極限

Similar documents
limit&derivative

( ) 2.1. C. (1) x 4 dx = 1 5 x5 + C 1 (2) x dx = x 2 dx = x 1 + C = 1 2 x + C xdx (3) = x dx = 3 x C (4) (x + 1) 3 dx = (x 3 + 3x 2 + 3x +

sekibun.dvi

1 1 3 ABCD ABD AC BD E E BD 1 : 2 (1) AB = AD =, AB AD = (2) AE = AB + (3) A F AD AE 2 = AF = AB + AD AF AE = t AC = t AE AC FC = t = (4) ABD ABCD 1 1

9 5 ( α+ ) = (α + ) α (log ) = α d = α C d = log + C C 5. () d = 4 d = C = C = 3 + C 3 () d = d = C = C = 3 + C 3 =

1 29 ( ) I II III A B (120 ) 2 5 I II III A B (120 ) 1, 6 8 I II A B (120 ) 1, 6, 7 I II A B (100 ) 1 OAB A B OA = 2 OA OB = 3 OB A B 2 :

A(6, 13) B(1, 1) 65 y C 2 A(2, 1) B( 3, 2) C 66 x + 2y 1 = 0 2 A(1, 1) B(3, 0) P 67 3 A(3, 3) B(1, 2) C(4, 0) (1) ABC G (2) 3 A B C P 6

(1) θ a = 5(cm) θ c = 4(cm) b = 3(cm) (2) ABC A A BC AD 10cm BC B D C 99 (1) A B 10m O AOB 37 sin 37 = cos 37 = tan 37

18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C

(1) 3 A B E e AE = e AB OE = OA + e AB = (1 35 e ) e OE z 1 1 e E xy e = 0 e = 5 OE = ( 2 0 0) E ( 2 0 0) (2) 3 E P Q k EQ = k EP E y 0

さくらの個別指導 ( さくら教育研究所 ) A a 1 a 2 a 3 a n {a n } a 1 a n n n 1 n n 0 a n = 1 n 1 n n O n {a n } n a n α {a n } α {a

#A A A F, F d F P + F P = d P F, F y P F F x A.1 ( α, 0), (α, 0) α > 0) (x, y) (x + α) 2 + y 2, (x α) 2 + y 2 d (x + α)2 + y 2 + (x α) 2 + y 2 =

A (1) = 4 A( 1, 4) 1 A 4 () = tan A(0, 0) π A π

i

5. F(, 0) = = 4 = 4 O = 4 =. ( = = 4 ) = 4 ( 4 ), 0 = 4 4 O 4 = 4. () = 8 () = 4

BD = a, EA = b, BH = a, BF = b 3 EF B, EOA, BOD EF B EOA BF : AO = BE : AE, b : = BE : b, AF = BF = b BE = bb. () EF = b AF = b b. (2) EF B BOD EF : B

4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx

D xy D (x, y) z = f(x, y) f D (2 ) (x, y, z) f R z = 1 x 2 y 2 {(x, y); x 2 +y 2 1} x 2 +y 2 +z 2 = 1 1 z (x, y) R 2 z = x 2 y

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)

高校生の就職への数学II


f(x) = x (1) f (1) (2) f (2) f(x) x = a y y = f(x) f (a) y = f(x) A(a, f(a)) f(a + h) f(x) = A f(a) A x (3, 3) O a a + h x 1 f(x) x = a

x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x

1

ac b 0 r = r a 0 b 0 y 0 cy 0 ac b 0 f(, y) = a + by + cy ac b = 0 1 ac b = 0 z = f(, y) f(, y) 1 a, b, c 0 a 0 f(, y) = a ( ( + b ) ) a y ac b + a y

( ) x y f(x, y) = ax


70 : 20 : A B (20 ) (30 ) 50 1

85 4

a n a n ( ) (1) a m a n = a m+n (2) (a m ) n = a mn (3) (ab) n = a n b n (4) a m a n = a m n ( m > n ) m n 4 ( ) 552

1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1

I y = f(x) a I a x I x = a + x 1 f(x) f(a) x a = f(a + x) f(a) x (11.1) x a x 0 f(x) f(a) f(a + x) f(a) lim = lim x a x a x 0 x (11.2) f(x) x

zz + 3i(z z) + 5 = 0 + i z + i = z 2i z z z y zz + 3i (z z) + 5 = 0 (z 3i) (z + 3i) = 9 5 = 4 z 3i = 2 (3i) zz i (z z) + 1 = a 2 {

.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =,

2016.

