f(k + ) f(k)} = {f(k) f(k + )} k= k= = {f() f() + f() f() + f() f() + + f(n + )} = f(n + ) f() = (n + ) (n + ) = n(n + ) a n = a + (n )d n a k = k= n n n n {a + (k )d} = a + d (k ) = a k= = a n + d n(n ) k= k= k= = n {a + (n )d} = n(a + a n ) n + d k= k

78 5 78 n 6 J n = f(η k ) n =, n =, n = η k (k =,,, n) k= J n F () F () (6 ) 6, [, ] f() =, F () = n n J n = f(η k ) = f (η k ) n, η k + k + k = n. (i) n = k= k= n J = f(η k ) n = f(η ) = f k= (ii) n = n ( J = f(η k ) n = f(η ) + f(η ) = f k= = (iii) n = Comment + 7 = ( ) ( ) = = = F () F () ) = = F () F () n J = f(η k ) n = f(η ) + f(η ) + f(η ) k= ( ) ( ) ( ) = f f f 7 = = ( 7 + f ) = = F () F () n =, =, F () F () = jf ( ff ) F () + jf ( ) F ( ff ) + jf () F ( ff ) = F (η ) + F (η ) + F (η ) = f(η ) + f(η ) + f(η ) (η ) F () F ( ) = f(),, =, = [ k, k+ ] = f(),, = k, = k+, f(η k ) δ k {η k } = = = J q q 7 q 7 q 7 7 q 9 7

79 5 79 f() = e, [, ] { k } n k = k n (k =,,,, n), = k+ k n n η k (k =,,,, n) [ k, k ] J n = f( k ), I n = f(η k ), I n J n (e ) k= k= lim I n J n = (6 ) f() [, ] () f( k ) f(η k ) f( k ) f( k ) (k =,,,, n) n n {f( k ) f(η k )} {f( k ) f( k )} k= k= n n {f( k ) f( k )} = {f( k ) f( k )} = {f( n ) f( )} = {f() f()} k= k= J n I n = n k= {f( k) f( ηk )},, J n I n = k= n n {f( k ) f(η k )} {f( k ) f( k )} {f() f()} = (e ) k= I n J n (e ) (e ), lim I n J n =. = f() = e f( k ) f( k ) f( k) f(η k ) e Comment k k η k η η η η η 5 J n I n ( n = 5 ) f() δ k = k k J n I n = n n n {f( k ) f(η k )}δ k {f( k ) f(η k )}δ k {f( k ) f( k )} k= lim I n J n = k= k=

80 5 8 5 S n = a + a + a + + a n S n = n n {a n } S n n (( )) n a n = S n S n = ( n n ) { n (n ) } = n (n ) n = a n = n (n ) (n =,,, ) ( ) a n n =,,, n a n 7 5 S n n 8 a n > ( ) S n S n = a n a n > S n > S n, a n < S n < S n. S > S > S > > S 6 S 6 < S 7 < S 8 < S n n n = 6 ( ) 6 F () = (t ) sin t dt F ().( ) d F d = d d (t ) sin t dt = ( ) sin F () + F () F () = ( ) Comment

81 5 8 7 ( ) t =, dt = d, I = + ( ) d t. J = I = ( ) +( ) ( t) ( t) + t ( dt) = d I = J I + J = ( t) t + ( t) dt = + ( ) + ( ) d = ( ) + ( ) d d = I = ( ) 8 n =,,, ( ) I = n k= k k + (n k + ) I = + n + + (n ) + + (n ) + + (n ) (n ) + + (n ) (n ) + + n n + J = n k= (n k+) (n k+) +k J = n n + + (n ) (n ) + + (n ) (n ) (n ) + + (n ) + I + J = I = J n k= k k + (n k + ) + n k= (n k + ) n (n k + ) + k = k= + n k + (n k + ) n (n k + ) + k = = n k= I = J = n ( ) Comment

