f (x) dx)x(f=yO0)0AOoooooo :Copyright : e-mail: strantoneas@gmail.com 6974593717 27310 28791 . 1 1.3 1 5 2 28 51 1. 53 155 2 80 3 x0\ 101 4 106 5 o x0\ 128 6 136 7 148 8 159 182 2. 185 1 187 2 206 3 233 4 242 5Rolle 247 6 261 7 272 8 281 9 291 10 305 11 de L Hospital 315 12 323 326 3. 331 1 333 2 344 3 357 4 F(x) =( )xf t dt 368 5 378 6 407 423 428 445 1 __________________________________________________________ - . - . -. - . . - . . . . - , . - . , , . , 16 - , . , . , - , . , . , . - , , , , . - , . - , , . , -, . , . , - . - . Bolzano . - , [, ]. T 2 , - . , . - , , - . - Rolle, , , - . - - . . . . . - , - . , , . - - , . , , - . - . , , . , 2010 , - . - . . , , . WILLIAM THRUSTON , 3 1. - ` ] _ \ ^ - . ( ). - - 3 4 - 16 . - i. , - . - , . - . - , . , - , , 16 . -, . - - . . - . 1:& 5 1 , , , -, . GOTTFRIEDLEIBNIZ(16461716) 16 . x = 242 . - - . H - , + , , . , - , . , . , . -, , . C R , : 6 1: - - , R, (0) (1) . i, i2 = 1. z C -z = + i, , R . + i,, R - i . O z Re(z). O z Im(z). C + 0i, i 0 + i. C (z1 + z2) + z3 = z1 + (z2 + z3 ) (z1.z2).z3 = z1.(z2.z3 ) z1 + z2 = z2 + z1 z1.z2 = z2.z1 z + 0 = 0 + z=z z.1 = 1.z = z z + (z) = (z) + z = 0z . 1z = 1z.z = 1,z 0 z1.(z2 + z3)=z1.z2 + z1.z3 z + z1 = z + z2 z1 = z2 z.z1 = z.z2 z1 = z2 , z 0 z + z1 = z2 z = z2 z1 z.z1 = z2 z = 21zz, z1 0 + i: + i = + i = = . 0 = 0 + 0i,: + i = 0 = 0 = 0. z = + i z= i - z. R C R C. - . . + i < + i < = = 0. z = + i- -(, ) . -z. -1:& 7 - - . z = + i - , OM , (, ). z= i z , (, ). M O (, ). z1 = + iz2 = + i: z1 + z2 = ( + i) + ( + i) = ( + ) + ( + )i. z2 = + i z1 = + i: z1 z2 = ( + i) ( + i) = ( ) + ( )i. . 1 2 1(, )2(, ) + i + i- , ( + i) + ( + i) =( + ) + ( + )i (+, +).(1) ,OM, = 1OM, + 2OM, ,: + i + i . , ( + i) ( + i) = ( ) + ( )i (, ).(2) ,ON, = 1OM, 2OM, ,: + i + i . z1 = + i z2 = + i: z1.z2 = ( + i).( + i) = + i + i + (i)(i) = + i + i + i2 = = + i + i = ( ) + ( + )i. z1 = + iz2 = + iz2 0: 8 1: - - 12zz = i i++ = ( )( )( )( ) i i i i+ + = 2 2( ) ( ) i + + + = 2 2 ++ + 2 2 +i. > 1z1 = z, z2 = z.z , -z = z1.z. z 0 , z0 = 1, z = 1z . 4 : i = i4+ = i4.i = (i4).i = 1.i = i = 1 , 0, 11 , 2, 3i i = = = =. C R .. (z w)2 = z2 2zw + w2,(z + w)(z w) = z2 w2, (z w)3 = z3 3z2w + 3zw2 + w3,. - . = 1 + (1),S = 2 (1 + ) = 2 [21 + (1)] = 1 . 1, S = 1( 1)1 , 1. ! R C. : x, yR x2 + y2 = 0x = y = 0. (x2 + y2 = 0x2 = y20.x20, x = 0y2 = 0 y = 0) z, wC z2 + w2 = 0 z, w-. , z = 1w = iz2 + w2 = 12 + i2 = 1 1 = 0. 1.1. : z = + i z = i-: (i) z+ z = 2a (ii) z z = 2i (iii) z z = 2 + 2 :(i) z +z= ( + i) + ( i) = + i + i = 2. (ii)z z= ( + i) ( i) = + i + i = 2i. (iii)z.z= ( + i)( i) = 2 (i)2 = 2 2i2 = 2 + 2. (i) :Re(z) = = 2z z +. (ii) :Im(z) = = 2z zi. 1.2. : z : (i) zR z= z. (ii)zI z= z. 1:& 9: z = + i, , R : (i)z= z i = + i 2i = 0 = 0 zR . (ii) z= z i = (+i) i = i 2 = 0 = 0 zI. 1.3. :z1 = + iz2 = + i , : (i) 1 2z z += 1z+ 2z (ii)1 2z z = 1z 2z(iii)1 2z z = 1z. 2z(iv)12zz = 12zz ,z2 0 : (i) 1 2z z +=) ( ) ( i i + + +=i ) ( ) ( + + += (+) (+)i = + i i = (i) + (i) = 1z +2z . (ii)(1) 1 2z z =( ) ( ) i i + +=( ) ( ) i + = () ()i = i + i= (i) (i) = 1z 2z . (2) 1 2z z = 1 2( ) z z + = 1z+( )2z = 1z 2z . ( )2z =) ( i + =i = +i = (i) = 2z . (iii)1 2z z =) )( ( i i + +=i ) ( ) ( + + = () (+)i = = i i = (i) i(i) = (i)(i) = 1 2z z . (iv) (1) 12zz = i i+ + = ( )( )( )( ) i i i i+ + = 2 2( ) ( ) i + + + = 2 2 ++ 2 2 +i. 12zz = i i = ( )( )( )( ) i i i i + + = 2 2( ) ( ) i + + + = 2 2 ++ 2 2 +i. 12zz = 12zz. (2) 1z= 122zzz 1z= 12zz 2z12zz = 12zz. 1.3.1. : : 1 2...z z z + + + =1z+ 2z+ + z . 1, 2, , R1 1 2 2... z z z + + + = 1.1z+ 2.2z+ + .z . 1 2...z z z =1z. 2z. . z . z1 = z2 = = z = zz=... z z z `_ =... z z z `_ =()z . 10 1: z = ( )1z = 1z = ()1z =() z , z 0. z2 + z + = 0, , R 0 Vieta z2 + z + =0 2z + za + = 0 z2 + 2z2 = z2 + 2z2 + 22 = 224 22z + = 2244 22z + = 24 = 2 4 . : > 0 :z1,2 = 2. = 0 :z = 2. < 0 24 = 2( 1) ( )4 = ( )222(2 )i = 22i - :22z + = 22i . :z1,2 = 2 i , - . :z1 + z2 = z1 . z2 = . _________________________________________________________ ,. , . z + i Re(z) , Im(z) , . 1.4. x, y : (2x + 5y 6) + (5x 6y 2)i = (7+60i). : (2x + 5y 6) + (5x 6y 2)i = 7 60i 2 5 6 75 6 2 60x yx y+ = = 2 5 15 6 58x yx y+ = =. D = 2 55 6 = 12 25 = 39 Dx = 1 558 6 = 6 + 290 = 296 Dy = 2 15 58 = 116 + 5 = 111. 1:& 11x = xDD = 29637 = 8 y = yDD = 11137 = 3. 1.5. z = (+i)(i+) (+i)(1+2i) + 2i. ) z +i. ) , zI. ) , :zR . : )z = (+i)(i+) (+i)(1+2i) + 2i = 2i + + i2 + 2i (+2i+i+2i2) + 2i= 2i + + 2i 2i i + 2 + 2i = (2) + (2+22+1)i. )z 2 = 0 = 2R . )zR 2 + 2 2 + 1 = 0 ( 1)2 + 2 = 0 1 = 0 = 0 = 1 = 0. 1.6. w = (3i)(i+2) ii11+. Re(w)Im(w). : w = (3 i)(i + 2) 11ii+ = 3i + 6 i2 2i 2(1 )(1 )(1 )ii i+ = 3i + 6 + 1 2i 21 21 1i i ++ = 7 + i 1 2 12i = 7 + i 22i = 7 + i + i = 7 + 2i. Re(w) = 7Im(w) = 2. . -: i :4 - 0, 1, 2, 3. i 1, i, 1i . zzC , z2, z3, -. , -, i. : (1 i)2004 . (1 i)2 = 1 2i + i2 = 1 2i 1 = 2i. (1 i)2004 = [(1 i)2]1002 = (2i)1002 = 21002.i1002 = 21002.i2 = 21002. , -, . 1.7. ) i1962. ) w = i + i2 i3 + i1961 + i1962 - 12 1: +i. ) w4= 4. : ) 1962:42, i1962 = i2 = 1. ) 1962 i i. w = i1962( ) 11ii = i196211ii+ = i1 11 i + = 21ii+ = 2 (1 )(1 )(1 )i ii i + = 2 (1 )1 1i i + = 2 (1 )2i i = i(1i) = i + i2 = 1i. ) 2w = (1+i)2 = (1)2 + 2(1)i + i2 = 12i 1 = 2i. 4w= ( )22w= (2i)2 = 4i2 = 4. 1.8. 4, (1+i)(1+i2) = 0. : 1 : 4, : = 4+1, N , (1+i)(1+i2) = (1+i4+1)(1+i8+2) = (1+i1)(1+i2) = (1+i)(11) = 0. = 4+2, N , (1+i)(1+i2) = (1+i4+2)(1+i8+4) = (1+i2)(1+i0) = (11)(1+1) = 0. = 4+3, N , (1+i)(1+i2) = (1+i4+3)(1+i8+6) = (1+i3)(1+i2) = (1i)(11) = 0. 2 :(1+i)(1+i2) = 1 + i + i2 + i3. - 1 = 1 = i1, 4. ,(1+i)(1+i2) = 14( ) 11ii = 411ii = 1 11i = 0 1.9. z = 32+ 12i. ) z3. ) z2002 ,z2019 . ) , : (+i)z2002 + z2z2019 iz =3 + i. : )z3 = 33 12 2i + =( )3132i + = 18( 3 +i)3 = ( ) ( )3 22 3 13 3 3 3 38i i i + + + = 18(3 3 +9i3 3 i) = 18.8i = i. )z2002 = z3 . 667+1 = ( )6673z z = i667 z = i3 z = i z = i 3 12 2i + = 12 32 i. z2019 = z3 . 673 = ( )6733z= i673 = i1 = i. )2z = 23 12 2i = 232 23 12 2i + ( )212i = 34 32i 14= 12 32i . -1-:S=111 .2002:3677:43.2019:31:& 13(+i).z2002 + 2z z2019 iz =3 + i (+i)1 32 2i +1 32 2i .ii 3 12 2i + = 3 + i 1 32 2i (+i+i)i3 12 2i + = 3 + i 1 32 2i (+2i)i 3 12 2i + =3 + i 2 + i 32i +3 32i+ 2= 3 + i 2 +( ) 32 i+ =0 + = 0. , . 1.10. :) =(+i)4 (i)4. ) =(+i)4+2 + (i)4+2. : )(1) (+i)2 = 2 + 2i + (i)2 = 2 2 + 2i = p (i)2 = 2 2i + (i)2 = 2 2 2i = p. , = (+i)4 (i)4 = [(+i)2]2 [(i)2]2 = p2 (p)2 = p2 p2 = 0. (2) = (+i)4 (i)4 =(+i)4 (i2i)4 = (+i)4 [i(+i)]4

= (+i)4 i4(+i)4=(+i)4 1.(+i)4 = (+i)4 (+i)4 = 0. )(1) (+i)2 = p(i)2 = p. , = (+i)4+2 + (i)4+2 = (+i)2 (2+1) + (i)2 (2+1)

= [(+i)2]2+1 + [(i)2]2+1 = p2+1 + (p)2+1 = p2+1 + (p2+1) = p2+1 p2+1 = 0. (2) = (+i)4+2 + (i)4+2 = (+i)4+2 + (i2i)4+2 = (+i)4+2 + [i(+i)]4+2

= (+i)4+2 + i4+2 (+i)4+2 = (+i)4+2 1.(+i)4+2= (+i)4+2 (+i)4+2 = 0. -. z -, : z= z z z= z z. z = w + w z = w .w z. z = w w z. , - 14 1: z = +i = 0 zR = 0 z. 1.11. )z = zz12 zz12 z . )zi w = ( ) i ww + i1 z . : ) z= 1 12 2z zz z =12zz 12zz =12zz 12zz= 1 12 2z zz z =z. zI. )zi w = ( )i ww + 1i zi w = i ww + 1i z = i2w +w z = w +w .zR . -. C : 1 . , . 2 (z2 + z + = 0, , R 0)- = 2 4 - - . , 2 , Horner. z = x + yi ,x, yR x, y. 1.12. , , : )zi 2 + 1 = z i )zi 1 z ii 2+ = z + i 112. : )2zi + 1 = z i z + 2 i = (2i)z i(2i) z + 2 i = 2z iz 2i + i2z 2z + iz = 2 + i 2i 1 z + iz = 3 i(1 i)z = 3 + i z = 31ii+ z = (3 )(1 )(1 )(1 )i ii i+ + + = 3 3 11 1i i + + + = 2 42i + = 2(1 2 )2i + = 1 + 2i. )1zi 2z ii+ = z + 112i (1 )(1 )(1 )z ii i+ + ( )(2 )(2 )(2 )z i ii i + = z + 112i 2z iz + 2 2 15z iz i = z + 112i 5(z + iz) 2(2z iz 2i 1) = 10z + 5(i 11) 1:& 155z + 5iz 4z + 2iz 4i + 2 = 10z + 5i 55 9z + 7iz = 57 + i (9 7i)z = 57 iz = 579 7ii = (57 )(9 7 )(9 7 )(9 7 )i ii i + + = 513 399 9 781 49i i + ++ = 520 390130i + = 520130 + 390130i = 4 + 3i. 1.13. ) , , : z3 3z2 + 4z 2 = 0 ) : z3 + 4z = 3z2 + 2z10 32z + 32 = 0 . : ) z3 3z2 + 4z 2 = 0: 1 , 2. z3 3z2 + 4z 2 = 0 (z 1)(z2 2z + 2) = 0 z 1 = 0z2 2z + 2 = 0 z = 1z = 1 + iz = 1 i [ = (2)2 4.1.2 = 4 8 = 4z = 2 22i = 2(1 )2i = 1 i. ] ):z3 + 4z = 3z2 2z3 3z2 + 4z 2 = 0 z = 1z = 1 + iz = 1 i . . z1 = 1:101z 32z1 + 32 = 110 32.1 + 32 = 1 32 + 32 = 1 0 z2 = 1 + i:22z = (1 + i)2 = 1 + 2i 1 = 2i

