I4 - : April, 4 Version :. Kwhir, Tomoki TA (Kondo, Hirotk) Google http://www.mth.ngoy-u.c.jp/~kwhir/courses/4s-biseki.html pdf 4 4 4 4 8 e 5 5 9 etc. 5 6 6 6 9 n etc. 6 6 6 3 6 3 7 7 etc 7 4 7 7 8 5 59 F6 69 C7 79 B8 89 A9 S
I4 - A4 A4 Cfe Dvid Cfe Dvid : 3:3 () C: () R: (3) Q: (4) Z: (5) N: (6) : () α: () β: (3) γ, Γ: (4) δ, : (5) ϵ: (6) ζ: (7) η: (8) θ, Θ: (9) ι: () κ: () λ, Λ: () µ: (3) ν: (4) ξ, Ξ: (5) o: (6) π, Π: (7) ρ: (8) σ, Σ: (9) τ: () υ, Υ: () ϕ, Φ: () χ: (3) ψ, Ψ: (4) ω, Ω: () x X x X x X () X {x X x } N = {n Z n > } (3) X Y X Y ( (4) A := B A B e := lim + n. n n) (5) : ( ) { n } : M n n M
I4 - : April, Version :. 4 8 -. -. r > r n (n ) (Hint. n + n > ( + n) n e (n )
I4 - : April 8, Version :. (4/) f(x) () x f(x) () f(x) = (3) b f(x) dx π 3.4 3.46 () f(x) = sin x f(47 ) = sin 47 =? () f(x) = x 5 x 3 + = x =? (3) π = 4 x dx π 5,6 x = π
I4 - x = ± x = i = 5 8 9 N 3 N = {,, 3, 4, 5,...} N Z Z = {, ±, ±, ±3, ±4, ±5,...}. 4 Z Q { } p Q = p, q q. q 5 Q R 6 3 89 Peno, 858 93) 4 Z 5 Q R 6 869 (Méry835 9) (Dedekind 83 96) 87
I4-3 R x = X, Y x X : x X X Y : X Y X Y : X Y X Y 4 N, RN R N Z Q R,, 3,... n n,, 3,... { n } n n { n } n= { n } n { n } n N 7 n n, 4, 8, 6,... { n } : 7 n =,,,......,,,,,,... { n } n= { n} n=
I4-4 { n } n { n } α n n α lim n = α n α (n ) n α { n } (limit) { n } α { n } α n α n α 8 { n } α n α n α ϵ ϵ-n [].4 [] B n n 9 lim n = n (n ) n { n } lim n ( n) = lim n = n (n ) n - { n }{b n } lim n n = α lim n b n = β () lim n ( n ± b n ) = α ± β () lim n nb n = αβ n (3) β lim = α n b n β 8 9 M n n > M
I4-5 n b n 3 n b n 3 = 6 () n n b n n b n 6 ϵ () ± + ( n +b n ) (α+β) n α + b n β (n ) lim n ( n+b n ) = α+β () n b n αβ = n b n αb n +αb n αβ n α b n + α b n β b n β n b n β b n β b n β + n n b n αβ n α ( β +)+ α b n β (n ) (3)() lim (/b n) /β b n β n b n β n β / > n b n β β / b n β β / = β /(> ) b n β = b n β b n β (n ) b n β ( β /) β { n } : n n n+ { n } : n n n+ n N, n n+ n n < n+ n > n+ { n } : M n n M { n } : M n n M { n } (bounded) M, n, 3 n M x, y x + y x + y. for ll 3 there exists
I4-6 : M 5 -. () =, n+ = n + () =, n+ = 3 n + 4 n + 3 -. α > lim n α /n = α = α () α > n := α /n n () () n β β > () lim n n = β = α < /α >
I4 3- : My, 4 Version :. e (4/8) { ( n = + ) n } n.5937446...,.74838...,.769393...,.78459..., n =,,,,.78683...,.78846...,.78869...,.78879..., e e { n } : n n n+ n n+ { n } : M n n M n M A x = M { n } : n = n
I4 3- =.9999 =.9, =.99, 3 =.999,... n n < A α = lim n.999999... n n = n α = n = (n ). n =.9999 - =, n+ = n + = 3 > k+ > k k+ k+ = k+ + k + = ( k+ k ) k+ + + k + >. n n+ > n x = x + (> ) α = + ) α > = α > k α k+ = α + (α k ) k + = α + + k + <. n α > n { n } A β = lim n n β = β + β = α = + -e n = ( + n) n e ( e := lim + n (=.78888459 ) n n) ) α α = lim n α = lim n+ = n n n + = α + y = x + lim n
I4 3-3 - { n } n = nc k ( + n) n = ( n k= n nc k n k k= ) k = k! n n n n n = ( k! ) ( n ( ) k = + n n n k + n ) n n nc k k= ( k n < ( k! ) ( ) n + n + = k! n + n + n n + n n + n k + n = n+ C k ( n + ) k ) ( ) k n ( k n + n ( ) k n+ ( ) k n < + n+c k < + n+c k n + n + { n } k= n+ = n+c k (n+) k = k= ( + ) n+ = n+. n + ( ) k n + n n < 3 ( ) k nc k = ( n k! ) ( ) ( k ) n n n k! n ( ) k n = + nc k + n k= k! = 3 k = k n k= k < + + + + 3 + = 3 ) { n } A { n } = < n < 3 < e 3