第121回関東連合産科婦人科学会総会・学術集会 プログラム・抄録

1W II K =25 A (1) office(a439) (2) A4 etc. 12:00-13:30 Cafe David 1 2 TA appointment Cafe D

( )

No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y

mobius1

function2.pdf

I II

II K116 : January 14, ,. A = (a ij ) ij m n. ( ). B m n, C n l. A = max{ a ij }. ij A + B A + B, AC n A C (1) 1. m n (A k ) k=1,... m n A, A k k

[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s

1 (1) ( i ) 60 (ii) 75 (iii) 315 (2) π ( i ) (ii) π (iii) 7 12 π ( (3) r, AOB = θ 0 < θ < π ) OAB A 2 OB P ( AB ) < ( AP ) (4) 0 < θ < π 2 sin θ

春期講座 ~ 極限 1 1, 1 2, 1 3, 1 4,, 1 n, n n {a n } n a n α {a n } α {a n } α lim n an = α n a n α α {a n } {a n } {a n } 1. a n = 2 n {a n } 2, 4, 8, 16,

微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.

1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C

6 2 2 x y x y t P P = P t P = I P P P ( ) ( ) ,, ( ) ( ) cos θ sin θ cos θ sin θ, sin θ cos θ sin θ cos θ y x θ x θ P

名古屋工業大の数学 2000 年 ~2015 年 大学入試数学動画解説サイト

CG38.PDF

Chap9.dvi

2009 I 2 II III 14, 15, α β α β l 0 l l l l γ (1) γ = αβ (2) α β n n cos 2k n n π sin 2k n π k=1 k=1 3. a 0, a 1,..., a n α a

OABC OA OC 4, OB, AOB BOC COA 60 OA a OB b OC c () AB AC () ABC D OD ABC OD OA + p AB + q AC p q () OABC 4 f(x) + x ( ), () y f(x) P l 4 () y f(x) l P

4 4 θ X θ P θ 4. 0, 405 P 0 X 405 X P 4. () 60 () 45 () 40 (4) 765 (5) 40 B 60 0 P = 90, = ( ) = X

2014 S hara/lectures/lectures-j.html r 1 S phone: ,

1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0

1 n A a 11 a 1n A =.. a m1 a mn Ax = λx (1) x n λ (eigenvalue problem) x = 0 ( x 0 ) λ A ( ) λ Ax = λx x Ax = λx y T A = λy T x Ax = λx cx ( 1) 1.1 Th

Chap11.dvi

1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =

1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f =

x = a 1 f (a r, a + r) f(a) r a f f(a) 2 2. (a, b) 2 f (a, b) r f(a, b) r (a, b) f f(a, b)

VI VI.21 W 1,..., W r V W 1,..., W r W W r = {v v r v i W i (1 i r)} V = W W r V W 1,..., W r V W 1,..., W r V = W 1 W

II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R

研修コーナー

入試の軌跡

1 12 ( )150 ( ( ) ) x M x 0 1 M 2 5x 2 + 4x + 3 x 2 1 M x M 2 1 M x (x + 1) 2 (1) x 2 + x + 1 M (2) 1 3 M (3) x 4 +

29

1

tnbp59-21_Web:P2/ky132379509610002944

x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s

直交座標系の回転

パーキンソン病治療ガイドライン2002

: α α α f B - 3: Barle 4: α, β, Θ, θ α β θ Θ

5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h 0 g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)

日本内科学会雑誌第97巻第7号

untitled

N cos s s cos ψ e e e e 3 3 e e 3 e 3 e

.1 A cos 2π 3 sin 2π 3 sin 2π 3 cos 2π 3 T ra 2 deta T ra 2 deta T ra 2 deta a + d 2 ad bc a 2 + d 2 + ad + bc A 3 a b a 2 + bc ba + d c d ca + d bc +


18 ( ) ( ) [ ] [ ) II III A B (120 ) 1, 2, 3, 5, 6 II III A B (120 ) ( ) 1, 2, 3, 7, 8 II III A B (120 ) ( [ ]) 1, 2, 3, 5, 7 II III A B (