102z = ( )522z = (2i)5 = 32 i5 = 32i 102z 32z2 + 32 = 32i 32(1 + i) + 32 = 32i 32 32i + 32 = 0 z3 = 1 i:103z 32z3 + 32 = 102z 322z+ 32 = 102 232 32 z z +=0 = 0. :1 + i , 1 i . 1.14. ) z : (1+i)z + 4 z= 8+10i. ) w = z +z + z z+z z +2 23 35 236 +i. ) w14 . : )1: z = x + yix, yR .: (1 + i)z + 4 z= 8+10i (1 + i)(x + yi) + 4(x yi)= 8 + 10i x + yi + xi y + 4x 4yi = 8 + 10i (5x y) + (x 3y)i = 8 + 10i = = 10 38 5y xy x) 5 ( = + = 50 15 58 5y xy x 14y = 42 y = 4214 y = 3 x 3(3) = 10 x + 9 = 10 x = 1. z = 1 3i. 13421 122 1220 16 1: 2: z z i 4 ) 1 ( + +=i 10 8+(1 i)z+ 4z = 8 10i. : (1 ) 4 8 104 (1 ) 8 10i z z iz i z i+ + =+ + = D = 1 44 1ii+ = (1 + i)(1 i) 16 = 1 + (1)2 16 = 1 + 1 16 = 14. Dz = 8 10 48 10 1ii i+ = (8 + 10i)(1 i) 4(8 10i) = 8 8i + 10i + 10 32 + 40i = 14 + 42i. z = zDD= 14 4214i + = 1 3i. )z + z= 2 ,z z= 6iz.z= 1 + 9 = 10 , w = 2 23 35 236z z z zz z+ + + + = 23 2 2( ) 2 5 2( ) 3 3 36z z z z z zz z z z z z+ + + + + = 23( ) 3 2( ) 3 ( ) 36z z z zz z z z z z+ + + + + = 23( ) 3 2( ) 3 ( ) 36z z z zz z z z z z+ + + + + = 232 3 10 2( 6 ) 3 10( 6 ) 36 i i+ + + + = 34 30 2216 180 36 i i+ + + = 36216 180 36 i i + = 3636 36i + = 3636(1 ) i + = 11 i + = 1(1 )(1 )ii i+ = 11 1i + = 12 12i . )w = 21(1 )2i = 14 (1 i)2 = 14 (1 2i + i2) = 14 (1 2i 1) = 14 (2i) = 12i w14 = (w2)7 = ( )712i = 712 i7 = 712 i3 = 712 (i) = 712 i. ,w 14 = 141w = 7112i = 72i = 2128 ii = 1281i = 128iI. -. (x,y) z = x+yi w ..., -, , , . z. x, y . (x,y) z = x + yi , - - . 1.15. z=x+yix, yR w = (zi)( z +1). 1:& 17) w +i. )wR, z . )wI , z C. ) C - , . , . : ) w = (z i)( z + 1) = z z + z i z i = x2 + y2 + x + yi i(x yi) i = x2 + y2 + x + yi xi y i = (x2 + y2 + x y) + ( x + y 1)i . ) wR x + y 1 = 0 y = x + 1 z : y = x + 1. ) wI x2 + y2 + x y = 0 x2 + 2.x.12 + 14 + y2 2.y.12 + 14 = 14 + 14

( )212x ++ ( )212y = 12. z C: ( )212x ++ ( )212y = 12. ) C(12,12). x = 12y = 12 + 1 = 12. C . C . y = x + 1: x2 + y2 + x y = 0 x2 + (x+1)2 + x (x+1) = 0 x2 + x2 + 2x +1 + x x 1 = 0 2x2 + 2x = 0 2x(x+1) = 0 x = 0x+1=0 x = 0x = 1. x = 0y = x + 1 = 0 + 1 = 1 x = 1y = x + 1 = 1 + 1 = 0. C (1,0)(0,1).() = ( ) ( )2OA OB = 1 12 = 12 .. 1.16. f (z) = z ii z4+2zC . ) f. ) z*C {12i}:( )fz1 = ( )fz1 . ) z f (z)R . : )iz+2 0 iz 2 z 2i z 22( ) ii z 2i. f C {2i}. 18 1: ) z 0 ,z1 2i z 12i z 22ii z 21i z1 2 i z1 2 i z1 2 i z 12i z 21i. z*C {12i}: f (z1) =1412iziz +=1 42izzi zz+=1 42izi z+ f (z1) =( )1412iziz +=1 42i zzi zz +=1 42izi z +=4 12izi z+. 1( ) fz =1 42izi z + =1 42izi z+ += 4 12i zi z+= f (1z). )z = x + yix, yR(x,y) (0,2).: f (z)R ) (z f= f (z) ( )42z iiz+ = 42z iiz+ 42z iiz+ + = 42z iiz+ ( z+ 4i)(iz + 2) = (i z + 2)(z 4i) iz z + 2 z + 4i2z + 8i = iz z + 4i2z + 2z 8i 2iz z 6z + 6 z+ 16i = 0 iz z 3(z z ) + 8i = 0 i(x2 + y2) 3.2yi + 8i = 0 x2 + y2 6y + 8 = 0 x2 + (y3)2 = 1. x = 0y = 2:02 + (23)2 = 0 + 1 = 1. z C: x2+(y3)2 = 1 (0,2). ___________________________________________________________________ 1.17. R :(2 1) + (10 9 10)i = 0. [.=-1] 1.18. x, y, : 3 2xi 1yi + = 9 15ii+ [.x=5 , y=2] 1.19. ) : i i+ + i i+ = 22 22 2 + , R(, ) (0,0). ) z = 33ii+ + 33ii+, Re(z). ) w = 121 153121 153ii+ + 121 153121 153ii+ , Im(w). [.)1, )0] 1.20. z = 5 13i. z = x(1 + i) + y(1 i). [.x=-4, y=9] 1:& 191.21. z1 = 1 + 2iz2 = 12 i. w w = z1 + z2, R ,: ) w = 5+2i) w = 112+76i. [.)=3 , =4)=1/4 , =-23] 1.22. , , *R 6 = 4 = 5 z = w , z = 2( ) + (2 )i w = 23 + 2i. 1.23. z1, z2C Re(z1.z2) = Re(z1).Re(z2) z1R z2R . 1.24. xR, z = 49xix i++ . [.x=6] 1.25. w = 2 x iy i+x, yR . ) x, ywR . ) w , 2. [.)x+2y=0] 1.26. z = 42x iyi+xZy*N . z . : ) x = 8y. ) z = 4y

)z 2 , x, y. [.(x,y)=(-8,1) (-4,2)] 1.27. 1, 2, 34 z1, z2, z3z4. 1234 z1 + z3 = z2 + z4 . 1.28. w = 1659200919902i ii . [.i] 1.29. + i ,, R : i) z = ( )963 77 3ii++ ( )351 33ii+ ii) w = ( )1336 55 6ii++ ( )1679 22 9ii+ [.i)z=1-i ii)0] 1.30. : ) i i+ + ( ) i i+ = i + i ) i i + + ( ) i i+ = (i) + (i). 1.31. : 20 1: )S1 = i + i3 + i5 + + i1777. )S2 = (1 + i) + (2 + 3i) + (22 + 5i) + + (299 + 199i). )S3 = (20 + i) + (17 2i) + (14 + 4i) + + (31 217i). )S4 = i + (2 + 3i) + (4 + 5i) + (6 + 7i) + + [(2 2) + (2 1)i] ,*N . [.)i )2100-1+104i )99-87381i )(-1)+2i] 1.32. z =2 2 i + 2 2 . ) z2 ,z4 ,z8. ) :z1792 = 21792. 1.33. z =2 3 + i 2 3 + . ) z2 ,z4 ,z12. ) :z1980 = 21980. 1.34. z = 12 i 32. ) z3,z1006,z1531 . ) w = i.z1006 + z1531 + 1254 3i z + i + i. ) w2010 = 22010. [.)w =2i] 1.35. zz + 1z = 1, : ) :z3 = 1. ) w = z2015 + 20151z. [.)1] 1.36. : ) (3 + 4i)2024 (4 3i)2024 = 0 ) (5 + 2i)2010 + (2 5i)2010 = 0 ) (2 + i)1992 (1 2i)1992 = (1 i)2138 + (1 + i)2138

1.37. i3+1 = 1. [.=4+1] 1.38. = (2 + i2)(3 i) ,N. 1.39. z = 12i, : i) 1z+1zii) z2 + 2ziii) zz + zz iv) z3 + 3zv) z4 + 4zvi) 3zz 3zz [.i)2/5 ii)-6 iii)-6/5 iv)-22 v)-14 vi)48i/125] 1.40. z = (5 + 9i)2010 + (5 9i)2010 . 1.41. , R*N , z = ( + i) ( i) -. 1.42. xR, w = ( )2020x ix i++ ( )2020x ix i+ . 1:& 21 1.43. z = + i, R. z ,z 3 4 + 10 = 0 42 + 62 = 5. 1.44. z1, z2C , z2 0 Re12zz = 1 2 1 22 22z z z zz z+ , Im12zz = 1 2 1 22 22z z z ziz z. 1.45. z1, z2C, : ) z2 + 2z ) 12zz + 12zz) 1 zz+ + 1 zz+ 1.46. z1, z2C , w = z12z+ 1z z2 , u = z12z 1z z2 . 1.47. 3 52z iz = ( )3 52z iz+ ,z*C zR. 1.48. 2 93i zi z = 2 93i zz ,z*C zR. 1.49. 45i ziz = ( )45z iz+ ,z*C z. 1.50. z1, z2, z C z2 0 1 22z z zz+ = 12zz (z1 + z2). z R. 1.51. i z ww+ = 21 i ( )wi w ,zC w*C z -. 1.52. 1 21z z = 1 21 2z zz w z w + ,z1 , z2 , w*C w . 1.53. z, ww 0 z.w =1. : ) z + w 0 ) 1 zwz w++,z wz w+1 zwiz w+ . 1.54. zC , w = iz 2u = 3(z 2 + 6i) - . [.z=3-5i] 1.55. z , 1zi + 13 2zi = 13 234 6ii+. [.z=5-2i] 22 1: 1.56. : i) x2 8x + 25 = 0 ii) 36x2 + 48x + 25 = 0iii) x2 2 3 x + 7 = 0iv) x4 3x2 4 = 0v) x4 + 14x2 + 45 = 0vi) x3 x2 + 8x + 10 = 0 [.i)4 3i , ii)-23 12i , iii) 3 2i , iv) 2, iv) 3i, 5 i vi)-1, 1 3i] 1.57. : ) x2 24x = 3 ) x2 +24x = 5 [.) 2, i ) i, 2i] 1.58. z (1 3i)iz (1 + i) z = 6 4i. [.z=2-i] 1.59. z i.z + z + i = 3. 1.60. zC 2 z + (1i 3 )z = i. 1.61. C z2 + 2 z + 1 = 0. [.1, 1+2i, 1-2i] 1.62. (1 ) 42 (1 2 ) 6 3i z iw iz i w i+ = + + = . [.z=2-i , w=i] 1.63. 2 1 4(1 2 ) 2 1 3z iw ii z i w i+ = ++ = . [.z=1+i , w=2+i] 1.64. x2 3x + 3 + i = 0 1 + i. 1 i. ; 1.65. 2x2 + x + = 0 , , R, 2 i . [.=-8, =10] 1.66. x2 + x + 5 = 0 , , R, 1 + 12 i - . [.=4, =-8] 1.67. 4x2 x + = 0 , , R, 12 3i. - . [.=4, =13] 1.68. z = 1i z5 + z3 + = 0 ,, R , = 2 = 8. 1.69. 2, , - 13 i. [.9x2-6x+10=0] 1:& 231.70. 2z3 z + 1 = 0128z14 + 2z 1 = 0. [.12(1 i)] 1.71. (z) = z2 2z + 2Q(z) = z3 + z2 + z 2, R. i) z1 , z2(z) 121z+ 122z = 27. ii) P(z) Q(z), , . (2001) [.ii)=-3,=4] 1.72. ) z1, z2 z2 + 2z + 2 = 0, 201z 202z= 0. ) z1 (), , 1z . [.)=4] 1.73. z : z= z2. [.)0, 1, -12+32i, -12-32i] 1.74. P(z) = z3 (3+i)z2 + (3+2i)z 1 3i. i , 1i 2 + i . 1.75. f (z) = z4 + (2 3i)z3 + (1 6i)z2 + (2 3i)z 6i) :f (i) = f (i) = 0. ) f (z) = 0. [.)-2, -i, i, 3i] 1.76. z : ) z2 = 3+4i) z2 = 34i) z2 = 1630i[.)2+i,-2-i )1-2i, -1+2i )5-3i, -5+3i] 1.77. z. : ) z20,zR. ) z2 < 0,z = i , *R . ) z3 > 0,z 3 i , < 0. 1.78. z = 1 + 3i z1 , z2 y = 3 x y = 2x + 5. [.z1=1+2i, z2=-2+i] 1.79. z = + i ,, R , - x y = 2 f () = i.z , N . ) f (1600),f (1994),f (1729) . ) z () 2.. [.)z=1-i] 1.80. z2 + 2z + = 0 > > 0. ) . ) . 24 1: 1.81. z = x + yi,x, yR R :( )22z z + + ( )22z zi i = + (1)i. : ) Im(z) = 0 , = 1. ) = 0 , z2 + 1 = 0. ) 0 1. ) z , . [.) , =1] 1.82. z , z2 1 , . OA OB , ,, . [.x=0 , x2+y2=1] 1.83. z . , 1, z, 1 + z2 . [.y=0 (x-1)2+y2=1] 1.84. z . , 1, z, i.z . [.(x-12)2+(y+12)2=12] 1.85. z Re(z2 z) = 1 Re(z). [.x2-y2=1] 1.86. z z2 + 2z + 4 ( )2Im( ) z= 2Re(z) 4.Im(z) [.(x-12)2+(y+1)2=54] 1.87. z : z3 + 3z= 2.Re(z). [.O yy x2-3y2=1] 1.88. zw, w = 42i zi z+. ) z w. ) z wI. [.)z 2i)x2+(y+1)2=9 (0,2)] 1.89. z -: ) Re(z 1z) = 12 Re(z) ) Im(z 1z) = 2.Im(z) [.) yy (0,0) x2+y2=2 ) xx (0,0) x2+y2=1] 1.90. zw, w = z + iz , z 0. - z , Re(w) = Im(w). [.y=x (0,0) x2+y2=1] 1:& 25 1.91. z 211zz ++ . [.y=0 (-1,0) (x+1)2+y2=2] 1.92. z , 21i zz+ . [.y=0 (-1,0) (x+1)2+y2=1] 1.93. f (z) = 31z ii z+zC. ) f. ) z*C {i} 1( ) fz= f (1z). ) z f (z)R. [.)C-{-i} )x2+(y+2)2=1 (0,-1)] 1.94. f (z) = iz + 2 z 3izC. ) f 0. ) - f. ) f xx. [.)-1-2i )-32- 32i )x-2y-3=0] 1.95. f (z) = iz + 1 zC. ) - f. ) (), - f , .. [.)12+12i] 1.96. z = (2 i) ,*R . ) 1z + i. ) (x,y) z , 1z, x + 2y = 25, x 2y = 2. ) . [.)5x2-20y2=4] 1.97. ) z, wz+wRzw , . 26 1: ) P(x) = .x + 1.x1 + + 1.x + 0 ,iR0 i .zC P(z) = + iP( z ) = i ,, R. 1.98. , C : i) zC : z, z = 0, z ii) z, w, :(z + w) (z.w). 1.99. ) (1 + i)z2 + (1 5i)z 4 + 2i = 0 1 + i1 + 2i. ) (z2 + z 4)2 + (z2 5z + 2)2 = 0. 1.100. , : (1 + 2i) + (2 i) = 0. [.2] 1.101. (1 + i)(1 + i+1)(1 + i+2)(1 + i+3) = 0. 1.102. : x4 16x3 + 80x2 168x + 135 = 0 (2 + i2)(2 i) ,N. 1.103. z z= z3. [.)0, -1, 1, i, -i] 1.104. 13 195 72 2112z wz wz w = =+ = . [.(i,i), (-i,-i)] 1.105. S = i + 2i2 + 3i3 + + i. : )S i.S)S _____________________________________________________ 1.106. . - . I. z = + iw = + i, . II. , C : = = 0 2 + 2 = 0. III. 2 3i < 2 + 3i. IV. 232i =322i =i3=i. V. , :i = i = . VI. z1, z2C Re(z1 + z2) = 0Re(z1) + Re(z2) = 0. VII. z1, z2C Re(z1 z2) = 0Im(z1 + z2) = 0z1 = 2z . VIII. z1, z2C Im(z1.z2) = Im(z1).Im(z2). 1:& 27IX. z1, z2C , z2 0 Re12zz = 12Re( )Re( )zz. X. zC (z z )20. XI. z1 = + i , z2C z1 + z2 = 2 ,z2 = 1z . XII. z1 = + i , z2C z1.z2 = 2 + 2 ,z2 = 1z . XIII. Re(z) = 1 z x = 1. XIV. 1, 2 z1z2 xx 12 z1 = 2z . XV. Im(z) = 4 z y = 4. XVI. zw , - 2z w +. XVII. x2 4x + = 0 , R, 1 + 2i1 2i. XVIII. x2 + x + = 0 , , R, 32i, 133 2i . 1.107. . - . I.( )23i = 1, : .1 .3 .2 .6 .5 II. 2 + 3i3 + 2i . x = 2 . y = 3 . y = x . y = 3 .x = 0 III. (2x + y 8) + (x y + 2)i < 0: .x = 2 ,y = 4 .x = 3 ,y = 2 .x = 3 ,y = 3 .x = 1 ,y = 3 IV. z2 + 8z = 0 ,R : .3 + 4i .4 + 3i . + i .4 3i .8 + 2i V. x2 + x + 13 = 0 , R : .3 + i .1 3i .3 + 4i .2 2i .3 + 2i 28 1: 2 , , - , . ARTHURCAYLEY (1821 1904) - . - -, , d(, ) = . R - z C. z, z , . - . 2.1. : z = x + yi(x, y) . - z -, z=OM= 2 2x y + . 2.2. : z : (i)z=z=z =z (ii) 2z=z z :z = x + yi , x, yR (i) z= x yi ,z = x yi z= x + yi. z=z=z =z = 2 2x y + . (ii) z z = (x + yi)(x yi) = x2 (yi)2 = x2 y2i2 = x2 + y2 = 2z . 2:292.3. :z1 , z2 : (i) 1 2z z =1 2z z (ii) 12zz = 12zz,z2 0. : (i) : 1 2z z = 1 2z z