I4 3-4 - > n () lim n /n = () lim n n/n = (3) lim n n! = () () n /n = + h n h n > ) n = ( + h n ) n = n nc k n k h k n = + nh n + k= n h n. (3) N < < h n < n. n(n ) h n + + h n n < N N n > N n n! = N N! N + N + n < N N! = N N! n(n ) h n. ( ) n N (n ). 5 9 3-. R () f : R R, f(x) = π + x () f : R R, f(x) = sin + x 3-. [] () f : [, 7] R y = f(x) = [x] f(x) f () g : [ 7, ] R y = g(x) = [/x] g(x) g (3) h : [ 7, ] R h(x) := g(x) h(x) h {,,, π } {x R x } ) x > x /n > h n >
I4 4- : My 9, 4 Version :. (5/) b (, b) := {x R < x < b} (, ) := {x R < x} [, b] := {x R x b} [, ) := {x R x} (, b] := {x R < x b} (, b) := {x R x < b} [, b) := {x R x < b} (, b] := {x R x b} (, ) := R + : (, ), [, ], (, ], [3, ) y = f(x) x f(x) ) (function) ) I I x f(x) I f f : I R, I f R, f : x f(x), x f f(x) I f(x) = x I ( 5, 5) f : ( 5, 5) R, f : 3 9, f. f : I R I f f(i) = {f(x) R x I} f I f ) y = f(x) x x y y = f(x) ) function
I4 4- : f : I J I f f(i) f(i) J f : I J 3) f(x) = x I = ( 5, 5) f(i) = [, 5). x f : I R f : ( 5, 5) [, ) f : ( 5, 5) [, 5) [y] y k y < k + k f(x) = [x] f : R Z f : [ 5, 5] {, ±, ±, ±3, ±4, ±5} f x x f(x) A x f(x) A x f(x) A lim f(x) = A f(x) A (x ) x x = f() x x sin x lim x x = x = sin / = / 3)
I4 4-3 ϵ-δ ϵ lim f(x) = A lim f(x) = A x x A = A = ± lim x + x = = lim x x. x f(x) lim f(x) = f(x) (x ) x f(x) lim f(x) = f(x) (x ) x ± ± lim ( x x = lim x ) x = x x < f(x) A lim f(x) = A f(x) A (x ) x x f(x) A x x > f(x) A lim f(x) = A f(x) A (x + ) x + x f(x) A 3- lim x f(x) = Alim x g(x) = B () lim x {f(x) + g(x)} = A + B () lim x f(x)g(x) = AB (3) B lim x f(x) g(x) = A B
I4 4-4 f : I R I x f(x) f() lim f(x) = f(). x f I f I f I f(x) = cg(x) = x h(x) = (, ) (, ) f(x) = g(x) = x x 3-(3) 3-f(x)g(x) x = f(x) ± g(x) f(x)g(x) x = g() f(x)/g(x) 3-3f : I Jg : J R h = g f : I R, h(x) := g(f(x)) cos x x + R x xx cos x cos x 3- x xx x x = x x x + 3-3 x cos(x) x + 3- x cos x x +. f : [, ] [, ] f() := x f(x) := [ ] x [y] y f /x x =. g : R [, ] g() := x g(x) := sin x g x x = h(x) := xg(x) = x sin(/x) x =
I4 5- / : My 6, 4 Version :. (5/9) 3-3-3 I R I f : I R x, x I x x f(x ) f(x ). x x f(x ) f(x ) f(x ) = f(x ) x = x f(x) = x f : [, ] [, ] f : [, ] [, ] f : [, ) [, ) x < x x < x x < x x x f(x ) f(x ) f : I R x, x I x < x f(x ) < f(x ) x < x f(x ) > f(x ) f : I R ) f f(x) = x f : (, ] R f : [, ) R ) ) y = f(x) )
I4 5- f : I f(i) = J x f y = f(x) y x f f : J I f : [, ) [, ), f(x) = x X = f(x) f : [, ) [, ) 3) f (X) = X f (X) = X n f(x) = x n [, ) x n X = f(x) = x n [, ) n X X n X n x > p q ( ) x p q := x p q x p/q p p := / p P Q P/Q = p/q x P/Q = x p/q ( x P/Q) qq = ( x /Q ) P qq = x P q ( x p/q) qq = ( x /q ) pqq = x pq P q = pq x P/Q x p/q x P q = x pq qq x r s x r x s = x r+s, (x r ) s = x rs 3) [, ) f