82 5 8 9 () a a d () a a d () () sin d ( + ) + d (5 ) () = a = a, ( a. a ( a ) a d = = a 8 ) + = ( a ( ) ),, I ` a = a, a = a ( ) a a = a sin θ, d = a cos θdθ, a θ I = = a 8 a ( a ) a ( a sin θ cos θdθ = [ θ + ( + cos θ)dθ = a 8 sin θ ) ] = a 8 cos θdθ a. () a a d = a a t a. a a d = (a ) a d t = a dt = d, t ( ) dt = a a [ ] a tdt = t = a ( ) () sin d = sin sin d = ( cos ) ( cos ) d cos = t sin d = dt, t. sin d = sin sin d = ( t )( dt) = ( t )dt = [ ] t t = ( ) Comment Z sin d =, cos. Z Z sin sin h d = (cos ) ( cos ) d = cos cos i =

83 5 8 I n = R sin n d (n =,,, ) I n+ = Z sin n+ d = Z Z sin n+ ( cos ) d = Z h i sin n+ ( cos ) Z + (sin n+ ) cos d = (n + ) sin n cos d = (n + ) sin n ( sin )d ( Z Z ) = (n + ) sin n d sin n+ d = (n + )(I n I n+) = (n + )I n (n + )I n+ (n + )I n+ = (n + )I n. I n+ = n + n + In ( ) () Z sin d = I = I = Z sin d = = () ( + + ) d = = { + d ( + ) + d + } d ( + ) d = tan θ d = cos θ dθ, θ. ( + ) d = = = ( + tan θ) cos θdθ = [θ + ] sin θ cos θ dθ = ( + cos θ)dθ = 8 + cos θ cos θ dθ + d = ( + ) [ ] + d = log( + ) = log, [ d = ] =, ( + ) + d = ( + log ) = 8 + log + 7 ( ) Comment + d = tan θ d, d, = t, + = t + d

84 5 8 I = sin d, J = sin tdt = f() = sin = sin t I = J ( ) sin = sin = sin t = sin t + h t = t = h sin t = h [, ] n = < < < < n =, = n = h g() = t k = g( k ) = k (k =,,,, n) = t < t < t < < t n = {t k } t [, ] n t = = h I n, J n n n I n = sin k = sin k h, J n = k= n k= k= n sin t k t = k= sin k (h) I n = J n h I n I, J n J Comment I = lim h I n = lim h J n = J g() = I n = J n g() I n J n I n J n I n J n

85 5 85 I = + d = g(θ) = tan θ = f() = + θ = f(g(θ)) = f(g(θ))g (θ) ( ) cos θ k = θ k θ = + = cos θ k tan θ k tan(θ k + θ) θ k θ θ k cos θ θ f(g(θ)) = f(tan θ) = +tan θ = cos θ θ [, ] n = θ < θ < θ < < θ n = = cos θ, g (θ) = cos θ, f(g(θ))g (θ) = cos θ cos θ =., θ = n k = g(θ k ) { k } [, ] n k = k k cos θ θ (k =,,, n). δi k δj k { δik cos θ θ δi k δj k (k =,,, n) n n δi k δj k k= k= δj k = θ = θ n k= + ( k ) k n θ θ = k= lim θ k= n δi k = lim θ k= n δj k + d = dθ ( = )

86 5 86 C : = f() = + a + b l : = g() = p + q + r =, = Q = f() = g() S a, b, p, q, r.(7 ) f() g() =, = ( ), =, f() g() = + a + b (p + q + r) = ( ) ( ) < < ( ) ( ) < S = = f() g() d = [ ( ) ( ) ] + { ( ) ( ) } { } ( ) d = ( )d ( ) d = [( ) ] = 8 ( ) < f() = sin, g() = a cos = f() = g() a = f(), = g() S a.(7 ) = t { { f(t) = g(t) sin t = a cos t f (t) = g (t) cos t = sin t = a cos tan t =. sin t = ± 5, cos t = ± 5 ( ). α α + = sin ( a = sin t + cos t = ± ) ( + ± ) = ± a > 5 α a = 5 ( ) tan t = sin t > t, t < [, ] t α, cos α = 5, sin α = 5. a = 5 f() g() = sin + cos 5 = { 5 sin 5 + cos 5 } 5 = 5 {sin sin α + cos cos α} 5 = 5 {cos( α) } < f() g() = α, α + α+ { } [ ] α+ S = 5 cos sin d = 5 sin + cos α α { } = 5(α + ) sin(α + ) + cos(α + ) ( 5α sin α + cos α) = 5 ( )