21 2z z = ( )21 2z z (z1.z2) ( )1 2z z = 2 21 2z z z1.z2.1z.2z= z1.1z.z2.2z , . (ii) : 12zz = 12zz 212zz = 212zz 12zz 12zz = 2122zz

1 12 2z zz z= 1 12 2z zz z , . 2.4. : :1 2...z z z =1 2...z z z . z1 = z2 = = z = z z=... z z z `_ =... z z z `_ = z . 2.5. : z1 , z2 1 2z z 1 2z z + 1z +2z . : 1(z1) , 2(z2)(z1 + z2) z1 , z2z1 + z2 . : 1 2 OM OM , , 1 2 OM OM +, , 1 OM, +2 OM,. 1 2z z 1 2z z + 1z +2z . 2.6. : . : 1(z1) , 2(z2)N(z1z2) z1 , z2z1 z2- . -12: (12) = 2 1M M, =ON, = 1 2z z . 30 1: 0z z = z -z0 . - (z0) . 1z z = 2z z z z1z2. - (z1)(z2). 1z z + 2z z = , > 01 2z z < - z 12z1z2. 1 2z z z z = , > 01 2z z > - z - 12z1z2. _________________________________________________________ . : . , z z = + i . z , z= z . z w , -. w w. 2.7. z , w: )z = 3 45(2+ ) (1 )2 (1+2 )i ii i )w =i7(1+ ) 2 (3 )1i ii 5 )9+512i= 0. : ) z= 3 45(2 ) (1 )2 (1 2 )i ii i+ +=3 45(2 ) (1 )2 (1 2 )i ii i+ +=3 45(2 ) (1 )2 (1 2 )i ii i+ +=3 452 12 1 2i ii i+ + 2:31=( ) ( )( )3 455 22 5=( )( )4222 5 =42 5 =25. )(1+i)2 = 12 + 2i + i2 = 1 + 2i 1 = 2i. (1+i)6 = [(1+i)2]3 = (2i)3 = 8i3 = 8i. (1+i)7 = (1+i)6(1+i)= 8i(1+i) = 8i8i2 = 88i. w = (1+i)7 2 (3 )1i ii 5=8 8i 2 (3 ) (1 )(1 ) (1 )i i ii i + + 5 = 8 8i 22 (3 3 )1 1i i i i + + 5=8 8i 2 (4 2 )2i i + 5 =8 8i 4i 2i2 5=5 12i. w=5 12i = 2 25 ( 12) + =25 144 +=169= 13. ) 9 + 512i = 0 9 = 512i. 9= 512i9= 1512 = 91512 = 12. 2.8. w w i = 3 , : ) z1 = 3w+4+4wi3i)z2 = w22wi1 : )1: z1 = 3w+4+4wi3i = 3(wi) + 4i(wi) = (wi)(3+4i). 1z=( ) (3 4 ) w i i +=3 4 w i i += 2 23 3 4 += 3.5 = 15. 2: z1 = 3w+4+4wi3i z14+3i = 3w+4wi z14+3i = w(3+4i) w = 14 33 4z ii ++. w i = 3 14 33 4z iii ++ = 3 214 3 3 43 4z i i ii + + = 3 13 4zi + = 3 13 4zi + = 3 12 23 4z+ = 3 15z = 3 1z= 15. ) z2 = w2 2wi 1 = w2 2wi + i2 = (wi)2. 2z= 2( ) w i = 2w i = 32 = 9. . 2z =z z . -. , > 0 , 32 1: 2.9. zC, :z+4=z 2 1 z2= 3. : 4 z +=2 1 z 24 z += 22 1 z (z+4) ( 4) z += (2z1) (2 1) z (z+4)( z +4) = (2z1)(2 z 1) z z+ 4z + 4 z+ 16 = 4z z 2z 2 z+ 1 3z z+ 6z + 6 z+ 15 = 0 z z 2z 2 z5 = 0 z z 2z 2 z= 5 z z 2z 2 z+ 4 = 9 z( z 2) 2( z 2) = 9 ( z 2)(z2) = 9 (z2) ( 2) z = 9

22 z = 9 2 z = 3. 2.10. zC, :z 1 +z 2 z+z 3 . : 1 z + 2 z z + 3 z ( 1 z + 2 z )2( z + 3 z )2 21 z + 2 1 z 2 z +22 z 2z + 2 z 3 z +23 z (z1)( z 1) + 223 2 z z + +(z2)( z 2)z z + 223 z z + (z3)( z 3) z z z z +1+223 2 z z + + z z 2z2 z +4z z +223 z z + z z 3z3 z +9 223 2 z z + 223 z z +4 23 2 z z + 23 z z +2, , . 2.11. z1 , z2 , , z . w w = 21 21 2+ z zz z + 22 32 3+ z zz z + + 211+ z zz z , . : :1z= 2z= =z = . 1 1:z= 2z = 2 z .z= 2 z= 2z 1 z + = 2+1 z. 211 z zz z +++ = 211 z zz z +++ = 211 z zz z +++ = 22 212 21 z z z z++ + = 2211211( )( ) z zz z z zz z++++ + = 211 z zz z+++ = 211 z zz z +++ . w= w , w . . z - 2:33 , : 2z = z2 z. 2z= z2 z 2.12. ) : z2= z2 zR . ) z : z3 =z 2z z3= z2 z . z . : )2z = z2 z z = z2 z z z2 =0 z( z z) = 0 z=0 z = z zR . )3z = z 2z z3 =z 2zz ( z 2z) = 3 2z 2 z z = 3 2z = 3+2 z z. z3 = z 2 z z(z 2 z ) = 3 z2 2 z z = 3 z2 = 3+2 z z. 2z= z2 . z . . , . . - . -0z z = 1z z = 2z z z z0 - z1z2. 2.13. zw : z w =z w + z = w . z = w. : 1 () zw .OA, OB, - . : z w =z w + OA OB , , =OA OB +, ,OA, OB, z = w OA,= OB,. OA, OB, . , z = w. 2 () z = +iw = +i. z = w 2z =2w 2 + 2 = 2 + 2. z w =z w + 2z w = 2z w + ( z w )2 = (z+w) ( ) z w + 34 1: 2z 2 z w +2w = (z+w)( z + w ) z z 22z + ww = z z + z w+ w z+ ww 22z = z w+zw 22z = 2Re(z w ) 2z = Re(z w )(1)z w= (+i)(i) = i + i + = (+) + ()i.(1) 2 + 2 = ( + )( + )2 + ( + )2 = 2 + 2 + 2 + 2 + 2 + 2 = 2(2 + 2) + 2( + )= 2( + ) + 2( + ) = 0. + = 0 + =0 = = z = w. 2.14.w*C w , 2w+32iw , 2w32iw - , . : z1 = w , z2 = 2w+32iwz3 = 2w32iw , . () = 1 2z z = 32 2w iww + = 32 2w iww+ = 3 32 2w iw= 3 32 2iw

=w3 32 2i=w9 34 4+=w 3 . () = 1 3z z = 32 2w iww = 32 2w iww+ += 3 32 2w iw+= 3 32 2iw +

=w3 32 2i+=w9 34 4+=w 3 . () = 2 3z z =3 32 2 2 2w iw w iw + = 3 32 2 2 2w iw w iw + + += 2 32iw =3 iw =w 3 . , () = () = () . 2.15. f(z) =2 z +4 z+ i zC . : ) . ) z f . ) z , . : ) f (z) =2 z +4 z i +=2 z +4 z i + (2 ) ( 4 ) z z i + +=2 4 z z i + +=2 4i += 2 22 4 + = 4 16 + = 20 = 2 5 . f2 5 . ) f (z) =2 z +4 z i +=2 z +( 4 ) z i = () + () () = 2 5 . f - z , () + () = (), -z . - = y yx x = 0 42 0+ = 2. y y = (x x)2:35y 0 = 2(x2) y = 2x 4. z = + (2 4)i ,[0,2]. ) 1 : z= 2 2(2 4) + = 2 24 16 16 + += 25 16 16 + . z () = 52 16 + 16 - . = 162 5 = 85 [0,2]. z z = 85 + (285 4)i = 85 + (165 4)i = 85 45i. z= () ( )2 28 45 5+ = 64 1625 25+= 8025 = 4 55. 2 : , . . = 1 .2 = 1 = 12. : y y = (x x) y 0 = 12(x 0) y = 12x. 122 4y xy x= = 1212 42y xx x= = 124 8y xx x= = 125 8y xx = = 1 8 42 5 585yx = ==. (85, 45). z = 85 45i z= 4 55. 2.16. z 4+3 z i 2. ) z . ) . : ) 4 3 z i + 2 (4 3 ) z i 2. - z (4,3) = 2. )1 () (, ), - . () = 2 2(4 0) ( 3 0) + =16 9 +=25= 5. ()()() () ()()() + () () ()() + 5 2 z 5 + 2 3 z 7. z 7, 3. 2 () 4 3 z i + 2 ( 4 3 ) z i + + 2. 4 3 z i + ( 4 3 ) z i + + = 4 3 z i + 2. ,4 3 z i + 25 z 2 2 z 5 2 2 + 5 z 2 + 5 3 z 7. z 7, 3. 36 1: 2.17. z 10+5 +2 i z 5 . ) z . ) . : ) 10 5 2 i z + + 5522( 5 ) z i + + 52525 z i + + 552( 5 ) z i 52. z (5,52) =52. )1 () (,), . ()()() . = K K y yx x = 5025 0

= 12. y y = (x x) y 0 = 12(x 0) y = 12x. 2 2125 5( 5) ( )2 4y xx y=+ + + = ( )22121 5 5( 5)2 2 4y xx x=+ + + = ( )22125 5( 5)2 4y xxx=++ + = 2212( 5)5( 5)4 4y xxx=++ + = 2 2124( 5) ( 5) 5y xx x= + + + = 2125( 5) 5y xx= + = 212( 5) 1y xx= + = 212( 5) 1y xx= + = 1 12 2 5 1 5 1y x y xx x= = + = + =` 1 1( 4) ( 6)2 2 4 6y yx x = = = =` 2 34 6y yx x= = = =` . (4, 2)(6, 3). z = 42i z = 6 3i. 2 () z = ( )( )5 52 25 5 z i i + + + 525 z i + + +525 i 52+25254+= 52+5 254