I4 5-3 4-x α {r n } () n r n Q. (b) n r n < r n+. (c) lim r n = α n lim n xr n x α x α {r n } α ()(b)(c) {r n } α > n α = =.44356... e {r n } = {.4,.4,.44,.44,.44, } e := lim n er n 4-. x > < x < x = r s r < s x r < x s y = x s r x s x r = x r (x s r ) > x r ( s r ) = {x r n } α < m n r n < m x r n < x m {x r n } lim n xr n r n {s n } ()(b)(c) r n s n = (α s n ) (α r n ) (n ) x r n x s n = x s n {x r n s n } {q n = r n s n } q n (n ) x q n q n < /m m m n m n (n ) -() x qn < x /mn q n < x qn = /x qn lim n xq n = x e x exp : R (, )exp x = e x > x x x = x = 4- > () x, y x y = x+y () > x x (3) < x x
I4 5-4 () r n rs n s r n + s n r + s lim n r n+s n = lim n r n s n ()x < y y x = x ( y x ) > x ( y x ) = x y x y y x < /n n y x n -() < y x < /n y x + < y x = x ( y x ) y x < x y y x y x x x (3) () 4-() exp : R (, ), x e x log : (, ) Rx log x > 4- () (3) x e x log : (, ) Rx log x log e x = log x log x ln x 4) 4-4-3 >, () x, y log x + log y = log xy () > x log x (3) < x log x xy x + y = (, ) θ P(θ) (cos θ, sin θ) cos θ θ (m + /)π (m ) OP(θ) tn θ tn θ := sin θ cos θ sin x xcos x xtn x x (S) sin : [ π/, π/] [, ] (C) cos : [, π] [, ] (T) tn : ( π/, π/) (, ) = R 4) exponentil function logrithmic function. ln logrithmus nturlis nturl logrithm
I4 5-5 (S)(C)(T) (S ) Sin : [, ] [ π/ π/], x Sin x (C ) Cos : [, ] [, π], x Cos x (T ) Tn : (, ) ( π/ π/), x Tn x Sin x xcos x xtn x x Sin xcos xtn x Arcsin xarccos xarctn x x x x Sin = Sin = π 4 Sin = π 4 x [, ] Sin x + Cos x = π x [, ] 6 5-. 5) sinh x := ex e x, cosh x := ex + e x, tnh x := sinh x cosh x () cosh x sinh x = cosh x = (cosh x), sinh x = (sinh x) () sinh(x ± y) = sinh x cosh y ± cosh x sinh y (3) cosh(x ± y) = cosh x cosh y ± sinh x sinh y 5-. x 5 3x 4 + = (, ), (, ), (, 3) 5) x x x
I4 5-6 Π Π 4 Π Π 4 Π 4 Π Π 4 Π : sin x Sin x Π 3 3Π 4 3 Π Π 4 Π 3Π 4 Π Π 4 Π Π 4 Π 4 Π 3 3 : cos x Cos xtn x Π Π 4 5 5 Π 4 Π 3: Tn x
I4 6- : June, 4 Version :. (5/6) f(x) = y = f(x) f(x) = α () f( ) <, f(b ) >, b f(x) f(x) < b () n < b n (n ) f( n+bn ) < ( n+, b n+ ) := ( n+bn, b n ). f( n+b n ) > ( n+, b n+ ) := ( n, n+b n ). f( n+b n ) = α := n+b n () { n } n < b {b n } < b n α = lim n b n n = b / n b n = (b n n ) + n + α = α (n ). f f(α) = lim f( n ) f(α) = lim f(b n ) f(α) = n n b n α mx { n α, b n α } b n n = b n. f(x) = x, (, b ) = (.,.) n, b n / n n n b n....5.5.5 3.375.5 4.375.4375 5.465.4375 6.465.488 7.446.488 n n b n 8.446.4797 9.446.46.446.454.446.4455.446.443 3.448.443 4.448.445 5.448.44 n n b n 6.44.44 7.44.44 8.44.44 9.44.44.44.44.44.44.44.44 3.44.44
I4 6- ( n, b n ). 5- f : [, b] R f() f(b) f() f(b) l f(c) = l c [, b] g(x) = f(x) l g(c) = c [, b] 5- f : [, b] R [, b] n n =,,,... X n n = X = {, (3 + b)/4, ( + b)/, ( + 3b)/4, b} X n X n+ M n = mx{f(x) x X n } f(x n ) = M n x n [, b] {x n } [, b] { x n(k) }k x = lim k x n(k) [, b] {M n } M n(k) = f(x n(k) ) (k ) f(x n(k) ) f(x) < {M n } M = lim n M n M = f(x) f(y) > M y n X n f(y ) > M y f(y ) M n M 5-3I R f : I f(i) = J J f : J I > f : (, ) (, ), f(x) = x f : (, ) (, ), f (x) = log x
I4 6-3 J x < x f(x ) < f(x ) y < y f (y ) < f (y ) f ϵ []p75 y = f(x) y = f(x) x = A = lim x f(x) f() x ( ) A f () y = f(x) I f I I f(x) y = f(x) I x I f (x) f : I R f f (x), y, df dx (x), d dx f(x), dy dx, Df(x) ( ) x = x, y = f(x) f() ( ) f(x) f() y A = lim = lim x x x x