取扱説明書[NE-202]

Part () () Γ Part ,

日本内科学会雑誌第98巻第4号

さくらの個別指導 ( さくら教育研究所 ) A 2 2 Q ABC 2 1 BC AB, AC AB, BC AC 1 B BC AB = QR PQ = 1 2 AC AB = PR 3 PQ = 2 BC AC = QR PR = 1

S I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt

CALCULUS II (Hiroshi SUZUKI ) f(x, y) A(a, b) 1. P (x, y) A(a, b) A(a, b) f(x, y) c f(x, y) A(a, b) c f(x, y) c f(x, y) c (x a, y b)

Note.tex 2008/09/19( )

arctan 1 arctan arctan arctan π = = ( ) π = 4 = π = π = π = =

7

x V x x V x, x V x = x + = x +(x+x )=(x +x)+x = +x = x x = x x = x =x =(+)x =x +x = x +x x = x ( )x = x =x =(+( ))x =x +( )x = x +( )x ( )x = x x x R

Gmech08.dvi

_0212_68<5A66><4EBA><79D1>_<6821><4E86><FF08><30C8><30F3><30DC><306A><3057><FF09>.pdf

ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d

f : R R f(x, y) = x + y axy f = 0, x + y axy = 0 y 直線 x+y+a=0 に漸近し 原点で交叉する美しい形をしている x +y axy=0 X+Y+a=0 o x t x = at 1 + t, y = at (a > 0) 1 + t f(x, y

31 33

gr09.dvi

(4) P θ P 3 P O O = θ OP = a n P n OP n = a n {a n } a = θ, a n = a n (n ) {a n } θ a n = ( ) n θ P n O = a a + a 3 + ( ) n a n a a + a 3 + ( ) n a n

1 26 ( ) ( ) 1 4 I II III A B C (120 ) ( ) 1, 5 7 I II III A B C (120 ) 1 (1) 0 x π 0 y π 3 sin x sin y = 3, 3 cos x + cos y = 1 (2) a b c a +


Transcription:

si θ = ) θ 0 θ cos θ θ 0 θ = ) P T θ H A, 0) θ, 0 < θ < π ) AP, P H A P T PH < AP < AT si θ < θ < ta θ si θ < θ < si θ cos θ θ cos θ < si θ θ < θ < 0 θ = h θ 0 cos θ =, θ 0 si θ θ =. θ 0 cos θ θ θ 0 cos θ) + cos θ) θ + cos θ) θ 0 si ) θ si θ θ + cos θ) θ 0 θ + cos θ) = ), θ 0 0 ) si θ si θ θ θ cos θ θ cos θ θ cos θ θ

. i) θ = π { si θ = si π = = 0.5 θ = π =.459 si θ θ 0.5 = 0.5598 0.559 0.9549 cos θ = cos π = = 0.805 = 0.974 θ = ).459 = 0.5598 ) = 0.74548 0.74548 0.4888 cos θ θ 0.974 θ = π θ ii)θ = π si θ = si π = si45 0 ) = 4 = 0.5889 θ = π =.459 = 0.799 si θ θ 0.5889 0.799 0.9885 cos θ = cos π = + 4 =.70508 = 0.04074 θ = ).459 = 0.799 ) = 0.08587 cos θ θ 0.04074 0.08587 0.4975 θ si θ θ cos θ θ = = si π π π π = cos = = si, = = cos = = = cos 0, ) = cos.)

) 0 si si ) 0 si si ) ) 0 cos 4) 0 cos cos

4 = = e = 4 = =.5 A0, ) = a A0, ) A = a a A0, ) a e ) f) = e f fh) f0) e 0) h 0 h h h 0 h = e A0, ) h 0 e h h = e ) = e f ) h 0 e +h e h h 0 e h h e = e Ph, e h ) e ) = e A0, ) h e h AP eh h g) = a h 0 A g ) h 0 a +h a h h 0 a h h a ) = h, 4 ) = 4 4 h h 0 h h 0 h a = e log e a = e log a h 0 a h h h 0 e h loge a h h 0 e h loge a h log e a a ) = a log a. log e a = log e a ) e =.7888845. e.