= 52+5 52 = 6 52 = 3 5 . z3 5 - . - : z + 5 + 52i = (552i) z = 5 52i 5 52i z = (55) + (5252)i , > 0 2:37525 z i + += 52 ( )5 5 52 2 2( 5 5) 5 i i + + += 52 5 5 52 2 25 5 5 i i i + += 52 525 i = 52 22 25254 += 52 25 254 = 52 5 52 = 52 = 15 = 15. z = (515 5) + (5 12 5 52)i = 63i. z = ( )( )5 52 25 5 i z i + + + 5 52 25 5 i z i + + 5 5 52 2= 4 52 = 2 5 . z2 5 - . -: z+5+52i = (552i) z = 552i 5 52i z = 5(+1) 5( 1)2+i , < 0 525 z i + += 52 5 5 52 2 25 5 5 i i i + += 52 525 i += 52 22 25254 += 52 25 254 = 52 5 52 = 52 = 15 = 15. z = 5(15+1) 52(15+1)i = 42i. 2.18. zw : z4 2 w i 3 1. ) z, w ; ) z w . ) z, w z w . : ) 4 z 2 3 w i 1. z - (4,0) 1 = 2, - w - (0,3) 2 = 1. )1 () , (,1)(,2). -, , . () = 2 2(4 0) (0 3) + =16 9 +=25= 5. ()()() () () () ()() + () + () () 1 2()() + 1 + 2 5 2 1 z w 5 + 2 + 1 2 z w 8. z w 8, 2. 2 () z w = ( 4) (4 3 ) (3 ) z i i w + + 4 z + 4 3i + 3i w 2 + 5 + 1 = 8. 38 1: z w = ( 4) (4 3 ) (3 ) z i i w + + 4 3i ( 4 z + 3i w ) =4 3i 4 z 3i w 5 2 1 = 2. z w 8, 2. ) = K K y yx x = 0 34 0 = 34. y y = (xx) y 0 = 34(x4) y = 34x + 3. 2 2334( 4) 4y xx y= + + = ( )23343( 4) 3 44y xx x= + + + = ( )2233412 3( 4) 44y xxx= + + = 223343( 4)( 4) 44y xxx= + + = 223349( 4)( 4) 416y xxx= + + = 2 233416( 4) 9( 4) 4 16y xx x= + + = 233425( 4) 4 16y xx= + = 23344 16( 4)25y xx= + = 3 33 34 42 4 2 44 45 5y x y xx x = + = + = =` 3 28 3 123 34 5 4 528 125 5y yx x = + = += =` 6 65 528 125 5y yx x = == =` . (285,65)(125,65). 2 2334( 3) 1y xx y= + + = 2 23343( 3 3) 14y xx x= ++ + = 2 23349116y xx x= ++ = 2 233416 9 16y xx x= + + = 233425 16y xx = + =23341625y xx = +=( )3 4 3 43 34 5 4 54 45 5y yx x= + = += =` 12 185 54 45 5y yx x = == =` .(45,125)(45,185). z, w z w z = 125+65iw = 45+125i z, w z w z = 28565iw = 45+185i. 2.19. z z1 1 z2= 1, z. : 1 z 1 2 z = 1, z (1,0) = 1 (2,0) 2:39 = 1. , z

. , - . = 900 (, ) . = 2, = 1 = 2 2 = 2 22 1 =4 1 =3 . z 3 , 1. ___________________________________________________________________ 2.20. : ) z1 = 56( 3 )(1 )ii+) z2 = (1 ) (1 2 ) (1 3 )(1 ) (2 ) (3 )i i ii i i+ + +

[.)4 )1] 2.21. : )z1 = 7 + i(2i) (12i)(3+i) 6i )z2 = 32(1 )(1 )ii+ 1 )z3 = 21ii+ + ( )21 21ii++)z4 = 2(1 3 )1ii+ (3 ) (3 )2i ii + [.) 10) 5) 61 /2 ) 122 ] 2.22. z, : ) z2 + 169i = 0 ) z3 125i = 0 ) z4 + 81i = 0 )z5 132i = 0)z2004 + i = 0[.)13 )5 )3 )12 )1] 2.23. zw 32z = 2w = 9 29 6zz+. 13w+= 1. 2.24. z = 3 + 2(1 2)i ,R. ) z 4x + 3y = 6. ) z z 1,2. 2.25. z = ( 3) + 2(2 + )i ,R. ) z 2x y + 10 = 0. ) z z 2 5 . ) z . 40 1: 2.26. z i , . z= 1, w = 2( ) z z i i z+ 1. 2.27. z00z 1 - 1 + z + z2 + + z1 = 0. 2.28. zCz 1 : z= 1 11zz+ . 2.29. zCz , *+R z z + z= . 2.30. z1, z2C , : ) 21 22 z z ++ 22 12 z z = 5(21z+ 22z ). )21 2z z + +22 1z z =(2 + 1)(21z+ 22z ) , R. 2.31. zC, :)2z i z += 22z + 2.Im(z2))2z i z = 22z 2.Im(z2). 2.32. zC :2 1 z=2 z z= 1. 2.33. zC : 16 z+= 4 1 z + z= 4. 2.34. z:(5z 1)5 = (z 5)5. ) 5 1 z=5 z) z= 1 )w = 5z + 1 , (w) . 2.35. zC : 8 z = 3 z 1 z+= 3. 2.36. zC : 10 z= 3 2 z 1 z= 3. 2.37. zC : 3 z+= 2 z i + 3 3 4 z i += 2 10 . 2.38. zC, : )5z i =5 iz+ z= 1 )z i =iz + z= 1R{1,1}. 2.39. zC , : ) 9 z=9 1 z z= 1 ) 4 1 z=4 z z= 1. ) z =1 z z= 1R{1,1}. 2.40. zC , : ) 49 z += 7 1 z + z= 7)2z += 1 z + z= 0 1. 2:412.41. z (4 z)10 = z10

x = 2. 2.42. z, w :100z w += 100z w . : )z w +=z w )z w + z w = 0 ) Re( z w) = 0 2.43. z, w*C . :z w +=z w zw . 2.44. z12iw = 22 1z ii z+. w 1 z 1. 2.45. zC,z2 + z + 1 = 0. z=1 z+=1. 2.46. zC , z=1 z += 1, :z2 + z + 1 = 0. 2.47. z1, z2. 21 2z z (1+21z )(1+22z ) 2.48. zC z 12 , :3(1 ) i z iz +< 34. 2.49. zC : 1 z +3 z + 6 +z+2 z . 2.50. zC , :Re( ) Im( )2z z + z Re( ) z+Im( ) z . ; ; 2.51. z1 ,z2 z3 1 , 24. : )z1 + z2 + z30)1 2 3z z z + += 1 2 31 4 16z z z+ + . 2.52. z, wCz2 + w2 = 0 , z w +=z w . 2.53. z1, z2 0 -: i)1 2z z += 1z+ 2z12zz > 0ii)1 2z z = 1z+ 2z12zz < 0. iii)1 2z z = 1 2z z +12zz < 0 iv)1 2z z = 1 2z z 12zz > 0. 2.54. ) z2 (3 + 2i)z + 6i = 0. ) f (z) = 1z z + 2z z z1, z2, . f. [. )3,2i ) 13 ] 42 1: 2.55. zC, : 1 z ++2 z+ z+3 z+ 2.56. zC , : 2 z++3 z+ 1 z++4 z+ 2.57. z1, z2, z3*C 1z= 2z= 3z= 1z1 + z2 + z3 = 1, : 11z + 21z + 31z = 1. 2.58. z1, z2, z3*C 1z= 2z= 3z= z1 + z2 + z3 = , : 11z + 21z + 31z = 1. 2.59. z, w, u 1. z + w + u zw + wu + uz . 2.60. z1, z2 , w = 1 21 2( ) z zz z++ ,N, . 2.61. z1, z2, z3C 1z= 2z= 3z z1z2 + z2z3 + z3z1 = 0, z1 + z2 + z3 = 0. 2.62. . z1, z2, z3C 1z= 2z= 3z z1+ z2+ z3 = 0, : ) z1z2 + z2z3 + z3z1 = 0 ) 21z +22z +23z= 0 ) 31z+ 32z+ 33z= 3z1.z2.z3. . w1, w2, w3 , , 1 2w w = 2 3w w = 3 1w w . 21w+ 22w+ 23w= w1w2 + w2w3 + w3w1. 2.63. z1, z2, z3C z1 + z2 + z3 = 0z1.z2 + z2.z3 + z3.z1 = 0, 1z =2z =3z . 2.64. ) z1, z2, z3Cz1 + z2 + z3 = 021z+ 22z+ 23z= 0 :

1z= 2z= 3z . ) , , , -w1, w2w3, : 21w +22w +23w=w1.w2 + w2.w3 + w3.w1 . . 2.65. zw : z w +=z+w z=w . z = w. 2.66. , , z1 = 1 + 2i , z2 = 4 2i , z3 = 1 6i . . 2:432.67. ) , C , z3 = 8. ) . 2.68. z1, z2C,z2 0 , , z1+z2 ,z1 z2 ,z1 + z2i 3 , . 2.69. zC 0z z = ,z0C,R 2z= 2Re(0z z ) + 2 20z . 2.70. z , : )z= 3 )2 z i = 1 )3 4 z i + = 4)2 4 z i + += 6)12 3 2 z i = 3)2 6 z i = 5) 22 1 z iz = 9 2.71. z, : )z< 2)3i z 1 ) 1 z 3) 2 0, :

1z z + 2z z = 1 2z z . z, z1, z2 ; 2.90. ) z1, z2 :21 2z z ++ 21 2z z = 221z+ 222z . . ) z1, z2C 1 2z z = 1z= 2z , 1 2z z +=31z . ) zC 1 z +=z= 1, :1 z=3 . ) z1, z2C1z= 2z=2, :2112zzz++ 21 2 122z z zz = 6. 2.91. z1, z2C : 1 2z z +=21z1 2z z =22z . 2.92. z1, z221z +22z= 21 2z z , 1 2z z += 1 2z z . . 2.93. zC 2z= 21 z , Re(z2) = 12. 46 1: 2.94. z*C z= 1zz+ , Re(z2) = 12. 2.95. z, z , 1zz1 -. [.1 3 i2 2] 2.96. z , 1 z=3 z=i z . [.2+2i] 2.97. w : 2(w4+4w ) = (w2+2w )2 , wR. 2.98. ) wC , : w2+2w= 0 Re(w) = Im(w) Re(w) = Im(w). ) z, w : (zw)2 + (z w)2 = zw2z+ ( w z )2. z w . 2.99. zC, : i)z z ++z z = 2 z zR ii) z z ++z z > 2 z z ( ) C R 2.100. C : ) 2z 2z + 2i = 0 ) 2z 4z = 1 + 8i. [. )1+i )3-2i,1-2i] 2.101. C : ) 2z z2 8i = 2) 2z + z2 + 6i = 2. [. )4-i,-4+i )1-3i,-1+3i] 2.102. C : )z 2z + 3 + 6i = 0)z+ z2 + 1 = 0. [. )4+3i )1+ 52i,-1+ 52i] 2.103. z :1 z + i 1 z +=z i + i. [. z=0 z=-1-i] 2.104. z1, z2 1 , 1 2z z < 1 21 z z . 2.105. z1, z2C , 1 21 21z zz z++ z1, z2 1. 2.106. z 3 4 z i + = 2. z. [.3, 7] 2:472.107. ) zC z2 + z + 1 = 0 z= 1. ) w*C w 1w = i , w= 1. ) z1, z2*C : 12zz 21zz = i. 1z= 2z . 2.108. z : 4 z 52 3 z i 52. [.z=2+32i] 2.109. z : z= 112iz= 2. [. z=-12+12i] 2.110. ) zC ( )( )2 22 2 z i z i + 0 21 z + 2 z . ) -z 2 z 21 z + . 2.111. ) zC ( )( )2 21 2 1 2 z z + 0 21 z 2 z . ) -z 2 z 21 z + . 2.112. zC z 1. :2 3 z 4. z . 2.113. zC z 2. :2 4 z 6. z - . 2.114. zC z 3. :2 5 z i 8. . 2.115. zC 1 z 1. :1 1 + z 3. . 2.116. zC 4 z i + 1, : 2 5 1 2 z 2 5 +1. . 2.117. zC 3 z = 2 2 z 2, 1 z 10 . . 2.118. zC 1 z + =1 2 z + 1, 3 z 2. . 48 1: 2.119. z z i = 1, 1 z + . 2.120. ) z1 , z2 , z3 , z4 : (z1 z4)(z2 z3) + (z2 z4)(z3 z1) + (z3 z4)(z1 z2) = 0. ) , , , z1 , z2 , z3 , z4 , - : ().() ().() + ().(). : - . 2.121. z, w 1. u, v u = z w zw + 1 v = z w zw + ,R. u, v . 2.122. zC z = 1 + i = 2211z i zz i z+ + +, R ( 1)2 + 2 = 2 243 z z + +. 2.123. z1, z2, z3 1 3z1 + 4z2 + 5z3 = 0. : )Re(z1.2z ) = 0.)21z+ 22z= 0. 2.124. z, wC , z.w = 1 z + w= 2, z + w . 2.125. 0, C 2z 2iz + 2(1+i) = 0. [.z1,2=+(-12 1- -2 )i , 0 2 -1] 2.126. 3 52 401z wz w + = =. [.(z,w)=(-1,1), (z,w)=(1,-1)] 2.127. 3 75 1101z wz w + = =. [.(z,w)=(i,i), (z,w)=(-i,-i)] 2:492.128. z1, z2C , :