I4 6-4 x y A y A x x f(x) f() x x f () = A f(x) f() r (x) := A (x ) x r () = (x = ) x r (x) f(x) f(x) f() = A(x ) + r (x)(x ) x g(x) x g(x) o(x ) f(x) f() = A(x ) + o(x ) f(x) = f() + A(x ) + o(x ). f(x) x = + 6 9 6-. () {tn x} = cos x (3) { Cos x } = x 6-. > () { x } = x log (4) { Tn x } = + x () cos 63.453994997 () 3 8 3.36588978 π = 3.4596535 3 =.735875
I4 7- : June 9, 4 Version :. (6/) y = f(x) x = ( ) A = lim x f(x) f() x A f () y = f(x) x f y ( ) x = x, y = f(x) f() ( ) f(x) f() y A = lim = lim x x x x x y A y A x x ) x ) ( ) f(x) f() A(x ) f(x) f() + A(x ) : ) X Y X Y X Y X Y ) f(x) = f() + A(x ) + o(x ) x r (x) f(x) = f() + A(x ) + r (x)(x )
I4 7- sin 47 47 = 45 + = π/4 + π/9 ( ) f(x) = sin x = π/4 f(x) f(π/4) + f (π/4)(x π/4) = sin(π/4) + cos(π/4)(x π/4). x = 47 = π/4 + π/9 ( π sin 47 = f 4 9) + π + π 9.7379. =.4435 π = 3.4596.733537.6% m 6mm 6 6 6 = 5 + = 5 + 5 ( ) f(x) = x = f(x) f() + f ()(x ) = + (x ). { x} = x 3) x = + /5 + 5 ( = f + ) + 5 5 = + 5. 6 5 + / = 5. (5.) = 6. 5.9995.% 6- f(x), g(x) () {f(x) + g(x)} = f (x) + g (x) () {f(x)g(x)} = f (x)g(x) + f(x)g (x) { } f(x) (3) g(x) = f (x)g(x) f(x)g (x) g(x) {g(x)} (4) {g(f(x))} = g (f(x)) f (x) y = f(x)z = g(y) dz dx = dz dy dy dx. 3) y = x x = f f () = /
I4 7-3 f g f() g(f()) f f () g g (f()). x = f(x)g(x) x f(x) = f() + f ()(x ) + o(x ), g(x) = g() + g ()(x ) + o(x ) 4) x f(x) + g(x) = f() + g() + {f () + g ()}(x ) + o(x ). {f(x) + g(x)} {f() + g()} x = {f () + g ()} + o(x ) x f () + g (). f(x) + g(x) x = f () + g () () f(x)g(x) = {f() + f ()(x ) + o(x )}{g() + g ()(x ) + o(x )} = f(α)g(α) + {f (α)g(α) + f(α)g (α)}(x α) + o(x α) 5) f(x)g(x) x = f (α)g(α) + f(α)g (α) () (3) f(x) = {/g(x)} = g (x)/g(x) () g(x) g() g(x) = = g ()(x ) + o(x ) = g () (x ) + o(x ). g() g(x)g() g(x)g() g(x)g() x /g(x) /g() x = g () o(x ) + g(x)g() x g () g(). (4) f() = b y = b g f(x) = f() + f ()(x ) + r(x)(x ) g(y) = g(b) + g (b)(y b) + R(y)(y b) lim x r(x) = lim y b R(x) = g(f(x)) g(f()) = g(y) g(b) = g (b)(y b) + R(y)(y b) = g (f()){f ()(x ) + r(x)(x )} + R(y){f ()(x ) + r(x)(x )}. 4) o(x ) ± o(x ) o(x ) 5) g(α) o(x )(x ) o(x ) o(x )
I4 7-4 g(f(x)) g(f()) = g (f()) f () + g (f())r(x) + R(y){f () + r(x)} x x y = f(x) b = f() g(f(x)) x = g (f())f () 6- f : I f(i) = J I f (x) () f : J I J () y = f(x), x = f (y) dx dy = dy dx d dy f (y) =. d dx f(x). () 5-3 x = y = f(x) b = f() x = x y = f(x) f() = y b x = f (y) f (b) f x y 5-3 f y x dx dy = lim x y y = =. y dy lim x x dx f () = y = f(x) = x 3 x = g(y) = y /3 g () d dx Sin x = x ( x < ) y = Sin x ( x < ) x = sin y ( y < π/) 6- = dx = cos y (> ) dy dy dx = cos y = sin y =. x 6-3y = f(x) d dx {log f(x) } = f (x) f(x) = y y.
I4 7-5 z = log y dz = /y 6-(4) dy dz dx = dz dy dy dx = y y. y = x x (x > ) log y = x log x 6-3 x y y = log x + x = log x +. x y = y(log x + ) = x x (log x + ) 6-4 α {x α } = αx α {e x } = e x {log x } = x {sin x} = cos x {cos x} = sin x {tn x} = cos x = tn x + { Sin x } { = Cos x } { = Tn x } = x x + x Sin x Cos x x < { x } = x log = e log x = e x log { x } = e x log log = x log 6 6 7-. () (sin x) cos x x + () 3 (x ) (3) x x + Sin x (4) f(x)g(x)h(x) 7-. () sinh x sinh : R R, y = sinh x = ex e x () R Hint. (3) x = sinh y y Hint. cosh x sinh x =.