87 5 87 C : = l : = + t (t=\ ) () t () =, = S (8 ) 6 t + t t = t = t t S S = + 6 = + (t + t) () l C = + = + t ( + t) = = t + t = t + t = + t ( t ) t = + t ( t + t, t ) t ( ) () = t + t = t t + = t =, t = 6 t t = 6. = t + t d = ( ) t dt, t 6. Comment S = = 6 d = 6 ( t t + t ( t ) ( t t ) [ t dt = ) dt 8 log t ] 6 t = 9 log ( ) a b = (, t) l : = ± b a + t t S = d, = cos θ Z d = log + p + C, Z p d = p log + p + C (..)

88 5 88 C : { 5 = sin θ + cos θ = sin θ + cos θ ( θ ) (8 ) (θ) = sin θ + cos θ, (θ) = sin θ + cos θ { (θ + ) = sin(θ + ) + cos(θ + ) = ( sin θ + cos θ) = (θ) (θ + ) = sin(θ + ) + cos(θ + ) = (sin θ + cos θ) = (θ) ((θ), (θ)) ((θ + ), (θ + )) θ d dθ = cos θ sin θ, d dθ = cos θ sin θ. tan α = α, C θ = α θ α d dθ + d ( dθ + ) ( ) ( ) ( 5 ) ( 5 ) θ = θ = ((θ), (θ)) C θ = S = = = d dθ dθ = (sin θ + cos θ)(sin θ + cos θ)dθ { sin θ + cos θ + sin θ cos θ } { dθ = sin θ + cos θ } dθ { } ( cos θ) + ( + cos θ) dθ = [ θ + ] sin θ = ( ) ( ) ( ) ( ) ( ) ( ) sin θ + cos θ cos θ = = sin θ + cos θ sin θ Q(cos θ, sin θ) A = ( ) A = ( ) a b c d A ad bc C S = ( ) () = ( )

89 5 89 B(, ) Q A (, ) θ Q A(, ) A= B (, ) Comment M = ( a c d b ) M ad bc A(, ), B(, ) M A,B A (a, c), B (b, d). A, B S S = ad bc A B, : ad bc ad bc (Q.E.D.) B (b, d) B A a b c d A (a, c) C : j =sin θ cos θ =sin θ + cos θ + ( θ ) S ( ) C : j =sin θ cos θ =sin θ + cos θ ( θ ) ««««sin θ cos θ = sin θ + = cos θ cos θ sin θ S = ( ) () = ( + ) ( ) C C

90 5 9 C : { 6 = θ(sin θ + cos θ) = θ(sin θ cos θ) ( θ ) = S (5 ) = θ(sin θ cos θ) = θ sin ( θ = θ ( sin θ + cos θ ). cos ) = θ ( sin θ sin + cos θ cos ) = θ ( cos θ ) { = θ cos ( ) { θ = θ sin ( ) θ = r cos φ = r sin φ, r = θ, φ = θ r, δφ δs δs r δφ, dφ = dθ, φ θ. S = r dφ = θ dθ = [ θ ] = ( ) = C (, ) θ θ Comment

91 5 9 7 = sin = cos (55 ) = sin = cos sin = cos =, 5 = sin, = cos V,. = { V = = sin d [ sin ] V cos d } = [ + sin ] = ( cos )d ) ( + ) ( ( + cos )d = + ( V = + ) = + ( )