5 e 4 ) 0 e = e =t log+t)= t 0 log + t) t = t 0 + t) t = e ± + ) = e 0 e e +. log + ) log + ) = +, = = e, = log + ) 0, ), 0, 0) = log + ), = = log, = = = log, 0) = e = log = = = = e = + = = log + ) = = log = log ) = t log + t) t 0 t,, ) 4 = log + t) t = = log e + t) t = e t 0 t 0 + ) = e e A

= e = log log = = log = e a e a. a = XX > 0) = log X e a = e e a e a = log ) = > 0) = log a e a -. ) 0 e ) 0 log + ) ) 0 5 4) 0 e si log + ) 5) 0 logcos )

7. e ) e ; e + ).) a = ) + =,,, ). a = + =, a = + ) = 9 4 =.5, a = + ) = 4 7 =.70707, a 4 = + 4 ) 4 = 5 5 =.44405, a 00% ) )? 5.5 + ) + ) = + ) 4 4 4 : + ) = 4. = + 4 8 : + ) = + ) ). + : + ) + ) = + ). 4, + ) = a ) = + a a 0., 0.0, ), a + ) = e ) + ) + ) + ) e e.788 7 ) ) 5 5 ), 500 500 )

8 P, + ), P, + ) ), P, + ) ) k,, P k, + ) ) k,, P k k =,,, ) = ) + = { +, ) } f ) = { + ) } { f ) + ) } = e ) = f ) = e 4) ) + = + ) e ) + = e = + ) = + = + P P P P A0, ) A0, ) ) - + = e. 5) ) + ) ) + ) ) + ) 4) P, + ) A0, ) AP P = +.) P = e Q, e ) Q A, = + A = e 5) ),) 00% ) 00%.

9. ) + ) ) ) 0 4) 5) + + 4 + + + + + + + ) + + ) ) + ) ) 7) = log = e,, =, =,, = < < < < < e) = log a =, b = e b a e a b

0 4 ) ) f), g) = a fa) = ga) = 0, = a g )=\ 0 f ) a g ) a f) g) a f ) g ) g a)=\ 0 a f) g) a f) fa) a g) ga) a = f a) g a) 0 0 ) : ) log cos ) 0 ) si ) f) =, g) = log f) = g) = 0. g ) =, g )=\ 0. log f) f) g) g) = f ) g ) = = f ) =, g ) = ) f) = cos, g) = f0) = g0) = 0 g ) = g 0) = 0. g ) = 0 = 0, 0 cos cos ) si 0 ) 0 = 0 si = ) f) = si, g) = f0) = g0) = 0 g ) = g ) = 0 = 0 si si ) cos 0 ) 0 cos ) si 0 ) 0 = ) 0 0 ) ) =. 0 )

4 ) 0 ) 0 cos XY Pt, cos t) t t = 0 P C =\ 0 A, cos ), A cos = A ) C A Q Q C A ) Q t = c 0 < c < < c < 0), Q C f c) g c) A ) cos = f c) g c) 0 < c < < c < 0) c 0 ) 0 c 0. c 0 f c) g c) 0 cos c 0 f c) g c) c 0 si c c t 0 si t t = t = c Y Y t = c A, cos ) Q C X 0 A, cos ) Q C X ) ) t = 0 g t) = 0 C : f c) g c) t = 0 g t) = 0

4 ) ) a f) g) Y { X = gt) C : Y = ft) t = a C :gt), ft)) Ag), f)) =\ a g )=\ 0 = a = a g)=\ 0. ga), fa)) = 0, 0), f) g) = A ) A Q t = t = c t = a X C A Q Q C A Q t = c 0 < c < < c < 0), Q C f c) g c) A ) f) g) = f c) g c) < c < < c < ) f c. a c a c) c a g c) a f) g) c a f c) g c) a f ) g ) Q.E.D. 4.) ) 0 e + ) ) 0 e + e ).

4 ) 4, ) ) 0 e + ) {e + )} 0 ) 0 e = ) 0 e + e {e + e } e e 0 ) 0 0 {e e } ) 0 e + e = 0 e + e 0 e + e e e ) 0 e = Commet ) ) 0 e + ) e + + e 0. ) ) e + e + ) 0 e + e = + ) + ) = 0 = 0 ) e + + e + e 0 ) ) + + + + ) = =.K.) 0 0,,, 0 0 MuPAD.)