I4 8- : June 6, 4 Version :. (6/9) 5- f : [, b] R f() = f(b) (, b) 7- f : [, b] R (, b) f() = f(b) f (c) = c (, b) f f f : [, b] R M m 5- m < M m f() = f(b) < M f(c) = M c (, b) x c x c + f(x ) f(c) f(c) f(x ) f f(x ) f(c) (c) = lim f f(x ) f(c) (c) = lim x c x c x c x c f (c) = m < f() = f(b) M f(c) = m c (, b) f (c) = x = c x c f(x) < f(c) f(x) x = c f(c) x = c x c f(x) > f(c) f(x) x = c f(c)
I4 8- f(x) x = c f (c) = x = c x = c x c f(x) < f(c) f (c) f (c) f (c) = f(x) f (c) = f(x) = x 3 x = 7- f : [, b] R (, b) ( ) f (c) = f(b) f() b c (, b) (, f()) (b, f(b)) (c, f(c)) ( ) ) ( ) f(b) f() = f (c)(b ) (, f()) (b, f(b)) l A = f(b) f() F : [, b] R b F (x) := f(x) {A(x ) + f()} l F (x) F (c) = c (, b) F (x) = f (x) A f (c) = A. 7-3 f(x) [, b] (, b) (, b) () f (x) > = f(x) () f (x) < = f(x) (3) f (x) = = f(x) )
I4 8-3 x < x b x x f : [x, x ] R c (x, x ) ( ) f(x ) f(x ) = f (c)(x x ) x x > f (c) (, b) f (x) = f (c) = f(x ) = f(x ) x x x > x > log( + x) f(x) = x log(+x) f x > x > f (x) = /(+x) > f f(x) > f() = x > log( + x) lim f(x) = lim g(x) = lim f(x) x x x g(x) sin x (sin x) lim =, lim x x x x sin x =, lim x ± x = ±, ) 7-4 f(x) g(x) x = () lim x f(x) = lim x g(x) = (b) lim x f (x) g (x) lim x f(x) g(x) f(x) lim x g(x) = lim f (x) x g (x) lim x e x cos x x = f(x) = e x cos xg(x) = x x f(x) g(x) f (x) g (x) = ex + sin x f(x) ()(b) lim x g(x) = lim f (x) x g (x) = ) (Johnn Bernoulli, 667-748) Guillume de l Hôpitl, 66-74
I4 8-4 () lim f(x) = ± lim g(x) = ± x x lim x lim x lim, x lim x + lim x lim x + x x = y = x x log y = x log x = log x. f(x) = log xg(x) = /x x + /x f(x) g(x) + x + f (x) g (x) = /x /x = x () (b) lim log y = x + f(x) lim x + g(x) = lim f (x) x + g (x) = y = xx (x +) f(x) f(x) = x + sin x, g(x) = x lim x f (x) lim x g (x) = lim x ( f(x) lim x g(x) = lim + sin x ) = x x + cos x g(x) f(x) f(x) = cos x, g(x) = + x lim x g(x) f (x) lim x g (x) = lim sin x = x f(x) lim x g(x) = lim cos x x + x = lim f (x) x g (x). ( ) X x = Y ( ) g(x) C () x = C f(x)
I4 8-5 ( ) ( ) d X g (x) = dx Y f (b) (x) ( ) x = ( ) C g(x) g (x) f(x) f x (x) X Y 7-6 f(x) g(x) [, b] (, b) g() g(b) (, b) g (x) f(b) f() g(b) g() = f (c) g (c) c (, b) (g(), f()) (g(b), f(b)) (g(c), f(c)) f(b) f() B = F (x) := {f(x) f()} g(b) g() B{g(x) g()} F (c) = c (, b) F (x) = f (x) Bg (x) = f (c) Bg (c)g (c) (, b) g (x) g g() g(b) f g ( ) c f() = g() = f g x = f (x) g(x) lim x g x = (x) g(x) x
I4 8-6 f(x) g(x) f(x) f() = g(x) g() = f (c) g (c) c x x c 6 3 8-. () sin x < x < tn x ( < x < π/) () + x < e x < ( < x < ) x 8-. sin x x sin x () lim () lim x x x x 3 ( x + b x ) /x (3) lim x x/x (4) lim x Hint
I4 9- : June 3, 4 Version :. (6/6) n y = f(x) f () (x) := f(x), f (n+) (x) := { } f (n) (x) f (n) (x) (n =,,, ) f(x) n n y (n), d n y dx n, ( ) d n f(x), dx d n dx n f(x), Dn f(x) f () (x) = f (x), f (3) (x) = f (x) f (n) (x) f (n) (x) f(x) n f (n) (x) f(x) n f(x) C n ) n f(x) f C y = x k y = e x sin xlog x C y = x 3/ C x = {fg} (n) = n nc k f (n) g (n k) k= =.44 = + 4 + + 4 3 + 4 +.4 ) n n n
I4 9- f(x) = x 3 x 3 = + 3(x ) + 3(x ) + (x ) 3 x =. = + / x =. (.) 3 = + 3 + 3 + 3 =.33 (.) 3 = + 3. + 3 (.) + (.) 3 = +.6 +. +.8 =.68.6.63.68 8- I I f : I R n x I f(x) = f() + f ()(x ) + f () (x ) + + f (n ) ()! (n )! (x )n () c x + f (n) (c) (x ) n () n! f(x) x = n = n () (n ) () () c = x c = x < c < x x < c < c x < θ < θ c = ( θ) + θx c = + θ(x ) n = 7-x = b f() = f(b) + f (c)(b ) f() f(b) = f (c)(b ). f(x) = x 3 I = (, ) = R = 4 f (x) = 3x, f (x) = 6x, f (x) = 6, f (4) (x) =