21 2z z + (1+) 21z + (1+1)22z . 2.129. z1, z2, z3C : 1 2 3(1 ) (1 )(1 ) z z z 1 1z 2z 3z . 2.130. z 1 z= 1, 11ii+ ,R. 2.131. z1, z2, , zz1 + z2 + + z i 11z iz i+ + 22z iz i+ + + z iz i+ 12. 2.133. z = + i,, z =1. 21z . _____________________________________________________ 2.134. . - . I. . II. :z= 3 z= 3z. III. z, wC2z+ 2w= 0z = w = 0. IV. z1.z2 = 22z z1 , z2 . V. z i =1 iz+ . VI. 1z z = 2z z zCz1, z2C, . VII. z, , :29 z =2 6 z , . 2.135. . - . 50 1: I. z = 32(1 2 )(2 )(1 )ii i+ : . 52 . 52 . 54 . 52 . 5 52 II. z = 3 32 2(2 ) (2 ) 2(1 2 ) (1 ) 2i i ii i i+ + + : . 4 . 202 . 5. 5 22 . 3 III. z = 12 + yi, yR, z= 13, y: . 13 .13 . 12 . 5. 2 3IV. 1z= 3z2 = 3 + 4i z1 + z2: . 5 . 8 . 9. 12. 14 V. 102 z i + = 10: . . . . . 51_________________________________________ _________________ 3.1. z , z iw = 21zz +. ) w , z z = 1. ) , , 21zz + = 33. ) z1, z2 (), - = 2 21 1 2 22 21 1 2 2( 1) ( 1)3z z z zz z z z+ + + . [.ii)3212i iii)32i -32i] 3.2. z1 = + iz2 = 1122zz+ , , R , 0. - z2 z1R . ) z2 z1 = 1. ) z1 . ) 21z . > 0, z1 (z1 + 1 + i)20 (1z + 1 i)20 = 0. 3.3. ) z1 , z2 , 21z +22z= 21 2z z 1 2Re( ) z z = 0. ) f : [, ] R [, ] z = 2 + i.f () , w = f () + i.2. 0. 2w+ 2z= 2w z -f (x) = 0 [, ]. (1995) 3.4. f, g [, ] , (, )g(x) ( ) g x 0 , x(, ). z1 = f () + i.g() ,z2 = g() + i.f ()1 2z z += 1 2z z , (, ), ( )( )f g + ( )( )f g = 0. 3.5. ) z1, z2 : z1 + z2 = 4 + 4i 2z1 2z = 5 + 5i. ) z, w1z z 2 2w z 2i) z, w, z = w ii) z w . [.)3+i, 1+3i )i)z=w=2+2i ii)4 2 ] 3.6. z = 1 + i.xw = 1 + x + i , , > 0xR , 2w z + 2w z . = e. 52 1: 3.7. f : [, ] R [, ]f (x) 0 , x[, ] zRe(z) 0 , Im(z) 0 Re( ) z>Im( ) z . z + 1z = f ()z2 + 21z = f 2(), : )z= 1) f 2() < f 2() ) x3.f () + f () = 0 (1,1). 3.8. z 14z = 14Re( ) z + . ) z . ) oz w = z 1 . [.)y2=x )z= z=21 12i] 3.9. f (x) =z ixw ++i z ixw ,z, w*C xC. : ) zwI. ) f (x) . 3.10. f R f (x) = 2 22 2x z x zx z ++, z- z = + i, , R , 0. ) limx+f (x),limxf (x). ) f, 1 z +>1 z . ) f. [.)0,0 )..f(-2+2 )=22++222 .. f(2+2 )=-22++222 )[-22++222 ,22++222 ] 53 1. x0\ x0\ - . - - . - , -, , , , , , . - . , - . - - - . . - , , . - , . - , - . - [,] - Bolzano. , - - - , - . 1: 55 1 16 , - , EulerIntroductio in analysin infinito-rum . Euler1 Introductio in analysin infinitorum 1748. (function) f (x). , , . , , . . , , , . 1.1. : \. - ()f, x y. y fx f (x). , :f : \ x f (x).

1 LEONARD EULER (17071783). 18 . - - . , . . Introductio in analysin infinitorum, Institutiones calculidifferentialis, Institutiones calculi integralis. , - , . , -. . 561:- - x - , y, fx, . f Df. f x, - f f (). : f () = {y / y = f (x) x}. 1.2. : f xy . (x, y) y = f (x), (x, f (x)) , xA, f Cf. f xy - . (x,y) y = f (x), (x, f (x)) , xA, f Cf. xA y\ , - f . - f . - : o f - Cf. o f f () - Cf.

H f , xx, - f, (x, f (x)) (x, f (x)), xx. 1: 57 H f Cf xx , xx Cf . , : 1. . f (x) = .x + 1.x1 + + 1.x + 0 ,0, 1, , \` . f \ 0 - . f (x) = 2x3 5x2 + 0,2x +2,g(x) = 23 x4 + 9x3 35 x2 + 6 . -. f (x) = x +

f (x) = x2 , 0

f (x) = x3 , 0 2. . f (x) = ( )( )PxQx ,(x)Q(x) . 581:- - f \ Q(x). f (x) = 5 3 243 6 82 3x x xx x + + ,g(x) = 323 1, 25x xx x + . f (x) = x , 0

3. . - x. - - . F(x,y) = 0. f (x) =2 3 x,g(x) = 35 x 2 . y2 2x + 3 = 0(y + 2)2 x + 5 = 0. f (x) =x

4.T . f (x) = x, f (x) = x,f (x) = x f (x) = x.

5. f (x) = x , 0 < 1. 1: 59 6. f (x) = logx , 0 < 1. . 1.3. : Kronecker - , - . : I.f :\ f f (x) =( ) f x xA. II.f, g :\ max{f, g}(x) =max{f (x),g(x)} = ( ) ( ) ( ) ( )2f x gx f x gx + + , xA min{f, g}(x) = min{f (x),g(x)} = ( ) ( ) ( ) ( )2f x gx f x gx + , xA III. f (x) = [x] , [x] , x. IV. x.sign(x) = 1 00 01 0xxx x.sign(x) = ( 1) 00 01 0x xx xx x < = > = 00 0 0x xxx x =x . , x.sign(x) =x sign(x) = xx = xx, x*\ . V. Dirichletf (x) = 1 0 ( )xx _\ _. . 601:- - 1.4. : fg : xf (x) = g(x). f = g. 1.5. :(i) f, g , . xf (x) = g(x), f, g . (ii) : , . f (x) = xg(x) = x3 xA = {1,0,1} , . 1.6. : f , - xA: xA f ( x) = f (x). - - , (x, f (x)) - , (x, f (x)), - yy. -, yy. 1.7. : f , - xA: xA f ( x) = f (x). - , (x, f (x)) - , (x, f (x)), - . , . 1.8. : f , - *\ xA: x + A f (x + ) = f (x). 1: 61 - - . - -. . , , . 1: f (x) = c . 2: f (x) = xg(x) = x = 2, h(x) = x(x) = x - = . 3: Dirichlet , . , _ x + _ x_f (x + ) = 1 =f (x). x + ( ) \ _x ( ) \ _ f (x + ) = 0 = f (x). , -, f, g . f, g f + g o = o(f + g)(x) = f (x) + g(x) , x. f, g f g o = o(f g)(x) = f (x) g(x) , x. f, g f . g o = o(f . g)(x) = f (x) . g(x) , x. f, g fg o = {x / xx , g(x) 0} o fg (x) = ( )( )f xgx , x. f (x) = 11xx+g(x) = 3xx. x f (x)g(x)-. fg f, g . x -1 , 0 , 2 , 3 , 4f (1) = 0 , f (0) = 1 , f (2) = 3 , f (3) = 2 ,f (4) = 53. x g g(f (1)) = g(0) = 0 , g( f (0)) = g(1) = 14 , g(f (2)) = g(3) = ; , g(f (3)) = g(2)621:- -= 2 , g(f (4)) = g(53) = 54. x f, Df =\{1} f(x) g, Dg =\{3}. - f g, ( )fgx Df x D 1131xxx + 13 3 1xx x + 12 4xx 12xx . x\ {1,2} g(f (x)) = g(11xx+) =11113xxxx++ =111 3( 1)1xxx xx++ =11 3 3xx x++ +=14 2xx+. 1.9. : f, g , , f g, g D f, 1 = {xA / f (x)B} (g D f )(x) = g(f (x)). f : f (A)g : B g(B). g D f xA f (x), 1 = {xA / f (x)}. g D f 1, f () B . 1.10. :, f, g g D ff D g , . , g D ffD g. ( ). f, g, h h D (g D f) , - (h D g) D f : h D (g D f) = (h D g) D f.( ). (x) = x f :I D f = f D I = f. ( ). _________________________________________________________ 1: 631.11. f x, y \ : f ( x y) = (x+4).f (y1) + (x+y)2 . : f . x = 1y = 1: f ( 1 1) = (1+4).f(11) + (1+1)2 f (0) = 5.f (0) + 4 4.f (0) = 4 f (0) = 1(1). x = 1y = 1: f ( 1 1) = (1+4).f (11) + (1+1)2 f (0) = 3.f(0) + 0 2.f (0) = 0 f (0) = 0(2). (1)(2) f . , f . 1.12. f, g : f (x) = 2+1 1xx xx xx x x g(x) = x x xx x2+ = 1 0 1( 1)2 0 1( 2)( 1) 1 22xxx xxxxx xx xx+ < + > = ()1 0 11 0 121 1 2xx xxxx x x< > . fg ( )fAg =( ,1) (1,0) (0,+). 641:- - 1.13. f (x) =x 1g(x) =x29 . : i)g f D ii)f g Dii)f f D iv)g g D: fDf = [0,+ ). g, x2 9 0 x3x3. g Dg = ( ,3] [3,+ ). i)( )fgx Df x D 01 3 1 3xx x

( )02 4 xx x 016xx x16. gD f g fDD = [16,+). xg fD D (gD f )(x) = g( f (x)) = g( x 1) = 2( 1) 9 x =2 1 9 x x + =2 8 x x . ii)( )gfx Dgx D 23 39 0x xx

x3x3. f D g f gDD = ( ,3] [3,+). xf gD D (f D g)(x) = f (g(x)) = f (29 x ) = 29 x 1 =4 29 x 1. iii)( )ffx Df x D 01 0xx 01xx 01xx x1. f D f f fDD = [1,+). xf fD D (fD f )(x) = f ( f (x)) = f ( x 1) =1 x 1. iv)( )ggx Dgx D2 2( )3 39 3 9 3 x xx x

23 39 9x xx

23 318x xx

23 318x xx

3 33 2x xx

3 33 2 3 2x xx x

x3 2 x3 2 . gD g ggDD = ( ,3 2 ] [3 2 ,+ ). xggD D (gD g)(x) = g(g(x)) = g(29 x ) =2 2( 9) 9 x =29 9 x = 218 x . 1.14. f (x) = x xx x2, 15 6 , 1 < g(x) =x 1 + . : i)g f D ii)f g D: gDg = [1,+). x 33+ x29+ 00 + 1: 65i)()f (x) = x 2 , x( ,1) g(x) =1 x+,x[1,+). ( )fgx Df x D 12 1xx< 11xx< . ()f (x) = 5 6x , x[1,+) g(x) =1 x+,x[1,+) ( )fgx Df x D 15 6 1xx 16 6xx 166xx 11xx x = 1. x = 1(gD f )(1) = g( f (1)) = g(1) =1 1 += 0. g fDD = {1}(gD f )(1) = 0. ii)()f (x) = x 2 , x( ,1) g(x) =1 x+,x[1,+ ). ( )gfx Dgx D 11 1xx+ < 11 1xx + < 10xx< 1 x < 0. x[1,0) (f D g)(x) = f (g(x)) = f ( 1 x+ ) =1 x+ 2. ()f (x) = 5 6x , x[1,+) g(x) =1 x+,x[1,+). ( )gfx Dgx D 11 1xx+ 11 1xx + 10xx x0. x[0,+) (f D g)(x) = f (g(x)) = f ( 1 x+ ) = 5 6 1 x+ . (f D g)(x) = 1 2 , [ 1, 0)5 6 1 , [0, )x xx x+ + +. 1.15. f (x) = xx 1. i)h ( f h D )(x) = xx x2+2 ii)g ( g f D )(x) = xx x22 12 2 +1. : fDf =\{1}. i) h, ( f h D )(x) = 22xx x + f (h(x)) = 22xx x + ( )1 ( )hxhx = 22xx x + , h(x) 1 h(x).( x2 x + 2) = x.(1h(x)) ,h(x) 1 h(x).( x2 x + 2) = x x.h(x) ,h(x) 1 h(x).(x2 x + 2) + x.h(x) = x ,h(x) 1 (x2 x + 2 + x).h(x) = x ,h(x) 1 (x2 + 2).h(x) = x ,h(x) 1 h(x) = 22xx + ,h(x) 1. h(x) 1 22xx + 1 x2 + 2x x2 x + 20 , = 1 8 = 7 < 0. x\ (f D h)(x) = f (h(x)) = f (22xx +) = 22212xxxx++ = 222222xxx xx++ + = 22xx x + , h(x) = 22xx + . 661:- -ii) g, (gD f )(x) = 22 12 2 1xx x +

f (x) = y1 xx = y x = y yx x + yx = y (1+y)x = y y = 10.x = 1. y1x = 1 yy +(1)x 1 1 yy + 1 yy+1 01 , (1) y1. (gD f )(x) = 22 12 2 1xx x + g( f (x)) = 22 12 2 1xx x + g(y) = 22 112 2 11 1yyy yy y + + + + =222 112 211(1 )y yyy yyy + +++ =2 22112 2 (1 ) (1 )(1 )yyy y y yy+ + + ++ = 2 2 2( 1)(1 )2 2 2 1 2y yy y y y y + + + + = 2211yy +. g(x) = 2211xx + ,x\ . :(gD f )(x) = g( f (x)) = g(1 xx ) = ( )( )221111xxxx+

= 2 222 22(1 )(1 )(1 )(1 )x xxx xx + = 2 22 21 21 2x x xx x x + + + = 22 12 2 1xx x +. , g(x) = 2211xx + ,x\ . 1.16. f :[0,1] [1,+ ) : f 2(x) + x4 = 1 + 2x2.f(x) , x[0,1]. f. : x[0,1]: f 2(x) + x4 = 1 + 2x2.f (x) f 2(x) 2x2.f (x) + x4 = 1 [f (x) x2]2 = 1 f (x) x2 = 1f (x) x2 = 1 f (x) = x2+1f (x) = x2 1. x[0,1] f (x) = x2 + 10 x 1 1 x2+1 2 1 f (x) 2 x[0,1] f (x) = x2 10 x 1 1 x21 0 1 f (x) 0 f (x)[1,+ ) x[0,1]f (x) = x2 + 1. , f (x) = x2 + 1,x[0,1]. 1.17. f : \ \ : f (x+y) f (xy) + f (x) + f (y) = x2 + 4xy + y2 , x, y \ . i) f (0) = 0. ii) f. : 1: 67i)x = y = 0 : f (0+0) f (00) + f (0) + f (0) = 02 + 4.0.0 + 02 f (0) f (0) + f (0) + f (0) = 0 2f (0) = 0 f (0) = 0. ii) f . y = 0 : f (x+0) f (x0) + f (x) + f (0) = x2 + 4x.0 + 02 f (x) f (x) + f (x) + 0 = x2 + 0 + 0f (x) = x2. x, y\ : f (x + y) f (x y) + f (x) + f (y) = (x + y)2 (x y)2 + x2 + y2