92 5 9 8 = f() = sin ( ) V (6 ) = g () = g () = g () ( ) (.) [, ] = g (), [, ] = g (), [, ] = g () V = {g ()} d + {g ()} d {g ()} d + {g ()} d = sin d = cos d, = g (), = g (),, = g () V = = = (cos d) + (cos d) cos d + (cos d) (cos d) + cos d (cos d) + (cos d) + (cos d) (cos d) = = f() cos d = (sin ) d = sin sin d sin d = ( cos ) d = ( cos ) ( cos )d = cos + sin + C cos d = sin ( cos + sin ) + C = sin + cos sin + C V = [ sin + cos sin ] + [ sin + cos sin ] = + 6 = 8 ( )

93 5 9 [, ] n δ (δt > ), [, ], [, ],,,, δ, δ, δ, δ [, ],[, ] f(), [, ] f() δ f ( ) δ, δ f ( )( δ ), δ f ( )( δ ), δ f ( ) δ δ δv, δv {( ) ( ) }δ ( ) f ( )( δ ) ( ) f ( )δ = ( ( ) f ( )δ + ( ) f ( )δ ) = ( ( ) cos δ + ( ) cos δ ) δv {( ) ( ) }δ ( ) f ( )δ ( ) f ( )( δ ) = ( ( ) f ( )δ + ( ) f ( )δ ) = ( ( ) cos δ + ( ) cos δ ) δv δv ( ( ) cos δ + ( ) cos δ ) δ δv V, δv ( ( ) cos δ + ( ) cos δ ) δ V = cos d + cos d () V = sin d + ( sin )d = 8 cos d cos d

94 5 9 9 C : r = a( + cos θ) ( θ ) (6 ) r = a( + cos θ) (, ) θ a r(θ) = a( + cos θ) r( θ) = a( + cos( θ)) = a( + cos θ) = r(θ) C a = a V a = (, ), θ { = r cos θ = ( + cos θ) cos θ = r sin θ = ( + cos θ) sin θ d dθ = sin θ( + cos θ) θ (, ) C V V a = d dθ dθ = = ( + cos θ) sin θ sin θ( + cos θ)dθ ( + cos θ) ( cos θ) ( + cos θ)( cos θ) dθ cos θ = t sin θdθ = dt, θ t. V a = = ( + t) ( t )( + t)( dt) = ( 5t + t + ) dt = [ t 5 + t + t ( t 5 5t t + t + t + ) dt ] = ( + ) + = 8 V = a 8 = 8 a ( ) Comment C () ) ),

95 5 95 z ABCD A(,, ), B(,, ), C(,, ), D(,, ) z E V (7 ) z z = t ( t ) z = t ( t ) A D Q B Q C H R Q Q = H Q R Q = z = t ( t ) AB, CD z,q,r,,q =, Q, R Q H A,B,C,D = H H(,, t). H D z ABCD t = H HQ z = t S(t) (i) t HQ H ( S(t) = (RQ RH ) = HQ (= (RQ ) ) = t ) = ( t) (ii) t H HQ S(t) = (R RH ) = H (= (R ) ) =, V = S(t)dt = ( t) dt + dt = 8 + = ( ) z = t R H A D H = Q z = t z = A z D = H Q z = t t B C B C u ABCD = A,B,C,D = A,B,C,D, S(t) = (RQ ), S(t) = (R ) A B C D z E S(t) E V E V. V = V = ( ) 7 + ( ) = ( )

96 5 96 z ABCD A(,, ), B(,, ), C(,, ), D(,, ) z E V (7 ) z z -t B S A Q t D S R z = t A S B z = C z = t Q z z = t ( t ) AC, AD, AB, z,q,s,r z = t E S(t) z = t z R = RQ, RS { S(t) = (RQ) (RS) } A B z z = (, S(,, z) z = S z z = t t = = t + ( ) t +. S,, t A z, Q z t Q AD -t : t Q = t A + ( t) + t t D = t ( t) + t + t = 6 t t t,, { { ( S(t) = (RQ) (RS) } ) ( ) ( ) } { 6 t t + t (6 t) = + = } t Comment V = S(t)dt = { (6 t) } t = 6 ( ) S