I4 9-3 c x f(x) = x 3 = f() + f ()(x ) + f ()! (x ) + f () 3! = + 3(x ) + 6! (x ) + 6 3! (x )3 + (x )3 4! = + 3(x ) + 3(x ) + (x ) 3. ) (x ) 3 + f (4) (c) (x ) 4 4! f(x) = e x I = (, ) = R = n f (n) (x) = e x c x e x = e + e (x ) + e! (x ) + + e x = + x + x! + + xn (n )! + ec n! xn e (n )! (x )n + ec (x )n n! = c, c x sin x = x x3 3! + x5 5! x7 7! + + ( )m (m )! xm + ( )m sin c (m)! cos x = x! + x4 4! x6 6! + + ( )m (m)! xm + ( )m+ sin c (m + )! x m x m+ x < c < x e c n! xn - e x x n (n ) n! e x = + x + x! + + xn n! + x = ( e = lim + ) n = + + n n! + 3! + + n! + e ( n = + ) n b n = + + n! + 3! + + n! ) f(x) = nx k (x ) k k = f (k) ()/k! k= x =
I4 9-4 n n n e b n b n e..788885..788885.5.4688885.5.88885 3.373737.3479458.666666667.565679 4.44465.768755785.78333333 9.9484956 3 5.4883.996885.76666667.65679 3 6.56637.966554567.7855556.67935 4 7.546499697.77834.7853968.78658 5 8.56578454.5497345.787877 3.5867775 6 9.587479.377367.78856 3.885853 7.5937446.45393684.7888.736676 8 sin c m sin x = x x3 3! + x5 5! x7 7! + + ( )m (m )! xm + cos x = x! + x4 4! x6 6! + + ( )m (m)! xm + sin x x = π 3) 8. = π π3 3! + π5 5! π7 7! + π 3 3! + π7 7! + π! + = π + π5 5! + π9 9! + 7- n = 3 x = x x b b n f (k) () f(b) (b ) k k! k= (b ) n = f (k) (c) n! c b A I F (x) { } n f (k) (x) F (x) := f(b) f(x) + (b x) k + A(b x) n k! k= f(x) n F (x) F () = F (b) = b F : [, b] R F : [b, ] R 3)
I4 9-5 c (, b) (b, ) F (c) = F { n ( f F (x) = f (k+) (x) (x) + (b x) k f (k) ) } (x) (b x)k + na(b x) n k! (k )! k= = f (n) (x) (n )! (b x)n na(b x) n x = c A = f (n) (c)/n! 6 3 9-. x n n x () () x x (3) sin x (4) xe x Hint. () () 9-. x = n () ( x < ) () sin x (3) cos x (4) xex x
I4 - : June 3, 4 Version :. (6/3) 8- I I f : I R n x I f(x) = f() + f ()(x ) + f () (x ) + + f (n ) ()! (n )! (x )n () c x + f (n) (c) (x ) n () n! f(x) x = n () (n ) () x = sin x e e e x x = e x = + x + x! + + xn (n )! + ec n! xn c x x =, n = 6 e = + +! + 3! + 4! + 5! () + ec 6! () ( < c < ) () e () () + + 6 + + 5 + = + 43 6 =.76666. () < c < < e c < e 3 ) ) e = lim n ( + /n) n e 3 < ec 6! 3 7 = 4 =.4666
I4 - () < e () + /4.76666 < e <.783333 () e.6% ( + /n) n n = 6.56 x k k lim x e x =. k = x > 4 e x = + x + x! + x3 3! + ec 4! x4 ( < c < x) k 3 e x > + x + x! + x3 3!. x e x < x + x + x! + x3 3! (x ). log x x > c x c log( + x) = x x + x3 3 + ( )n x n + ( )n n nc n x n. < x n x = log( + x) = x x + x3 3 x4 4 + log = + 3 4 + ( + x) n = + nx + n (n ) x n (n ) + + (n ) n xn (n N)
I4-3 α R x < ( + x) α α(α ) = + αx + x + + ( ) α n α(α ) (α n + ) n! ( ) α n α(α ) (α n + ) x n + n! α n α α C n ( ) (+x) α α n (+c) α n x n n α = / + x = + x x 8 + x3 6 + 6 5. x = /5 + 5 + 5 8 5 + 6 5 3 = + + 4 3 6 = 5 + 5 5 + + 3 = 5 + 99 5 = 5.99. 6 = 5.99953 n x f(x) f(x) (x ) (x ) n f(x) = o((x ) n ) (x ) ) f(x) (x ) n x f(x) f(x) = o() n = x () 4x + x 3 = o(x) () sin x = o() (3) x sin x = o(x ) () 4x + x 3 = 4x + x (x ). x () n = sin x (x ). x sin x cos x sin x (3) lim x x = lim = lim =. x x x ) (x )
I4-4 f(x) = o(x ), g(x) = o(x 3 ) (x ) () x f(x) = o(x 4 ) () f(x) + g(x) = o(x ) (3) f(x)g(x) = o(x 5 ) () x f(x) f(x) + g(x) () x = f(x) x + x g(x) x 3 (x ). (3) f(x)g(x) x 5 = f(x) x x 4 = f(x) x (x ). g(x) x 3 (x ). f(x) = o(x)g(x) = o(x) f(x) g(x) = f(x) g(x) = o(x) o(x) = o(x) 9-f(x) x = C n f (n) f(x) = f() + f ()(x ) + + f (n) () (x ) n + o((x ) n ) (x ). n! x n f(x) = n k= f (k) () k! c x f(x) n k= f (k) () k! (x ) k = f (n) (c) n! (x ) k + f (n) (c) (x ) n. n! (x ) n f (n) () (x ) n. n! x c f (n) (c) f (n) () [] (x ) n (x ). C x e x = + x + + xn n! + o(xn ) = + x + x! + o(x ) = + x + o(x) = + o(). e x sin x = x + x + o(x ) sin x = x x 3 /3! + o(x 3 ) ) e x sin x = ( + x + o(x)) (x x3 3! + o(x3 ) = x x3 3! + o(x3 ) + x x4 3! + o(x4 ) + o(x ) + o(x 4 ) + o(x 4 ) = x + x + o(x ). o(x ) sin x = x + o(x) x 3 = o(x ) sin x 3
I4-5 e x sin x x () lim x x () lim x log( + x) x x () ex sin x x x = x + x + o(x ) x x = + o(x ) x = (x ). log( + x) x () x = x x / + o(x ) x x = + o(x ) x = (x ). cos x = x + o(x ) (x ) + t = + t/ + o(t) t = x + x = + x / + o(x ). cos x = x + o(x ) cos x + x = o(x ) o(x ) = o(x ) o O x = f(x), g(x) f(x) lim = f(x) = o(g(x)) x g(x) (x ) M > x f(x) M g(x) f(x) = O(g(x)) (x ) o O sin x = x x 3 /6+o(x 3 ) (x ) sin x = x+o(x 3 ) sin x x = x 3 /6 + o() o() (x ) x /6 + o() sin x x = x 3 sin x x = O(x 3 ). y = f(x) f(α) = α f(x) =. () y = f(x) x () α x (3) n (x n, f(x n )) x (x n+, )