4 5 Talor ) = a f ) a) = a f) = fa) + f a) a) + f a)! =,, a) + f a)! a) + + f ) a)! a) [ ] f) = fa) + f a) a) [ ] f) = fa) + f a) a) + f a)! [ ] f) = fa) + f a) a) + f a)! a) a) + f a)! a) ) a = 0, = 0 f ) 0) = 0 =,, f) = f0) + f 0) + f 0)! + f 0)! + + f ) 0)! [ ] f) = f0) + f 0) [ ] f) = f0) + f 0) + f 0)! [ ] f) = f0) + f 0) + f 0)! + f 0)! ) f ) a) f ) a) = f a), f ) a) = f a), f ) a) = f a). a).

5 Talor ) 5 5. f) = a a f) fa) a = f a) Q h) = f) g) = f) A P a f) fa) f a) a) a = 0 g) = f a) a) + fa) = a! ) = g) a a a f) g) a = 0 f) g) h) a a) = a = g) = f), f) = a = a f) fa) + a)f a) f) h) = f) {fa) + a)f a)} a ). ) ) a α, β 0 β a α = 0 β α β α ). β a α β α = k k )

5 Talor ) a) 4 0m 5m,.5m.) =, =. h),, ) i) f) = a, a h) = f) g) < = f) = g).. Q = f) = g) Q Q P A P P a 4 ii) f) = a f) = a = a Aa, 0) = 0 a+0 f) fa) a =, a 0 f) fa) a =. f f) fa) a) a a f) = a = a = a = 0 = a = = 0 = a Q = a Q = a Q Q Q Aa, 0) P P P Q Aa, 0) P P Aa, 0) P 4 = f a) a) + fa) = a = f).

5 Talor ) 7 5. = 0 = 0 f ) 0) = 0 f) = f0) + f 0) + f 0)! + f 0)! + + f ) 0)! 5.. e = 0 f) = e f ) = e, f ) = e, f ) = e f0) = f 0) = f 0) = f 0) = e + +! +! = 0 = f) ) = + f) A A 0 e + ) {e + )} e 0 ) 0 = 0 e = = 0 e + ) e + + [ ] f) B ) e + + B 0 0 {e + )} e 0 ) 0 { )} e + + e + ) ) 0 = = 0 ) e + + e + + + [ ]

5 Talor ) 8 0 a + b) ±., e + ) a = e = + = e + ) = = e + ) = = e + ) 0 a + b + c) ±, = e = + + ) = e + + ) = e + + = e : + + a = e + + ) = = e 0,, = + + + = + + = + = e e = 0,,

5 Talor ) 9 5.. si = 0 f) = si f ) = cos, f ) = si, f ) = cos f0) = 0, f 0) =, f 0) = 0, f 0) = si 0 +! + 0! +! si = 0 = f) ) = f) A, A 0 si si ) cos 0 ) 0 0 cos ) ) 0 si = 0, si + 0 = f) + 0 B, B 0 si si ) cos 0 ) 0 0 cos ) si 0 ) 0 = si si [ ] = = si = si = si = si : + 0 a

5 Talor ) 0 = si, = = = si = = si = si = 0,, 5.. cos = 0 f) = cos f ) = si, f ) = cos, f ) = si f0) =, f 0) = 0, f 0) =, f 0) = 0 cos + 0! +! + 0! cos = 0 = f) ) = f) A,, A 0 cos = cos [ ]

5 Talor ) 5. 5.. A a f) {fa) + f a) a)} a) F ), G), F a) = Ga) = 0. f a) A a f) {fa) + f a) a)} a) a f) {fa) + f a) a)}) a) ) = 0 f) a f ) f a) a) = f a) f) fa) + f a) a) + f a) f a), = a f) a). f) fa) + f a) a) + f a) a) a) f a) a f ) f a) a 5.. f a), f) B a a { fa) + f a) a) + f a) a) } a) 0 0 { f) fa) f a) a) f a) a) } { a) } a f ) f a) f a) a) a) 0 0 a {f ) f a) f a) a)} { a) } a f ) f a) a) = f a) { } f) fa) + f a) a) + f a) a) a) f a) f a) a f ) f a) a

5 Talor ) a f a) f) fa) + f a) a) + f a) f) = fa) + f a) a) + f a)! a). a) + f a) a) + f a)! a) a) 5.4., = 0 = 0 e + +! si! cos! +! + 5 5! + 4 4! + +! 7 7!! log + ) + 4 4 + + + + + ) α + α! + αα )! + + ) )! + + ) )! + + ) + αα )α )! + + αα ) α ))! α. α α = + + 8 + + + ) ) + )! ) ) + )α α =, + + + + = + + + + + ) = = + +, 0 + = 0.