97 II

98 98, δ k = k+ k, = ma(δ k ) =ma(δ k )=δ k = k+ k, > < ( = g(u) g (u) ) )?) ( ) ( ) ) ( )

99 99 6 () 6. ). ) β α, β lim α = β α β = o(α) = k ( k ) β α β lim α β α β = (α) ) lim = sin θ. lim θ θ ==\ sin θ cos θ θ lim θ θ = cos θ θ θ = cos θ θ cos θ lim θ θ cos θ = lim θ θ = o(), sin θ = (θ), cos θ = (θ ), cos θ = o(θ) 6. = f() = a = f( + ) f() ), lim = { sin (=\ ) f() = sin θ sin ( =) lim f() ==f(), f() = lim + = lim f(h) f() h + h = lim h + h sin h h = lim h + h sin h lim ) ) t θ. ) β α / o o,,order( ) (order) = o( ). DNA DNA= o( ). = ( ). = ( ). β α lim = k(=\ ) lim α β = (=\ ) α β β α k ) lim a f() =f(a) ) g() = sin, {a n} =,,,, n, g(a n ) = n sin n =. {b n } = n o q 5, 9,,, n+, g(b n ) = n + sin `n + q = n + n. n a n, b n sin.

100 6 () 6. ( ) = f() = a f(a + h) f(a) lim h h = lim a f() f(a) a = f (a) f (a)=\ f (a) = = ( ) f = f() = g() F = f(a) = a = g() F () =f() g() F () F (a) lim a a = lim a f() {f(a)+f (a)( a)} a { f() f(a) = lim a a a } f (a) = F f() = a 5) 6. (Talor ) F () =f() f(a) f (a)( a) ( a) f (a) f (a)=\ ( a) f (a) = ( a) f (a)=\ f (a) =, f (a)=\ () = f() f = f() = g() F = g() a a = F () =f() g() k( a) = F () =f() g() k( a) j f (a)=\ F () =f() f (a)( a) k( a) (k=\ ) f (a) =,f (a)=\ F () =f() f (a)( a) k( a) (k=\ ) a 5)

101 6 () f() =cos f () = sin, f () = cos, f () =sin. f () = =\, f ` ` =,f =. =F () = F () = =cos = + () f(),g() b g(a)=\ g(b).a<<b {f ()} + {g ()} =\ B c. f(a) f(b) g(a) g(b) = f (c) g (c), a<c<b A Γ = g(t), = f(t) t = a, t = b, t = c A,B, AB, f (c) g Γ (c) f(a) f(b) g(a) g(b) Γ A B AB ( ) {f ()} + {g ()} =\ f(),g() = a g ()=\ ( = a g (a) = ) f() f(a) lim a g() g(a) = lim f () a g () 6) f ()=\ >a a <η< η f() f(a) =f (η)=\ c a c a lim a+ g() g(a) f() f(a) = g (c) f (c) (a<c<) g() g(a) f() f(a) = lim c a+ g (c) f (c) = lim a+ g () f () <a lim a g() g(a) f() f(a) = lim g () a f () lim a g() g(a) f() f(a) = lim a g () f () (Q.E.D.) 6) g (a)=\ f() f(a) lim a g() g(a) = lim a f() f(a) a g() g(a) a = f (a) g (a)