= x2 + 2xy + y2 (x2 2xy + y2) + x2 + y2 = x2 + 2xy + y2 x2 + 2xy y2 + x2 + y2

= x2 + 4xy + y2. , f (x) = x2, x\ . 1.18. f :\ \ : f (x+y) f (xy) + f (x) + f (y) = 2x + 3y + 2 , x, y \ . : f . x = y = 0 : f (0+0) f (00) + f (0) + f (0) = 4.0 + 3.0 + 2 f (0) f (0) + f (0) + f (0) = 2 2f (0) = 2 f (0) = 1. y = 0 : f (x+0) f (x0) + f (x) + f (0) = 2x + 3.0 + 2 f (x) f (x) + f (x) + 1 = 2x + 2 f (x) = 2x + 1. x, y\ : f (x+y) f (xy) + f (x) + f (y) = 2(x+y) + 1 2(xy) 1 + 2x + 1 + 2y + 1 = 2x + 2y + 1 2x + 2y 1 + 2x + 1 + 2y + 1= 2x + 6y + 2. , f . 1.19. , \ , :f (x).f (y) = f (x) + f (y) + 3 , x, y \ . : f . y = x : f (x).f (x) = f (x) + f (x) + 3 f 2(x) 2f (x) 3 = 0 f (x) = 1f (x) = 3. x1 , x2\f (x1) = 1f (x2) = 3, x = x1 y = x2:f (x1).f (x2) = f (x1) + f (x2) + 3 1.3 = 1 + 3 + 3 3 = 5. f (x) = 1 , x\ f (x) = 3 , x\ . x, y\ : f (x).f (y) = f (x) + f (y) + 3 1.(1) = 1 1+ 3 1 = 1 . x, y\ : f (x).f (y) = f (x) + f (y) + 3 3.3 = 3 + 3 + 3 9 = 9 . , : f (x) = 1, x\ f (x) = 3,x\ . 1.20. f : (0,+ ) \ f (1) = 0 x, y >0 : f (xy)lny + f (x). : f . x = 1 : 681:- -f (1.y)lny + f (1) f (y) lny + 0 f (y) lny. f (x)lnx, x > 0 (1) y = 1x : f (x1x )ln1x + f (x) f (1)ln1 lnx + f (x) 0 lnx + f (x) lnxf (x) x >0(2) (1)(2)f (x) = lnx , x > 0. x, y(0,+)f (xy) = ln(xy) = lnx + lny = lny + f (x)lny + f (x). , f (x) = lnx,x > 0. 1.21. f : \ \ : .f (1+x) + .f (1x) = x2 + 5x 12, x\, (1,10)(3,6). : f . Cf

(1,10)(3,6) f (1) = 10f (3) = 6. x = 2x = 2 : .f (12) + .f (1+2) = (2)2 + 5(2) 12 .f (1) + .f (3) = 41012 106 = 18 5 + 3 = 9 .f (1+2) + .f (12) = 22 + 5.2 12 .f (3) + .f (1) = 4+1012 610 = 2 3+5 = 1. 5 3 93 5 1 + = + = .D = 5 33 5 = 259 = 16,D = 9 31 5 = 45+3 = 48, D = 5 93 1 = 527 = 32. = DD = 4816 = 3 = DD = 3216 = 2. :3.f (1+x) 2.f (1x) = x2+5x12(1). xx: 3.f (1x) 2.f (1+x) = (x)2+5(x)12 3.f (1x) 2.f (1+x) = x25x12(2) (1)(2): 223 3 (1 ) 2 (1 ) 5 122 2 (1 ) 3 (1 ) 5 12f x f x x xf x f x x x + = + + + = 229 (1 ) 6 (1 ) 3 15 364 (1 ) 6 (1 ) 2 10 24f x f x x xf x f x x x + = + + + = 5.f (1+x) = 5x2+5x60 f (1+x) = x2+x12. xx1: f (1+x1) = (x1)2+x112 f (x) = x22x+1+x112 f (x) = x2x12. x\: 3.f (1+x) 2.f (1x) = 3[(1+x)2 (1+x) 12] 2[(1x)2 (1x) 12]= 3(1+2x+x2) 3(1+x) 36 2(12x+x2) + 2(1x) + 24 = 3+6x+3x233x362+4x2x2+22x +24 = x2+5x12. , f (x) = x2x12 ,x\ . 1.22. f : \ \ ex+yf (x).f (y)f (x+y) , x, y\ . ) f (0) = 1. ) f (x) = 1( ) f x, x\ . ) f. 1: 69: )x = y = 0e0f (0).f (0)f (0+0) 1f 2(0)f (0). , f (0)1 f 2(0)f (0)(0) 0 f > f (0)1. f (0) = 1. )y = xe0f (x).f (x)f (0) 1f (x).f (x)1. , f (x).f (x) = 1 f (x) = 1( ) f x. ) f . y = 0 : exf (x).f (0)f (x) exf (x)f (x). f (x)ex x\(1) xx : f (x)ex 1( ) f x1xe exf (x). f (x)ex x\(2)(1)(2)f (x) = ex, x\ . , f (x) = ex, x\ . ___________________________________________________________________ 1.23. f (x) =3 22( 1) 5 1 1 x x xx x + + + . . [. =5] 1.24. f : \ \, : f (x) + x.f (y) = x + y , x, y\. 1.25. f \, : f (x + 1) + f (1 x) = 2x + 3 , x\. 1.26. )f (x) =2 x ++ 243xx )f (x) = ln(1 5 x ) )f (x) = 211xex )f (x) = 32xx x + l n( 1) xx+ [.)={-2} [2,3) (3,+ ) )=(4,5] )=[0,1) (1,+ ) )=(-1,0) (0,1) (1,+ )] 1.27. ) f (x) =2 1 x ) g(x) = 11xx+ ) h(x) = 4ln(4 ) xx x+) (x) = 13xx [.)=[1,5] )= \ -{x\ :x=2,] } )=(0, 2 ) )=(-3,-1] [1,3)] 701:- -1.28. )f (x) = 21 x + (x)) g(x) = 3 21 1xx [.)=[-1,-12) (-12,12) (12,1] )[-1,0) (0,1]] 1.29. ) f (x) = 28 9 24 x x + ) g(x) = ( )5 3log log5 2xx+ [. )=(- ,-11] [-5,-3] [3,+ ) )[4725,52)] 1.30. , , : i) f (x) = 262x xx + ii) g(x) = ln(3x 1) [. i)(3,0),(0,-3) ii)(8,0)] 1.31. : )f (x) = x2 3x + 2 g(x) = 6x )f (x) = x + 1 + 11 x + g(x) = x2 + x + 2[.)(3,2) )(0,2)] 1.32. , [f (x) x][f (x) + x] = 1 , x\ . 1.33. f (x) = x4 + x3 (3 + 1)x2 2x + 4 ,\. f (1,0). i) . ii) Cf . [.i)=2 ii)(-2,0), (-1,0), (12,0), (2,0) (0,4)] 1.34. f (x) = x2 x + + 3g(x) = x3 + (5 )x2 2x + 2 4. , , f, g - yy x = 2. [.(,)=(3,2) (,)=(-7,42)] 1.35. f, g :f 2(x) + g2(x) = 4f (x) + 6g(x) 13 , x\. [.f(x)=2, g(x)=3] 1.36. , , f (x) = x3 + (2 2)x2 2x g(x) = 4x2 + (2 + 4)x + 3 , x + 1 = 0. [.=2, =-1] 1.37. f (x) = 11 2x + 11 2x . ) f. ) f . 1: 71) . [.)=*\)f(x)=1] 1.38. f (x) = 2 3224 4x xx x +. i) f. ii) . iii) . 1.39. f (x) = ln(x 1) ln(1 1x). i) f. ii) . iii) . 1.40. . ) f (x) = 24 4 x x + 22 1 x x + + ) g(x) = 211xx +)h(x) = 2 31x xx )(x) = 222xx x )k(x) = 1 11 1x xx x+ + + )F(x) =2 x 1.41. . ) f (x) = 3 11 1x xxx >) g(x) =2 2x 1.42. f = g. f g, \ f (x) = g(x). i)f (x) = 223 10 84 4x xx x + g(x) = 4 32xx ii) f (x) = 11 x x + +g(x) = 1 x + xiii) f(x) = ln(11xx+) g(x) = ln(1 x) ln(1 + x) iv) f(x) = ln(21xx ) g(x) = 2lnx ln(1 x) 1.43. f, g f (12x) = 22 2 1xx x g(x) = 211xx +. f = g. 1.44. f (x) = 3 222x xx + g(x) = 3 2 222( 1)x x xx +. f = g. [.=1] 721:- -1.45. f (x) = 3 224 4 31x+ x xx x+ ++ +g(x) = x + 3. f = g. [. =1] 1.46. f, gf (23x) = 239 12 7xx x + g(x) = 3 24 2( ) ( 1) 46x x xx x + + + + ++ +. i) f. ii) , f = g. iii) , , h = fg 12 I, (x) = x. iv) h . [.i)f(x)=22-xx +3 ii)=5,=-3] 1.47. \ , f (x) = 3 22 123x+x xx ++ + 2x, . [.= \ -{-3,3}] 1.48. \ , - . 1.49. . f : \ , \, 0. g(x) = ( ) ( )2f x f x + h(x) = ( ) ( )2f x f x f = g + h. . f (x) = 231 x xx x -. 1.50. f : \ 0. f (0) = 0. 1.51. f : \ \ f (x + y) = f (x) f (y), x, y\ . f . 1.52. f , g : \, 0. : ) f, g , f + g ,f . g . ) f, g , f + g , f . g . ) , f . g . 1.53. f : \ \ x, y\ f (x + y) = f (x) + f (y). : )f (0) = 0. )f . 1: 73)f (x) = .f (x),x\ , (i) ` (ii) ] (iii) _. 1.54. f , f , . 1.55. f : \ \ , f (x) + f (x + 1) + f (x + 2) = 1, x\ . f . 1.56. f \ , (x2 + 1).f (x) x , x\ . [.f(x)=2xx +1] 1.57. f : \ \ : (x2 + ).f (x) + xx3 x\ > 0. i) f. ii) , -(2,1); [.i)f(x)=32x xx + ii)=2] 1.58. f (x) =1 ( 2, 0]1 (0, 2]x xx g(x) =22 [ 1, 0]2 1 (0, 2]x x xx x + + . f + g, . 1.59. f (x) =3 22 11 1x x xx x < g(x) =1 1 11 1x xx x+ >. - fg, . 1.60. f (x) = x2g(x) = lnx. , , g D ff D g. 1.61. f (x) = 2 11xx +g(x) = 1 xx. g D ff D g.g D f = f D g; 1.62. f (x) = 24 33x xx +g(x) = ln(x 1). ) f. ) g D f. ) g D f . ) x g D f xx; [.)2 0. 1.99. f :(0,+ ) \ ln(xy)f (x) + f (y)f (xy) ,x, y > 0. ) f (1) = 0. ) f (1x) = f (x) , x > 0. ) f. [.f(x)=lnx] 1.100. f (0) = 1 x, y\ f (x + y)ex.f (y). [.f(x)=ex] 1.101. f :(0,+ ) \ : f ( xe) lnxf (x) 1 , x > 0. [.f(x)=lnx+1] 781:- - 1.102. f (x) = 1 11x x x xx+ + +. ) f. ) f (x) =2 , xDf. [.)[1,+ )] 1.103. f :[1,+ ) \ f (x) = 2 22 1 x x + + 2 22 1 x x g(x) = 22 1 22 1 2xx x < . f = g. 1.104. f, g (f + g)(x).[(f + g)(x) 2] = 2[(f.g)(x) 1] , x, f = g. 1.105. f (x) = 2x2 1, x[1,1]g(x) = 4x3 3x, x[1,1]. g D f = f D g. 1.106. f \ x, y\ f (f (x) + y) = f (x + y) + 1. : i)f (x) = x + f (0) , x\. ii)f (x) = x + 1 , x\. 1.107. f : \ \ f (0) = 0 ( ) ( ) f x f y =x y , x, y\. f (x) = x , x\f (x) = x , x\. 1.108. f : \ \ f (x + y)f (x) + f (y), x, y\. ) f (x)0 , x\. )( ) ( ) f x f y f (x y), x, y\. 1.109. f :*+ \ \ f (x + y).f (x y) = f 2(x).f 2(y), x, y\. : )f (0) = 1. )f . )f (x) = 21( ) ( ) f x f x, x\ ` . )f (x) = 2x . 1.110. ) f , - . ) f 1, 2 f. T1 + T2, ] 0, . 1: 79______________________________________________________ 1.111. . - . I. f, (62,13)(32,12). II. f xx. III. f (x) = 2xx ,x 0 . IV. f f 3(x) = 8, x\. f . V. f f 2(x) = 9, x\. f . VI. (f.g)(x) = 0 , x\, -. VII. . VIII. f (x) = lnx2g(x)= 2.lnx . IX. f \ (x) = x , x\. (f D I )(x) = (I D f )(x), x\. X. f, g \ g D ff D g, . XI. f, g g D ff D g, f D g g D f. XII. Dg =\ , g fDD= Df . XIII. f, g g D f , g fDD Df. XIV. f : \ f D f. 1.112. . - . I. f (x) = 21 x x + + \. : . = 2 = 2 . < 2 > 2 . [2,2] . 4 4. (2,2) 801:- - 2 - Leibniz, . . C.L. SIEGEL , , , , , - , . , - , , -. . . 2.1. : f: , x1 , x2 x1 < x2 : f (x1) < f (x2). , x1 , x2 x1 < x2 : f (x1) > f (x2). 2.2. : f: , x1 , x2 x1 < x2 : f (x1)f (x2). , x1 , x2 x1 < x2 : f (x1)f (x2). 2:-812.3. : : f 1 21 2( ) ( ) f x f xx x > 0 x1, x2x1 x2. f 1 21 2( ) ( ) f x f xx x < 0 x1, x2x1 x2. f 1 21 2( ) ( ) f x f xx x0 x1, x2x1 x2. f 1 21 2( ) ( ) f x f xx x0 x1, x2x1 x2. 2.4. : : f(x) 0, g(x) 0 1f f, gf + g f . g fD g .f , >0 f, gf + g f . g fD g .f , >0 f , g f + g f . g fD g .f, >0f , g f + g f . g fD g .f, >0f, gf + g f . gfD g .f , ex e0 > ex 0 > x x > 0. fDf = (0,+). x1 , x2Dfx1 < x2 x1 > x2 1xe>2xe 1xe< 2xe1 1xe < 1 2xe ln(1 1xe) < ln(1 2xe) f (x1) < f (x2). f Df = (0,+). 2.13. f (x) = x2 + x 2 4x + 9 , . : x\ :f (x) = x2 +2 x 4x + 9 = x2 4x + 4 + 2 x + 5 841:- -= (x 2)2 + 2 x + 55. f (2) = (2 2)2 + 2 2 + 5 = 5f (x)f (2) , x\ . x = 2f f (2) = 5. 2.14. f : \ \ : x22x+4f (x)2x24x+5, x\ . f , . : x\ : x22x+4f (x)2x24x+5 x22x+1+3f (x)2x24x+2+3 (x1)2 + 3f (x)2(x22x+1) + 3 (x1)2 + 3f (x)2(x1)2 + 3. x = 1(11)2 + 3f (1)2(11)2 + 3 3f (1)3,f (1) = 3. x\ :f (x)(x1)2 + 30 + 3 = 3 , f (x)f (1). f f (1) = 3. 2.15. f , \ (f D f)(x) = x , x\ . f (x) = x ,x\ . : , , f () = . < .: < < f () f/ f () < f (f ()) f () < (f D f )()) < , . > . f (x) = x , x\ . 1-1. f1-1 x1 , x2f (x1) = f (x2) x1 = x2. x1 , x2 , - . f , 1-1. 1-1 , f () = f (). f (x1) = f (x2) x1 = x2(x1, x2) = 0 - x1, x2Df. 2.16. 1-1 i)f (x) = ln 2ln 1xx+ii)g(x) = x x 3x ,x(0,2] . : i)f :0ln 1 0xx> 0ln 1xx> 0ln lnxx e> 0 xx e> . 2:-85 fDf= (0,e) (e,+ ). x1 , x2Dff (x1) = f (x2) 11ln 2ln 1xx + = 22ln 2ln 1xx +(lnx1 + 2)(lnx2 1) = (lnx2 + 2)(lnx1 1)lnx1.lnx2 lnx1 + 2lnx2 2 = lnx1.lnx2 lnx2 + 2lnx1 2 3lnx1 = 3lnx2 lnx1 = lnx2 x1 = x2, f1-1. ii) gDg = (0,2]. x1, x2Dg x1 < x2 x1 < x2