I4-6 9-. x n (n =,,...) x n+ = x n f(x n) f (x n ) f(x) = α δ = δ(f, α) > x x α δ x n α () [].3.3(p43)[]..6 p5 f(x) = x = x = x n+ = x n x n x n = x n + x n x 6 x n x n..586.5.858.466666666666666667.45 3.4456867459839. 6 4.4435637468996.59 5.4435637395488 8.99 5 6.4435637395488.86 49 α x n+ α = O( x n α ) C 7 7 -. -. e x log t α lim t t α = () lim x cos x x sin x () lim x tn x sin x x 3 cos x + x (3) lim x x 4
I4 - : July 7, 4 Version :. (6/3) f : [, b] R ( < b) : Step [, b] {x k } N k= = x < x < x < < x N < x N = b Step N [] N k= f(x k )(x k+ x k ) f(x k )(x k+ x k ) k k < N {x k } N k= Step 3. {x k } N k= N mx x k+ x k k<n I ) I = b f(x) [, b] f(x) dx ) I []3.4 p 73 []3 (p 8)
I4 - f Step 3 I Step Step N /N f(x), g(x) α, β > b b f(x) dx := f(x) dx b. - b b b () {αf(x) + βg(x)} dx = α f(x) dx + β g(x) dx b b () x b f(x) g(x) f(x) dx g(x) dx c b c (3) f(x) dx = α f(x) dx + β f(x) dx. b I f : I R I x I F (x) := f(x) ) x f(t) dt -f F (x) = F (x) = f(x) d dx x f(t) dt = f(x) x f(t) dt x > [x, x + x] f(x) m x ( x)m x ( x) m x ( x) f(t) M x ( x) (x t x + x) -() x+ x x m x ( x) dt x+ x x f(t) dt x+ x = m x ( x) x F (x + x) F (x) M x ( x) x = m x ( x) F (x + x) F (x) x ) x M x ( x) M x ( x) dt
I4-3 f(x) x + m x ( x)m x ( x) f(x) x < lim x F (x + x) F (x) x F (x + x) F (x) lim = f(x) x + x = f(x) f(x) F (x) = f(x) F (x) f(x) -3F (x) G(x) f(x) F (x) G(x) {F (x) G(x)} = f(x) f(x) = F (x) G(x) -4 f(x) G(x) = x F (x) = f(t) dt + C (, C) x f(t) dt f(x) -3-4 f(x) f(x) dx x dx = x3 + C (C) C 3-5 f(x) F (x) b f(x) dx = -4 F (x) = x ( ) b F (b) F () = f(t) dt + C [ ] b F (x) = F (b) F (). f(t) dt + C ( ) f(t) dt + C = / dx x b f(t) dt.
I4-4 (Sin x) = x / [ ] / dx = Sin x = π x 6 = π 6. -6x = x(t) α t β = x(α), b = x(β) b f(x) dx = β α f(x(t)) dx dt dt. x dx = (Sin x + x ) x + C (C : ) x = sin t ( t π/) dx = cos t dt x dx = sin t cos t dt = cos t dt + cos t = dt = ( ) sin t t + + C = (t + sin t cos t) + C = (Sin x + x ) x + C. -7f(x), g(x) C b f (x)g(x) dx = [ ] b f(x)g(x) b f(x)g (x) dx. I n = cos n x dx (n =,,, ) I n = n cosn x sin x + n n I n (n ).
I4-5 I n = cos n x cos x dx = cos n x sin x = cos n x sin x cos n x sin x dt (n ) cos n x( sin x) sin x dt = cos n x sin x + (n ) cos n x( cos x) dt = cos n x sin x + (n )(I n I n ). I = dx = x + CI = cos x dx = sin x + C J n = π/ I = cos x sin x x + C I 3 = 3 cos x sin x 3 sin x + C cos n x dx (n =,,, ) J = π/, J = n J n = n n J n (n )(n 3) 3 π/ cos n x dx = n (n ) 4 (n )(n 3) 4 n (n ) 3 π (n) (n) 7 4 -. > () x dx () Sin x dx (3) + x dx (4) + x dx Hint. () Sin x (3) t = x + + x (4) + x ) -. () (3) π/ e cos 3 x dx () x log x dx x e x dx
I4 - : July 4, 4 Version :. (7/7) f(x) = mx m + + x + b n x n + + b x + b x ) - () () (3) (m N) (x + b) m x + b (x + cx + d) n (n N, c 4d < ) x + x x + = (x + ) = x + x + () +. x + () x + x x + C x dx = + x + log x + + C. 5x 4 x + x 6 = +. x 3 () x + () x 3 + b x + 5x 4 x + x 6 dx = log x 3 + log x +. () () (3) (3) u = x + c/ x + b (x + c/) c/ + b (x = + cx + d) n ((x + c/) c /4 + d) n = u + b (u + c ) n = u (u + c ) n + b (u + c ) n. )
I4 - b = c + b, c = c > c > 4d c 4 u (u du + c) n u (u + c) n du = (u + c) (u + c) n du = >. (u du + c) n log(u + c) (n = ) ( n + )(u + c) n (n ) t = c u (u + c) n du = c c n (t + ) n dt I n I n+ = t n(t + ) n + n n I n (n ) n = t = tn θ I = t + dt = + tn θ dθ cos θ = dθ = θ = Tn t. n -. I = x + x dx t = x t + = x t dt = dx t I = t + + t dt.. x + dx t = x + x + (t x) = x + x = ( t ) t x + = t x = t t + dx = t + t dt x + dx = t t + t + t dt = dt = log t = t log(x + x + ).