102 6 () ) sin lim ( sin ) cos = lim = lim ( ) ( cos ) sin = lim = lim ( ) 6 = 6 f() = a F () =f() f (a)( a), g() =( a) F () F (a) lim a g() g(a) lim a f() f(a) f f() f (a)( a) f () f (a) = lim = lim a ( a) a ( a) f (a) (a)( a) ( a) ( a) n f() f (a)( a) a ( a) = lim f() =f(a)+f (a)( a)+ f (a) = f (a) o f (a) = ( a) + o(( a) ) f (a)=\ F ( ) f() = a, G() =f() f (a)( a)! f (a)( a) g() =( a), G() G(a) lim a g() g(a) = lim a f() f (a)( a)! f (a)( a) ( a) f () f (a) f (a)( a) f () f (a) = lim = lim a ( a) a 6( a) = 6 f (a) f() f(a) f (a)( a) f (a) ( a) f (a) ( a) = o(( a) ) 6 f (a) =, f (a)=\ F ( ) f() = a n, Talor f() =f(a)+( a)f (a)+( a) f (a)! +( a) f (a)! + +( a) n f (n) (a) n! f() =e f() = f () = f () = f () = = f n () =, a = e =+ +! +! + + n n! + o(( a) n ) + o( n ) sin = ( ) n n+! 5! 7! (n +)! + o(n+ ) cos = ( ) n n!! 6! (n)! + o(n ) log( + ) = + + n +( )n n + o(n ) = n + o( n ) 5,, >, 5 =\ n = n+ = n + n+

103 6 () 6.5 ( ) f() = a lim a f() f(a) a = f (a) lim = f f (a) (a) lim f (a) = o( ) = f (a) + o( ) = A + o( ) A A = ɛ 7) lim = lim A + ɛ f() = a f (a) =A = f() = a lim A f (a) 8) = A lim = A = A + o( ) f A ( ) = df d = lim, f A = df d = lim f = A f = A = ɛ = ( ( ;) ) f A f f() = f =( + ) = +( ).( ) f () =. 6.6 f(x) > = f(x) X = a,x = ( >a), = S() f(x) [, + ] m, M, S = S( + ) S() m S M m S M Y = f(x) m f(),m f() ds d = lim S = f() f() S a + X 7) e error. 8) f (a)

104 6 () f() S, ( E) S f() ( E ) Y = f(x) f() f (), () ( ) ( ) f() f() = f (),lim = E S f() ( ) ds d = f() a + X C (, ) = r(θ)cosθ, = r(θ)sinθ C θ = α, ( θ>α) S(θ) r [θ, θ + θ] m, M, S = S(θ + θ) S(θ) m θ S M θ m S θ M θ m r(θ),m r(θ) ds dθ = lim θ S θ = r θ θ S r θ S r θ, r ( E) S r θ r θ r r(θ) r r (θ) θ, r θ ( ) r θ r ( θ) ( ) E θ. θ r r θ θ r(θ) θ r,lim θ r θ r θ = T θ S = {r(θ)} θ + o( θ) ds dθ = {r(θ)}

105 6 () 5 = f(x)( f (X) ) X = a X = ( >a) l() l = l( + ) l(), 9) ) + l ( ) +( ) = + ( ) = f ()+ɛ ( ɛ ) ( ) = +(f ()+ɛ) = +({f ()} +ɛf ()+ɛ ) = +{f ()} + ɛf ()+ɛ +{f ()} = +{f ()} ( + ɛf ()+ɛ ( + {f ()} ) = +{f ()} + (ɛ) l = +{f ()} + o( ). ) + o(ɛ) = f(x) dl d = +{f ()} l ( ) +( ) ( ) = a X +. lim lim =. ) S V f() S V ( θ) ) ( ) f () f() (?) 9). ) lim + ( + ) = lim ( + ) `+ ` + + `+ = lim + + «= o() ) f() = p lim h + f(h) h =. = + + `+

106 6 () = f(x),x = a, X =, = S() S f () () S f () ()