x1 < x2 x1 > x2 x1 < x2 x1 < x2 13x > 23x 1 23 3( )x x +< x1 x1 13x < x2 x2 23x g(x1) < g(x2). g (0,2], 1-1 . 2.17. i)f (x) = x2 + 3x 4 ii)g(x) = 21xx 1-1. : i) fDf =\. 1 : 41: 41f (4) = f (1) = 0. f 1-1. 2 :x1 , x2\f (x1) = f (x2) 21x+ 3x1 4 = 22x+ 3x2 421x 22x + 3x1 3x2 = 0 (x1 x2)(x1 + x2) + 3(x1 x2) = 0 (x1 x2)(x1 + x2 + 3) = 0 x1 x2 = 0x1 + x2 + 3 = 0 x1 = x2x1 + x2 = 3. x1 , x2 , x1 + x2 = 3, f 1-1. ii) gDg =\{1}. 1 : x1 , x2 . g(x) = 121xx = 1x2 = x + 1 x2 + x 1 = 0. = 1 4.1.(1) = 1 + 4 = 5 > 0. x1 , x2Dgx1x2g(x1) = g(x2) = 1. g 1-1. 2 :x1 , x2Dgg(x1) = g(x2) 2111xx = 2221xx 21x (x21) = 22x (x11) 21x x2 21x= 22x x1 22x 21x 22x+ 22x x1 21x x2 = 0(x1 x2)(x1 + x2) x1x2(x1 x2) = 0 (x1 x2)(x1 + x2 x1x2) = 0x1 x2 = 0x1 + x2 x1x2 = 0 x1 = x2x1 + x2 = x1x2. x1, x2Dg,x1 + x2 = x1x2 = . t2 (x1 + x2)t + x1x2 = 0 t2 t + = 0. 861:- - > 0 ()2 4.1. > 02 4 > 0 < 0 > 4. g 1-1. 2.18. f (x) = 2x x. i) f 1-1. ii) :

2 34 -2 -32 2 3- -22 + 2 = (23+3). : i) f =\. x1, x2\ x1 < x2 x1> x2 12x > 22x x1 > x2 (+)

12x x1 > 22x x2 f (x1) > f (x2) f \. 1-1. ii) 2 34 2 32 2 322 + 2 = (2 3 + 3)

3 2(2 4 32 ) + 3 2( 2)2 + + 2 = 3 32 + 3

3 2(2 4 32 ) + 3 2( 2)2 + = (23 42 + 3) (3 2 + 2) 3 2(2 4 32 ) + (23 42 + 3) = 3 2( 2)2 + (3 2 + 2) f (23 42 + 3) = f (3 2 + 2) 1 1 f 23 42 + 3 = 3 2 + 2 3 32 + 3 2 = 0 : 1 , 2. 3 32 + 3 2 = 0 ( 2)(2 + 1) = 0 2 = 0 2 + 1 = 0 = 2. Cf,y=xCf,1fC . , f1-1, - f 1, f (x) = y, x,xDf . f f 1 y = x, f (x) = x f 1(x) = x. - ff 1 f f (x) = f 1(x) f (x) = x. 04 + 2 4+ 00+ 13322 222 1110 2:-872.19. f : [1,+ ) \ f (x) = x2 2x 3. i) f . ii) f. : i)x1, x2[1,+ )f (x1) = f (x2) 21x 2x1 3 = 22x 2x2 321x 2x1 22x + 2x2 = 0 (x1x2)(x1+x2) 2(x1x2) = 0 (x1x2)(x1 + x2 2) = 0 x1 x2 = 0x1 + x2 2 = 0 x1 = x2x1 + x2 = 2. 1211xx x1 + x22. x1 = x2 = 1. x1 = x2 , f1-1, . ii) f (x) = y, x, xDf= [1,+). f (x) = y x2 2x 3 = y x2 2x 3 y = 0. 0 (2)2 4.1.(3y)0 4 + 12 + 4y0 4y16 y164 y4. x1,2 = 2 4( 4)2y + = 2 2 42y + = ( )2 1 42y + = 1 4 y + . x = 1 + 4 y + 1. ,f 1(y) = 1 + 4 y +y4 f 1(x) = 1 + 4 x +x[4,+). 2.20. f (x) = xxx+x2 3 552 =5. f , . : x1, x2 \{5}f (x1) = f (x2) 112 35xx + = 222 35xx + (2x13)(x2+5) = (2x23)(x1+5)2x1.x2 + 10x1 3x2 15 = 2x1.x2 + 10x2 3x1 15 13x1 = 13x2 x1 = x2. x 5 , f (x) = 2 2 35xx + = 2 2x 3 = 2x + 10 3 = 10. f1-1, . f (x) = y, x, xDf =\. y 2f (x) = y 2 35xx + = y 2x 3 = xy + 5y 2x xy = 5y +3(2y)x = 5y + 3 x = 5 32yy+. ,f 1(y) = 5 3 225 =2yyyy+f 1(x) = 5 3 225 =2xxxx+. 881:- - 2.21. ff (x) = x xx x226+7 36 + 2. i) . ii) f . iii) f. : i)f 6x2 + x 20. = 12 4.6(2) = 1 + 48 = 49. x1,2 = 1 492 6 = 121 7 6 112 12 2 1 7121 7 8 212 12 3xx += = = = = =.Df =\{23,12}. :6x2 + x 2 = 6(x+23)(x12) = 3(x+23)2(x12) = (3x+2)(2x1). :3x2 + 7x + 6 = 3(x+23)(x3) = (3x+2)(3x). = 724.(3).6 = 49+72 = 121x1,2 = 7 1212 ( 3) = 7 116 127 11 4 26 6 37 11 1836 6xx += = = = = = . xDff (x) = (3 2)(3 )(3 2)(2 1)x xx x+ + = 32 1xx. ii)x1, x2Dff (x1) = f (x2) 1132 1xx = 2232 1xx (3x1)(2x21) = (3x2)(2x11) 6x2 3 2x1.x2 + x1 = 6x1 3 2x1.x2 + x2 5x2 = 5x1 x1 = x2. f1-1, . iii) f (x) = y, x, xDf . f (x) = y 32 1xx = y 2xy y = 3 x 2xy + x = y + 3 (2y+1)x = y + 3(1). o2y+1 = 0 2y = 1 y = 12 (1) 0x = 12+3 0x = 52 oy12 (1) x = 32 1yy++. xDf: x23 32 1yy++23 3y+94y2 7y11 y117. x12 32 1yy++12 2y+62y+1 61. ,f 1(y) = 32 1yy++y\ {12,117}. f 1(x) = 32 1xx++x\ {12,117}. 2.22. f (x) = 32 1 2 + 2 = 4. x1 = x2 , f1-1 f . ) f 1 y = x : f 1(x) = x f (x) = x 10 + 4x x2= x x2 3x 10 = 0 x = 2 ()x = 5 x = 5. (5,5). 2.25. f : (fD f)(x) = 6x + f (x), x\ . i) f 1-1. ii) f . iii) , f. iv) 1 -. : i)x1, x2\ f (x1) = f (x2). :f (x1) = f (x2) f (f (x1)) = f (f (x2)) (f D f )(x1) = (f D f )(x2) 6x1 + f (x1) = 6x2 + f (x2) 6x1 = 6x2 x1 = x2. f 1-1. ii)x = 0:(f D f )(0) = 6.0 + f (0) f (f (0)) = f (0)1 1 f f (0) = 0. f (0,0). iii) f 1-1 . f(x) = y x = f 1(y). (f D f )(x) = 6x + f (x) f (f (x)) = 6x + f (x) f (y) = 6.f 1(y) + y 6.f 1(y) = f (y) y f 1(y) = 16 [f (y) y]. f 1(x) = 16 [f (x) x]. iv)f (x) = x+ , 0. x\ : f (f (x)) = 6x + f (x) f (x+) = 6x + x+ (x+) + = (+6)x + 2x + + = (+6)x + 26 + = = + 26 00 = =

0

2 30 == = . f (x) = 2x f (x) = 3x. 2.26.. ) f = f () , f (x) = f 1(x) ,x f (x) = x. ( Cf1fC , , - 1 3). ) f (x) = x2 2x + 2x[1,+ ). i) f 1 . ii) x2 2x + 2 = 1 +1 x. f (x) = 1x , x > 0. f , f 1 f (x) = f 1(x). ; : . )x0 f (x) = f 1(x), f (x0) = f 1(x0). f (x0)x0 , of (x0) > x0 f 1(f (x0)) > f 1(x0) x0 > f (x0). of (x0) < x0 f 1(f (x0)) < f 1(x0) x0 < f (x0). 2:-91f (x0) = x0 x0 f (x) = x. , x0 f (x) = x, f (x0) = x0 f 1(f (x0)) = f 1(x0) x0 = f 1(x0). f (x0) = f 1(x0) x0

