=0. f = 2 +3 { 2 +3 0 2 f = 1 =0 { sin 0 3 f = 1 =0 2 sin 1 0 4 f = 0 =0 { 1 0 5 f = 0 =0 f 3 2 lim = lim 0 0 0 =0 =0. f 0 = 0. 2 =0. 3 4 f 1 lim 0 0 = lim 0 sin 2 cos 1 = lim 0 2 sin = lim =0 0 2 =0. f 0 = 0. f 0 lim 0 0 1 = lim sin 0 sin1/ 0. =0. f 0 = 0. 5 =0. 1
=0. sin 1 0 f = 0 =0 1 + sin 1 0 2 f = 0 =0 3 f = 3 + 4 f 0 lim 0 0 1 = lim sin 0, =0. 2 =0. 3 f 0 lim = lim + 2 0 0 0 + 2 + 2, 0. =0, f 0 = 0.. 1+ + 2 2 logtan 1 3 arcsin 4 5 1 a 2 2 2 + a 2 arcsin a y = 1 + + 2 1/2 1+2 = 2 1+ + 2 2 y tan sec 2 = = tan tan = 2 sin 2 2
3 y 1 1 = 1 2 1 = 1 2 2 1 4 y =, log y = log. y /y = log +1. y = ylog +1= log +1 5 y = 1 2 a 2 2 a2 + 1 1 2 a2 = a 1 /a 2 a 2 log + 2 1 2 log 3 cos + sin 2 4 e 1+2 5 arctansec + tan 1+ 2 1 + 2 1 = 1 2 1 2 y = log log y = log log. y /y = log log + 1/. y = log log log + 1 log log 3 y = 2cos + sin sin + cos 4 y = e 1+2 2 5 y 1 = 1 + sec + tan sec tan + 2 sec2 = 1 2 n, =0. e +1 2 cos 2 3 1 + 3 log1 + 3
y = e +1, y n = ee y = e + e + e2 2 + e3 6 + e4 24 + 2 y = cos 2 = 1 2 cos 2 + 1 2, yn =2 n 1 cos 2 + nπ 2 y =1+ 2 + 4 4 + 3 y =1+ 3 log1 +, y = 31 + 2 log1 + + 1 + 2, y = 61 + log1 + + 51 +, y = 6 log1 + + 11, y n = 6 1n n 4! 1 + n 3 n 4 y = + 52 2 + 113 6 + 4 4 + =0. arctan 2 sin 3 3 e sin y = arctan 1 + 2 y n+2 +2n +1y n+1 + nn +1y n =0. y0 = 0, y 0 = 1, y 0 = y 4 0 = 0, y 0 = 2. 2 y = sin 3 = 3 4 sin 1 sin 3. 4 y n = 3 4 sin + nπ 2 y = 3 3 3n 4 sin 3 + nπ 2 y = 3 + 4
3 y = e sin. y n =2 n/2 e sin + nπ 4 y = + 2 + 3 6 + R, φ = dist, Z T = n=0 φ10 n 10 n R,. R, 0 φ10 n 1 2 n=0 φ10 n 10 n R. φ φ10 n R., T R. φ φ10 n + 1 = φ10 n, T. 0 <1 T. [0, 1 =0.a 1 a 2 a n 0 a n 9 0.a 1 a 2 a n 99 0.a 1 a 2 a n + 100.. φ10 n = { 0.an+1 a n+2 a n+1 4 1 0.a n+1 a n+2 a n+1 5 { 10 m a m =4, 9 h m = 10 m a m 4, 9 + h m =0.a 1 a 2 a ma m+1, a m = 5 { am 1 a m =4, 9 a m +1 a m 4, 9
a m a m =10 m h m φ10 n + h m φ10 n 0 n m = 10 n m a m a m =10 n h m n < m, a n+1 4 10 n m a m a m= 10 n h m n < m, a n+1 5 m 1 T + h m T = ± 10n m 1 h m = ±1 h m 10 n h m n=0 m. m h m 0, m 1 ±1 T 0 <1. n=0 n=0 R f = b k cos a k π, a, b R, b < 1, ab < 1 k=0 R C 1. n b < 1 f n = b k cos a k π R f k=0. f R. n ab < 1 f n = πab k sin a k π R k=0 g = πab k sin a k π k=0. f =g f R C 1. f = 3 2 1 i vi. 6
i, f f. ii f =0, f. iii f =0, f. iv f. v f. vi f. i R \{ 1, 1}. f f f = 4 3 2 2 1 2, f = 23 +6 2 1 3 ii f =0 = 3, 0, 3. f 3 < 0 = 3 f 3 = 3 3 2 f 0 = 0, =0 f0 = 0 f 3 > 0 = 3 f 3 = 3 3 2 iii f iv < 1 f < 0 1 <<0 f > 0 0 <<1 f < 0 lim 1 0 lim 1 0 1 < f > 0 =0 f 0 = 0 3 2 1 3 =, lim 1+0 2 1 =, f = + 2 1, lim ± lim 1+0 y = f. 3 = =1, 2 1 3 = = 1, 2 1 =0 y =, 2 1 7
v f y y y vi f.., f, f = 2 e, R 2 f = log, 0, f 2 = e 2 f = 1 log,, >0 < arctan <. 1+2 f = arctan, f, > 0 1+2 f 2 2 = > 0. f0 = 0, f 1 + 2 2, >0 f > 0. g = arctan, g, >0 g = 2 > 0. g0 = 0, g, >0 1+2 g > 0. 0 <<π/2 2 π < sin. 8