107 6 () ( ) [a, b] n, = a, = a + b a n, = a + (b a) n,, n = b, = k+ k ( ), η k [ k, k ] ) b a f()d = lim k= n f(η k ) F () =f() F k = F ( k+ ) F ( k ) F k = F (η k ) = f(η k ) η k [ k, k+ ] ) n n F k = {F ( k+ ) F ( k )} = {F ( ) F ( )} + {F ( ) F ( )} + + {F ( n ) F ( n )} k= k= = F ( n ) F ( )=F (b) F (a) n n F (b) F (a) = F k = f(η k ) k= k= nx f(η k ) k k= nx f(η k ) k k= = f() = f() = f() V k η η η η η η k k η k η k η k η k f() V k [ k, k+ ] ; α + β α + β lim k= n f(η k ) lim k= n f(η k ) = lim k= lim k= V k ma(v k ). ) lim k= n f(η k ) = lim ma(v k ) lim k= n (f(η k ) f(η k )) n (f(η k ) f(η k )) k= n f(η k ) n =ma(v k )(b a) ), ) k V k ( ).

108 6 () 8 n F (b) F (a) = F k = lim k= k= n f(η k ) = lim k= n f(η k ) = b a f()d F () =f() (F () f() ) ( ). F () =f() F (b) F (a) = n k= f(η k), f() ) f() =, F () = [, ] { k } n = < < < < n = = n. F ( k + ) F ( k )=( k + ) ( k ) =( k ) + k ( ) +( ) ( ) ( ) ( ) ( ) k k = n n + + = k +k + n n n n F k = F ( k + ) F ( k )=F (η k ) = f(η k ) k n < η k < k+ n F k = k +k + n = n (η k) k +k +. η k = n η k [ k, k+ ] n n f(η k ) = (η k ) = k= k= n k= k +k + n n = n ==F () F () ( ) n η k [ k, k+ ], [ k, k+ ] η k n I n. ( ) I n n n I = f(η k ) = (η k ) k= n ( ) I n = k= k <η k < k+ ( k n ) <η k < ( k+ n k= n { k } +k + n (η k ) ). k ( ) ( ) +k + k + n < k +k + n n (η k ) < k +k + k n n k + n < k +k + n (η k ) < n k + n ) f(), F () =f() F ().

109 6 () 9 n k= n n k + n < ( ) I n < k= n k + n n η k I n ( ) 5) F () =f() ( ) F k = f( k ) + ɛ ɛ lim ɛ = n n n F k = f( k ) + ɛ k= ( ) ɛ E n n n F k f( k ) = n n ɛ ɛ E = E(b a) k= k= k= k= k= n k= F k = F (b) F (a), E 6) F (b) F (a) = lim k= k= k= n f( k ) = b a f()d 7) ɛ f() =,F() =,a = < < < < n = b { k }, k= F ( k+ ) F ( k )=( k + ) ( k ) =( k ) + k ( ) +( ) n n n {F ( k+ ) F ( k )} = ( k ) + ɛ k (ɛ k = k +( ) ) k= n k= {F ( k+) F ( k )} = F ( n ) F ( )=F (b) F (a), a k b ɛ k = ( k + )) ( k + ) ( ma( a, b )+(b a)) k= E =(ma( a, b )+(b a)) n {F (b) F (a)} n ( k ) = n ɛ k lim k= k= k= k= n ( k ) = F (b) F (a). n ɛ k E b a k= = E(b a) d = F (b) F (a) 5) F () =f()! 6) k E ( ). 7) [ k, k+ ] f()

110 7 ( ) Γ :( ) + =, B, A(, ) Γ (, A.) Γ Q ABQ t Γ S ( ) Q B t A(, ) t t t + t Q Q = AQ = t t t Q, S S Q Q S = Q t + o( t) ( ) t t B A o( t) t ds dt = Q = t., S = t dt = [ t ] = 6 (.) BQ t Q BQ ( 9 ) t BQ = (cos t, sin t), Q = t(cos(t 9 ), sin(t 9 )) = t(sin t, cos t) = B + BQ + Q = (, ) + (cos t, sin t)+t(sin t, cos t) =(+cost + t sin t, sin t t cos t) (, ) j =+cost + t sin t =sint t cos t t,, S S = = Z d dt = Z Z t sin tdt + (sin t t cos t)t cos tdt = Z t ( + cos t)dt ( ) Z t sin t cos tdt + Z t cos tdt