5 Talor ) si si. ) si = 0! = π si π π π ) 48 0.008 ) si π = 0.5, π.459 = 0.5598 0.0598. log a a a??) ) log + ) = + 4 4 + 5 5 + ) log ) = 4 4 5 5 + log + ) = + + 5 5 + ) 0 < < + + = = { ) log + ) } = 0.958047 { 5 ) log + ) + ) } 5 = 0.90045 5 log = 0.94780 e = = + + =.5 e + +! +! + 4! + +! = + + +. = 4 + + + + 4.708 = 5 + + + + 4 + 0.77 = + + + + 4 + 0 + 70.7805555 = 7 + + + + 4 + 0 + 70 + 5040.78598 e.78888, +)! = 7 8! 0.00004805.78888.78598 = 0.000078 ) + e

4. e e = 0 e =t t 0 log + t) t = t 0 + t) t = e ± + ) = e ) e 0 = e + e e.)?) a = ) + = ±, ±, ±, ). a = + =, a = + ) = 9 4 =.5, a = + ) = 4 7 =.707, a4 = + 4 ) 4 = 5 5 =.4440, + ) ) ) ) = C 0 + C + C + + C = + + )! = + + )! ) ) +! + ) )! ) + +! + + )! a + ) = + + ) + ) ) + + )!!! < + +! +! + +! < + + + + +! = 4 > = = + ) < a e ) e + ) = e ) ) a a a e )

5. ) 0 si si si 0 si = ) 0 +0, cos, 0 cos 4) si si ) ) 0 = 0 si si ) si si = cos cos = cos + ) cos ) α ± β = cos cos si si ) cos si + si si ) = si si Commet cos cos 0 0 si si si θ θ, cos θ θ 0 4) si si = 4 ) 0 ) 0 si si = ) si si ) si ) = cos = ) = 9, cos cos cos 9 ) ) = 4 = 4 - ) ) 5 ) 0 4) 0 e log 5) 0 e log 5) log 5) log 5 = log 5 0 e si log + ) e si 0 si si log + ) = = 5) logcos ) 0 Commet 0 log + cos ) 0 log + cos ) cos 0 e +, log + ) 4) 0 5) 0 e si log + ) = logcos ) = log ). cos ) = ) = logcos ) =

- ) { + ) } = e ) ) + ) { + Commet ) } = e { + ) } = e ) 00% e ) 00% e ) 00% e. ) ) ) 4) 5) ) + + ) ) + + ) + + ) 0 + + ) + + 4 + + + 4 + = + + + 0 + + + ) + + ) ) ) + { + + + + = + ) = + + ) + ) ) = ) + ) = ) } ) + ) < + < ) < + < =, ) + = + =

7 7) = log, 0), = S a = T log e =, b S a, T b S = T a b. b a = S a = e b T = log A, 0), P + a, 0), Q + a, log + a )), R + a, a ) = log A = log S = log = = APQ < S < APR a log + a ) < S < a = T b loge b ) < T < b = = log R Q + a, log + a )) S = T, a log + a ) < S = T < b b loge b ) < T = S < a A, 0) a = P + a, 0) = log log + a ) < b a a < loge b ) loge b ) Se b, 0) b e a 0, b 0 log + a ) a t 0 log + t) t =, loge b ) = log e = Commet b a = S T log e = =\ 0 T log = 0 S

8 )http://miedmoss.com )http://www.geogebra.org )http://www.cabri.com 4)http://www.wolframalpha.com iteret ), MuPAD MuPAD free,.) ) geogebra free MuPAD ) cabri geogebra 4) Mathematica CATComputer algebra sstem) free Mathematica ) si 0 π) si ) plot plot si, =0..pi = si 0 π) d si d/d d/d si d π it it si), =0..pi si d 0 si si )/ as ->0 0 * ) *si si si si / si )/ ^ ^ log log π e, π e, pi it si, =0..pi si d 0 plot ^- =