f = sin 2 π, f, f cos sin = 2. g = cos sin g, 0<<π/2 g = sin <0. g0 = 0, g 0 <<π/2 g < 0. 0 <<π/2 fπ/2 = 0, f, f > 0. y 2 f = 0, 0. y 2 +y 2, y 0, 0 0, y =0, 0 2 fh, 0 f0, 0 0 0 f 0, 0 = lim = lim =0 h 0 h h 0 h f0,k f0, 0 0 0 f y 0, 0 = lim = lim =0 k 0 k k 0 k fh, 0 f0, 0 h f 0, 0 = lim = lim h 0 h h 0 h =0 f0,k f0, 0 f y 0, 0 = lim k 0 k k = lim k 0 k =0. sin y y sin 1, y 0, 0 2 f = 2 +y 2 0, y =0, 0 9
2 f 0, 0 = lim h 0 fh, 0 f0, 0 h f y 0, 0 = lim k 0 f0,k f0, 0 k = lim h 0 sin h h =1 = lim k 0 sin k k = 1 f 0, 0 = lim h 0 fh, 0 f0, 0 h f y 0, 0 = 0. h 0 sin 1 = lim 0 h h 0 h =0. sin5 + e y 2 e 2 +y 2 logy y 3 arctan 4 y y f = 5 cos5 + e y, f y = e y cos5 + e y 2 f = 2 logy+ 1 e 2 +y 2 f y = 3 f = y 2 + y 2, f y = 2y logy+ 1 y 2 + y 2 e 2 +y 2 4 f = y y y + log y, f y = y y y + log. a + by c + dy 2 log y 3 e a sin by + cos by 10
4 arcsin log y ad bc f = c + dy, f ad bcy 2 = c + dy 2 2 f = 1 log y, f y = log ylog y 2 3 f = ae a sin by + cos by, f y = be cos by sin by log y 4 f = 1 log y, f 2 y = y 1 log y 2. e log1 + y 2 1 2 y 2 f0, 0 = 0, f 0, 0 = 0, f y 0, 0 = 1, f 0, 0 = 0, f y 0, 0 = 1, f yy 0, 0 = 1 e log1 + y =y + 1 2 2y y2 + 2 f0, 0 = 1, f 0, 0 = 0, f y 0, 0 = 0, f 0, 0 = 1, f y 0, 0 = 0, f yy 0, 0 = 1 1 2 y 2 =1 1 2 2 + y 2 +. e a cos by 2 + 1 sin y f0, 0 = 1, f 0, 0 = a, f y 0, 0 = 0, f 0, 0 = a 2, f y 0, 0 = 11
0, f yy 0, 0 = b 2 e a cos by =1+ 1 2 a2 2 b 2 y 2 2 f0, 0 = 0, f 0, 0 = 0, f y 0, 0 = 1, f 0, 0 = 0, f y 0, 0 = 1/2, f yy 0, 0 = 0 + 1 sin y = y + 1 2 y R 2 f, y = 4 + y 4 2 2 +4y 2y 2. f f, y =43 + y, f y f y, y =4y3 y +, f = f y =0, y = 2, 2,, y = 2, 2,, y =0, 0. f 2 f 2, y = 432 1, f yy 2 f y 2, y = 43y2 1 f y 2 f, y =4, y f f, y Hf, y f f y f y f yy = y 2 f, y =4 y 43 2 1 4 4 43y 2 1. 2, 2, 2, 2, 0, 0 f 12
2, 2, 2, 2 det Hf 2, 2 = 384 > 0, f 2, 2 = 20 > 0 det Hf 2, 2 = 384 > 0, f 2, 2 = 20 > 0 det Hf0, 0 = 0. f, y ± 2, 2, f± 2, 2 = 8. f0, 0 = 0 = y 0 f, =2 4 > 0 =0, 0 y < 2 f0, y=y 2 y 2 2 < 0. 0, 0 f, y f0, 0 = 0 f0, 0 = 0. R 2 f, y = 3 + y 3 2 + y y 2. f =3 2 2 + y, f y =3y 2 2y +, f = f y =0, y =0, 0,, y =. f, y Hf, y = 1 3, 1 3 6 2 1 1 6y 2 1 0, 0 3, 1 f 3 det Hf0, 0 = 3 > 0, f 0, 0 = 2 < 0 13
f0, 0 = 0. f det Hf 1 3, 1 = 1 < 0 3 1 3, 1 = 1 3 27. 2 2 +2y +3y 2 =1 2 0,y 0 2 + y 2,. F, y, λ = 2 + y 2 λ2 2 +2y +3y 2 1 F =2 λ4+2y, F y =2y λ2+6y, F λ = 2 2 +2y +3y 2 1, F = F y = F λ =0 0,y 0. F + yf y, F λ =0 2 0 + y2 0 = λ F = F y =0 λ 2 1 2 0 +y2 0 0 + y 0 =0 0 +3 1 2 0 +y2 0 y 0 =0. 0,y 0 0, 0 2 1 1 2 0 +y2 0 1 3 1 2 0 +y2 0. 1 2 0 + y0 2 2 1 1 3 =0 14
.. 5+ 5 2 0 + y0 2 = 10 4l,., y, z +y +z = l, S =2y+yz+z. F, y, z, λ=2y + yz + z λ + y + z l, F = F y = F z = F λ =0 λ = 4 l. 3 = y = z = l 3.. 15