111 7 ( ) (( ) ) f() = f() Q Q = f() Q Q Q Q Q θ o( ) R H Q Q H θ R Q Q H θ R Q (a, f(a)) () Q = a +,Q Q Q S,Q Q H Q Q Q R, Q = r,q Q Q r + r, QR r r Q r r θ () r + r θ + Q, Q H () (r r) θ S Q H (r + r)+ (r + r) θ r + r r r r r. Q Q lim Q H = Q H=o( ) ( ) θ (r r) S Q H (r + r)+ ds d = dθ r d S = r θ + o( ) f () > θ (r + r) θ dθ d = dθ d(tan θ) d(tan θ) d =cos θ df () d =cos θ f ()=\ θ S = r θ + o( θ) ds dθ = r ( ) ( ) f() Q () Comment Q H ( ) Q R ( ) (Q Q.)

112 7 ( ) Γ : = f() = e +e ( log( + )) Q Q Q =( ) Q Γ A(, ) B(log( + ), ) Q Q(t, f(t)) Q Q f (t) Q = (,f (t)) = (, et e t ) = e +e ( ) Q et = e + t ( ) et + e = t = et + e t Q θ(t), t t θ θ t t Q S Q B θ S = Q θ + o( θ) ds dθ = Q A t =log(+ ) θ = t log( + t + ) ( e t + e t ) dθ S = Q dθ = θ tan θ θ tan θ ( ) dθ tan θ = u du = cos θ dθ =(+tan θ)dθ =(+u )dθ =(+{f (t)} e )dθ = t +e t dθ = ( e t +e t ) du, θ u S = ( e t + e t ) dθ = ( e t + e t ) ( e t +e t ) du = du = ( )

113 7 ( ), θ θ S = Q Q Q Q Q Q=Q Q =, et e t = ««Q + Q = t, et + e t +, e e t =(t +,e t ) = e +e = e = e S, α = log( + ) e α =+ Z α e + e S = d + + jz α+ ff e d + h e e i α = + + h e i α+ = eα e α + + (eα ) = ( + ) α α + = = Comment Q Q Q = Q Q Q S S S = Q dθ = Q dθ = Q = S S S

114 7 ( ) Γ : = f() = e +e ( log( + )) Q Q Q =( ) Q Γ A(, ) B(log( + ), ) Q Q(t, f(t)) (t >) Q f (t) ( Q = k, ) f (k ) (t) Q,k = f (t). ( e Q = (f t e t ) (t), ) =, = e +e ( ) et Q = e + t ( ) et + e = t = et + e t A Q B θ Q θ(t), t t Q, Q, θ θ Q S, Q s. Q Q Q R θ S Q, s ( QR Q ) Q θ ( Q R ) ) S Q s + Q θ θ ) ( ) Q s t Q log( + ) Q Γ θ [, ] n θ k(k =,,,,,n) θ = θ k+ θ k, n n S Q s + k= k= n k= θ Q θ ( ) θ θ Γ L R S = L Qds + Q dθ ds = ( ) et +{f (t)} e dt = + t = et + e t dt ) ( ) S = Q s + Q θ + o( θ)

115 7 ( ) 5 L log(+ ) e t + e t Qds = et + e t dt = = [ ] log(+ ) e t e t +t = 8 8 +log(+ ) = log(+ ) { ( + ) Q Q dθ = e t ++e t dt ( + ) +log(+ ) },, +log(+ ) + S = ( ) Comment ( ) S Q s + Q θ θ ) ( ) θ ds dθ = Q ds dθ + Q S = Z Q ds dθ dθ + Z Q dθ, R L Qds S = Z L Qds + Z Q dθ ( ) Q Q R S = f() = g() S = Q + o( ) s = + o( )., f () > θ S = Q s + o( θ) Q s Q Q R S θ S = Q R θ + o( θ) = Q θ + o( θ) R t t + t S = S + S = Q s + Q θ + o( θ) Q