I4-3 u = tn x/ sin x = u u u, cos x =, tn x = + u + u u dx = du + u + sin x I = dx + cos x + I = + u = u +u +u ( + u + u du + u + u = + u du + u )du = u + log( + u ) = tn x ( + log + tn x ). t ( ) C : x(t) p (t) = y(t) (A t B) x x(t), y y(t) A < t < B C ) C [, b] (A, B) x (t) := dx dt (t), y (t) := dy (t) dt C C = C[, b] ( ) C C[, b] : x(t) p (t) = ( t b) y(t) -3 C l(c) b {x (t)} + {y (t)} dt 3) t {t k } N k= = t < t < < t N < t N = b ) 3)
I4-4 p k := ( ) x(tk ) p (t k ) = C = C[, b] { p (t y(t k ) k )} N k= N k= p k+ p k p k+ p k = {x(t k+ ) x(t k )} + {y(t k+ ) y(t k )} c k, d k (t k+, t k ) x(t k+ ) x(t k ) = x (c k )(t k+ t k ), y(t k+ ) y(t k ) = y (d k )(t k+ t k ) p k+ p k = {x (c k )} + {y (d k )} (t k+ t k ). N mx k<n t k+ t k [] = N k= b C {x (c k )} + {y (d k )} (t k+ t k ) {x (t)} + {y (t)} dt y = x x C p (t) = ( ) t t ( t ) -3 u = t l(c) = l(c) = + (t) dt. + u du = u( + u [ ) log(u + ] + u ) = { 5 + log( + } 5).
I4-5 -3 y = f(x) [, b] b + {f (x)} dx. f (x) r C r r π r x + y = r y = ± r x y = x x r/ y = r x 8-4 l(c r ) = 8 r/ u = rx r r/ + (y ) dx = 8 + x r x dx = 8 / l(c r ) = 8r du. u [C r ] [C r ] = l(c / r) = 4 r du u r/ r r x dx sin x, cos x (, ) x sin x, cos x x dt t Sin x sin x cos x sin(π/ x) -3 C C[, b] : x(t) p (t) = y(t) ( t b) z(t) b {x (t)} + {y (t)} + {z (t)} dt
I4 3- : July 4, 4 Version :. 7/4 y = /x [, + ) x β > [, β] β [ x dx = x ] β = β + β β lim dx = lim ( β ) β x + =. β dx = x f : [, b) R b = + b f(x) dx := β lim f(x) dx β b f(x) [, b) b f(x) dx. f : (, b] R (, b] f(x) = /x [, + ) [ x dx = ] = + = x lim f(x) = /x (, ] dx := lim x α + α dx = lim x α + ( + α ) = +.
I4 3- - > () () x p dx p <. x p dx p >. () () p < < p < p =, p + + 3 + + + 3 + + + 3 + + n n + + 3 + n := + + 3 + + n + n dx (n ). x x dx < + x dx - n = + + 3 + + n (, b) ( < b + ) c (, b) b f(x) dx := c f(x) dx + b c f(x) dx = c lim α + α f(x) dx + lim β b β c f(x) dx
I4 3-3 dx = π x = + [ ] ( dx = Sin x = π ) = π x [ ] dx = Sin x = π x = π b g(x) f(x) f(x) g(x); b g(x) dx f(x) dx. -f(x) g(x) b b b f(x) dx f(x) dx f(x) f(x) -4 f(x) dx () dx () + x3/ b f(x) dx sin x x dx () = + dx + x3/ x > + x 3/ - dx x3/ x3/ - dx x3/ + x3/
I4 3-4 () sin x sin x x t = x dx = x x t ( dt) = dt. - - t x sin x x dx s Γ(s) := e x x s dx s Γ : (, ) R Γ γ n! -3s > Γ(s) := e x x s dx f(x) = e x x s s > c (i) [c, ) g(x) = x f(x) (ii) (, c] h(x) = x s f(x) f(x) (i): g(x) = e x x s x = xs+ e x - lim f(x) g(x). - c f(x) dx s + < N N x xs+ e x xn e x. x N f(x) = c x c x e x c g(x) x dx g(x) = x f(x) (ii): x > e x < f(x) = e x x s < x s s > - h(x) = x s f(x) (iii) Γ(s) = f(x) dx = c c f(x) dx f(x) dx + c c f(x) dx x s dx -4s > Γ(s + ) = sγ(s) s = n =,,,... n! = Γ(n + ). Γ(s + ) = Γ() = s [ ] e x x s dx = e x x s + e x sx s dx = + sγ(s). e x dx = Γ(n + ) = nγ(n) n! = Γ(n + ) Γ(s + ) = sγ(s) s
I4 3-5 p >, q > B(p, q) := x p ( x) q dx B β () B(p, q) = B(q, p). () B(p +, q) = p q B(p, q), B(p, q + ) = B(p, q) p + q p + q (3) B(, ) =, B (, ) = π (4) B(p, q) = π/ (cos x) p (sin x) q dx (5) B(p, q) = Γ(p)Γ(q) Γ(p + q). π/ (cos x) 9 (sin x) 7 dx = B(5, 4) = Γ(5)Γ(4) Γ(9) = 4! 3! 8! = 56. ()(4)