f (x) = f 1(x). ) i) x1, x2[1,+ )x1 x21 21 2( ) ( ) f x f xx x = 2 21 1 2 21 2( 2 2) ( 2 2) x x x xx x + + = 2 21 1 2 21 22 2 2 2 x x x xx x + + = 1 2 1 2 1 21 2( )( ) 2( ) x x x x x xx x + = 1 2 1 21 2( )( 2) x x x xx x + = x1 + x2 2 > 0 x1 1 ,x2 1x1 x2. f 1-1, . f (x) = y, x, x[1,+). f (x) = y x2 2x + 2 = y x2 2x + 2 y = 0 = (2)2 4.1.(2y) = 4 8 + 4y = 4y 4 = 4(y1). 0 4(y1)0 y10 y 1. x = 2 4( 1)2 1y = 2 2 12y = ( ) 2 1 12y 1 11 1x yx y= + = x = 1 1 y [1,+) x = 1 + 1 y [1,+). f 1(y) = 1 + 1 y y 1 ,f 1(x) = 1 + 1 x x[1,+ ) ii) x 10 x1. x2 2x + 2 = 1 +1 xf (x) = f 1(x) f (x) = x x2 2x + 2 = x x2 3x + 2 = 0 x = 1x = 2 . .0 < x1 < x2 11x > 21x f (x1) > f (x2). f (0,+). f1-1, . f (x) = y, x, x(0,+ ). f (x) = y 1x = y x = 1y (0,+ ). f 1(y) = 1yy > 0 ,f 1(x) = 1xx(0,+). f (x) = f 1(x) x(0,+). f , f (x) = f 1(x) f (x) = x, f . 2.27. f : \ (0,+ ) : f (x) + lnf (x) = x , x\i) f . ii) f. iii) f . : i)x1, x2\ f (x1) = f (x2) lnf (x1) = lnf (x2) f (x1) = f (x2)(+) f (x1) + lnf (x1) = f (x2) + lnf (x2) x1 = x2. f1-1, . 921:- -ii) f (x) = y, x, x\ . y > 0f (x) + lnf (x) = x y + lny = xx\ .f 1(y) = y + lnyy > 0 ,f 1(x) = x + lnxx(0,+). iii)x1, x2(0,+ )x1 < x2 lnx1 < lnx1 x1 < x2 (+) x1 + lnx1 < x2 + lnx2 f 1(x1) < f 1(x2) , f 1 f -. ff 1 f (x) = f 1(x) f (x) = x f 1(x) = x x + lnx = x lnx = 0 x = 1. f 1(1) = 1 + ln1 = 1 + 0 = 1. (1,1). ___________________________________________________________________ 2.28. f \ , . 2.29. f , . 2.30. ) f, g , f + g - . ) h(x) = logx + x (0,2). 2.31. ) f, g , f + g . ) (x) = 2x + x [0,]. 2.32. ) f, g xA f (x) > 0g(x) > 0 f .g . ) h(x) = x2.x (0,2). 2.33. ) f, g xA f (x) > 0g(x) > 0 f .g . ) (x) = xxe (0,4). 2.34. f, g \ , g D f: i) , f, g . ii) , f, g . iii) F(x) = 4(1 + ex)3 + 1G(x) = 3(5 2x3)5 9 . 2.35. f - . 2:-932.36. f (x) = (1) 1 x+ x + . f . [.01] 2.50. f (0,+ ), -g(x) = f (x) 2lnx , x > 0. ) g . ) (1,2) f, (i) f (x 1) = 2 + 2ln(x 1)(ii) ln(lnx)2 < f (lnx) 2[.)i)x=2 ii)1 0. f (x) = 0 . ) f (1) = 0 ) f (1x) = f (x) ) f1-1 ) f (x) + f (x2+1) = f (x+8) [.)x=2] 2.58. , , : ) f (x) = 3x 5 ,x 1 ) g(x) = 2 3 x ) h(x) = 2 11xx + 2.59. f f (x) = 223 10 83 13 12x xx x + +. i) . ii) f . iii) f . [.iii)f-1(x)=3x-2x-1 x\ -{1,25}] 2.60. , , : )f (x) = 1xxee + )g(x) = ln(ex 1) )h(x) = ln(x3 1) [.)f-1(x)= +3 xe 1x\ ] 2.61. f : [1, + ) \f (x) = 1 ex1. ) f . ) f 1(x) = 1 + ln(1 x) ,x( ,0]. ) ff 1 . [.)f:(1,0), (0,e-1e) f-1:(0,1), (e-1e,0)] 2.62. f (x) = (x 1)4 + 1x 1. ) f1-1. ) f 1. ) f 1 y = x. [.)(1,1), (2,2)] 2.63. f (x) = 6 5 336 3xxxx = . f -, . 2.64. f (x) = 13 2xx x +. i) f (x) = 12xx +. 961:- -ii) f . iii) f . [.iii)f-1(x)= ( )22x+1x-1 x (- ,-2) (-2,-12] (1,+ )] 2.65. f , g : \ \ (x) = x , x\ . g D f = f D g = , g = f 1 =f = g 1. 2.66. f (x) = x + , 0. i) f . ii) , f 1 = f . [.=1,=0 =-1,\ ] 2.67. f (x) = x x 02 , -f 1 = f . 2.68. f (x) = ln( 2 x +1). i) f . ii) f . iii) f 1 . [.f1(0)=2] 2.69. f (x) = x3 3x2 + 3x , . 2.70. f : \ \ , , (3,1)(1,2). i) f . ii) , :f (4 + f 1(2 + )) = 1. [.ii)-2,1] 2.71. f : ( ) f = \ \ \ , - (1,5)(6,4). i) f ; ii) f 1 \ . iii) f 1 (1 + f (x2 2x 4)) < 6. [.i) ii)x (-1,3)] 2.72. f : \ \ , [f (x)]5 + f (x) x = 0, -x\ . i) f1-1. ii) f 1. 2.73. f : \ \ , f ( \ ) =\ , (f D f )(x) + x = 0 , x\ . : ) f . ) f f 1 = f . ) f . ) f . 2:-972.74. f (x) = 2 1 1 22 2xx xx xx< >. i) f . ii) f,f 1 . 2.75. f (x) =1 x + 2x + 1. f 1(x) = 13 3 x + 23 x 1. 2.76. f : \ \ f ( \ ) = (1,1), . i) f , f 1 . ii) f ; 2.77. f :+ \ \ f (x.y) = f (x) + f (y) , x, y+\ . f : f 1().f 1() = f 1( + ) , , \ . 2.78. f : \ \ f (x + y) = f (x) + f (y) , x, y\ . f () = 0 = 0. : i)f . ii) f . iii) f 1(x + y) = f 1(x) + f 1(y), x, y\ . 2.79. , y = x. 2.80. ) f (x) = x5 + x + 1. f 1 f (x) = f 1(x). ) f (x) = x.ex , x 0 , g(x) = x.lnx, x 1h(x) = x.x , x[0,2] . (: 2.26) [.)x=-1 )(0,0),(e,e) (0,0),( 2, 2)] 2.81. f (x) = x x . ) f, . ) ff 1. [.)(-1,1),(0,0),(1,-1)] 2.82. f : \ \ : f (x2) f 2(x)14 , x\ . ) f (0). ) f 1-1. 981:- -2.83. , 2 1 + + 2 +3 = 2 1 + + 2 +3 = . 2.84. f (x) = 3 33 31 11 1x xx x+ + + . ) 1-1. ) . 2.85. f : \ \ f 1 = f. f (x) = x , x\ . 2.86. f (x) = lnx ex + x. ) f . ) f 1(x) = f (x). [.)x=e] 2.87. f \ . x\ (f D f )(x) = x + 2. : i) f . ii)f 1(x) = f (x) 2 , x\ . iii) f y = x. 2.88. ) f : g : A, f (A) = g() = . g D f = I g1-1g1 = f. ) f (x) = x2 x + 1, = [12,+ ) g(x) = 12 + 34x . i) f, g . ii) x2 x + 1 = 12 + 34x . [.)ii)x=1] 2.89. ) f : g : . :g D f = I f D g = I f , g. ( , ..) ) f (x) = 1 xx , x(1,1)g(x) = 1 xx + , x\, . 2.90. f : [0,1] [0,1]f (x) = 2( 1)( )1x x x+ ++. : i) = 0 ii)f 1-1. _____________________________________________________ 2.91. . - . 2:-99I.f : \ . 1 21 2( ) ( ) f x f xx x x1, x2x1 x2, f . II. . III. f (2,3), (1,2) . f . IV. f (x) = 5x ( ,0) (0,+ ). V. f (, ][, ) , (, ). VI. f , . VII. f : \ (0,+ ). f \ 1f \ . VIII. f , 1-1. IX. f1-1, . X. f1-1, g, h \ f (g(x)) = f (h(x)) x\, g = h. XI. 1-1, . XII. f . f 1 f. XIII. f (x) = 2x g(x) = log2x. XIV. 1-1fC , C ff 1 - . C , C y = x. XV. 1-1f y = x, f 1. XVI. ff 1 y = x. XVII. f (2,92,) f 1 (4,5 , 2) XVIII. f : f (A) . f (f 1(x)) = x , xA. 2.92. . - . I. f x0, x0. II. f x0, x0. III. f , 1-1. IV. f , . V. f1-1, . 1001:- -VI. f f 1. , , y = x. 2.93. . - . I. : . f (x) = x , x[2,2] . g(x) = x3 1 . h(x) = 23 ex . (x) = 221xx + . k(x) = ln(x 2),x > 2. 3:X0\ 101 3 x0\ 250. RICHARDCOURANT , . - , . . , - -x . , . -. x x0 - l. l . f (x) = 18 x3, x x0 = 2 - l. f (x) = 18 x3 f (x) = 3 22 21x xx= 22 ( 1)1x xx = 2x2 102 1:-- f (x) = 3 22 21x xx x x0 = 1 - 2. 1 - f. x 1, f - 1 1. f (x) = 2 11/ 2 1 1x xxx x f (x) = 2 1 1x xx x < > 3.1. : f (, x0) (x0, ). f x0 l, > 0 > 0, x(, x0) (x0, ) 0 < 0x x < , : ( ) f x l < . fx0, , 0limx x f (x). 0limx x f (x) = l, fx0

l. 3.2. : f x0 . 3.3. :()0limx x f (x) = l 0limx x [f (x) l] = 0 0limx x ( ) f x l = 0. ()0limx x f (x) = l 0limhf (x0+ h) = l. 3.4. : f (, x0) (x0, ), :

0limx x f (x) = l 0limx xf (x) = 0limx x+f (x) = l. 3.5. : : ()0limx x x = x0()0limx x c = c 3:X0\ 1033.6. : fx0 . x x0. f (x) = 11xx x0 = 1, (0,1) (1,2) . x(0,1)f (x) = ( 1)1xx = 1 1limxf (x) = 1limx(1) = 1, x(1,2)f (x) = 11xx = 11limx+f (x) =1limx+1 = 1. 1limxf (x). f (,1) (1, ) , < 1 > 1, (1,1) (1,5)(0,997 ,1) (1, 1,0002). . , , . - f (x) = xxx0 = 0 - (,0) (0,) f. - f (x) = 4xxx0 = 0 (4,0) (0,4) _________________________________________________________ 3.7. f (x) = (1x) x 0. : (,0) (0,) 1 1. l f x 0. x 0 . . 0limxf (x) = l. = 12, > 0 x 0 < 0 x < 0 < x < ( ) f x l < 12. ` x1 = 12

x2 = 212 + 0 0x(0,1) (1,2)1limxf (x) = 0. 4.1.2. : x0 . x0 - , . - , . 4:1071: f (x) = 5 27 8xx x + 1. , 1limx( x5 7x2 + 8) = 2 > 0, x(,1) (1, )x5 7x2 + 8 > 0. 2: f (x) = 31 5 12x xx+ +, 2limx(1 + 5x x3) = 1 < 0, 1 + 5x x3 < 0 x(,2) (2, ). x f (x) = 31 5 12x xx+ +

= 35 1 12x xx + = 35 22x xx + = 2( 2)( 2 1)2x x xx+ + = x2 2x 1. 4.2. : f, g x0 f (x)g(x) x0 , 0limx x f (x)0limx x g(x) . 4.2.1. : f (x) < g(x) x0 , , -, . -, 0limx x f (x)0limx x g(x). , -f (x) = x4g(x) = x2 , x*\ ,- x(1,0) (0,1)f (x) < g(x), 0limxf (x) = 0limxg(x) = 0. 4.3. : ( ) fgx0 1 , : 1. 0limx x (f (x) + g(x))=0limx x f (x) + 0limx x g(x) 2. 0limx x (.f (x))=.0limx x f (x), \3. 0limx x (f (x) . g(x))=0limx x f (x) . 0limx x g(x) 4. 0limx x ( )( )f xg x=00lim ( )lim ( )x xx xf xg x, 0limx x g(x) 0 5. 0limx x ( ) f x= 0lim ( )x xf x 6. 0limx x ( )f x= 0lim ( ) x xf x,f (x)0 x0 . 4.3.1. :(i) f, gx0, . 10522 242 1210 108 1:-- , 0limx(x.1x) = 0limxx.0limx 1x , 0limx 1x . (ii) , f, g, 1 . , f (x) = 21 x g(x) = 21 x 1limxf (x) = 01limxg(x) = 0. (f + g)(x) = f (x) + g(x) = 21 x + 21 x x{1,1} 1limx[f (x) + g(x)] (iii) , f, g , f, g . , f (x) = xxg(x) = xx, : 0limx[f (x) + g(x)] = 0limx0 = 0, 0limx[f (x).g(x)] = 0limx(1) = 10limx( )( )f xg x = 0limx(1) = 1, - f, g 0. (iv) 5. . , f (x) = xx0limx( ) f x= 0limxxx = 0limxxx = 0limx1 = 1, f0. 4.3.2. : 1.3. - . : 0limx x [f (x)] = 0limx x [ ( ) ( ) ... ( )] f x f x f x

=0 0 0lim ( ) lim ( ) ... lim ( )x x x x x x f x f x f x

= 0lim ( )x xf x , *` . , 0limx x x = 0x . 4.4. :P(x) 0limx x P(x) = P(x0), x0. : P(x) = .x + 1.x1 + + 1.x + 0x0\ . : 0limx x P(x) = 0limx x (.x + 1.x1 + + 1.x + 0) = 0limx x (.x) + 0limx x (1.x1) + + 0limx x (1.x) + 0limx x 0

= .0limx x x + 1.0limx x x1 + + 1.0limx x x + 0limx x 0

= .0x+ 1.10x + + 1.x0 + 0 = P(x0). 4.5. :Q(x) Q(x0) 0

0limx x ( )( )P xQ x = 00( )( )P xQ x. 4:109:Q(x0) 0 0limx x ( )( )P xQ x = 00lim ( )lim ( )x xx xP xQ x = 00( )( )P xQ x. - f , x- x0, - -hg. - , x0f hg. - hg, x x0, l. f l, x x0. , Sandwich . 4.6. : ( ) f, g , h. h(x)f (x)g(x) x0 0limx x h(x) = 0limx x g(x) = l , 0limx x f (x) = l. 4.6.1. :(i) 0limx(x.1x)*. `x 0: 1xx = x 1xx 1 = x x x.1x x . 0limx(x ) = 0limxx= 0 , 0limx(x.1x) = 0. (ii) 0limx(x.1x) = 0, *. ` 110 1:-- (iii) f (x) =x 1x. x 0: 1xx =x 1x x 1 =x x x 1x x . 0limx( x ) = 0limxx= 0 , 0limx( x 1x) = 0. 4.6.2. : : 0limx x f (x) = l 0limx x ( ) f x=l . l = 0, ,

0limx x f (x) = 0 0limx x ( ) f x= 0. . , x(, x0) (x0, ) ( ) f x f (x) ( ) f x . , 0limx x ( ( ) f x ) = 0limx x ( ) f x= 0 0limx x f (x) = 0. - . 4.7. : x\ x x . x = 0. 4.8. : (i) 0limx x x = x0(ii) 0limx x x = x0. 4.9. : (i) 0limxxx = 1(ii) 0limx 1 xx = 0. 4.10. : u = g(x)f (x).0limx x g(x) = u0 ,g(x)u0 x00limu u f (u) = l , 0limx x f (g(x)) = 0limu u f (u) = l. 0limx x f (g(x)) , - : g(x)u0 x0 . 4.10.1. : 0limxln( xx). f (x) = lnx u = g(x) = xx > 0 x(2,0) (0,2), u0 = 0limxxx = 1.4:1110limxln( xx) = 1limulnu = ln1 = 0. _________________________________________________________ . f x0, . x x0 (, x0) (x0, ) Df . x 0x+ (x0, ) Df . x 0x (, x0) Df . , . 00 , , -. 00 f (x) = ( )( )P xQ x , P(x) , Q(x), 0limx x f(x), f(x) xx0. = ( ) ( ) ( )1 2 1... + + +. f , - xx 0limxxx = 1. - , .. , - x0. 4.11. f (x) =+1 xx x22 3 . , , : i) 1limx f (x)ii)3limxf (x) . : i) f, 21 02 3 0xx x+ . -x 1 3+ x+1 0 ++ x22x3 + 0 0 + 112 1:-- . fDf = {1} [3,+ ). 1 , .. (2,1) (1,0) - Df . 1limxf (x). ii) 3 (3,) Df . , 3limxf (x) = 3limx+( 1 x + 22 3 x x ) = 3limx+1 x + 3limx+22 3 x x =3 1 + 23 2 3 3 =40= 2. 4.12. : )2limx323 10 43 12 x xx )3limx 327 1327 xx . : ) f (x) = 323 10 43 12x xx . f, 3x2 12 0 3x2 12x2123x2 4x 2x 2. Df =\{2,2} = ( ,2) (2,2) (2,+), 2limxf (x). xDf: f (x) = 323 10 43 12x xx = 22( 2)(3 6 2)3( 4)x x xx + +

= 2( 2)(3 6 2)3( 2)( 2)x x xx x + + + = 23 6 23( 2)x xx+ ++. 2limxf (x) = 2limx23 6 23( 2)x xx+ ++ = 23 2 6 2 23(2 2) + ++ = 2612 = 136. ) f (x) = 32727 x 13 x . f, 327 03 0xx 3273xx 3273xx 33xx x3. Df =\{2} = ( ,3) (3,+), 3limxf (x). xDf:f (x) = 3 3273 x 13 x = 227( 3)( 3 9) x x x + + 13 x

= 2227 ( 3 9)( 3)( 3 9)x xx x x + + + + = 2227 3 9( 3)( 3 9)x xx x x + + = 223 18( 3)( 3 9)x xx x x + + + =2( 3)( 6)( 3)( 3 9)x xx x x + + = 263 9xx x + +. 3limxf (x) = 3limx263 9xx x + + = 23 63 3 3 9 + + = 93 9 = 13. 301042 6124 3620 13183 318 160 4:1134.13. : ) 1lim xxx+5 28 3 ) 3lim xx x xx1 + 2 +7 6+3 ) 2limxx xx3 2344 2

: ) f (x) = 5 28 3xx+ . f, 5 08 08 3 0xxx+ 588 3xxx 588 9xxx 581xxx . Df = [5,1) (1,8], 1limxf (x). xDf:f (x) = 5 28 3xx+ = ( ) ( ) ( )( ) ( ) ( )5 2 5 2 8 38 3 8 3 5 2x x xx x x+ + + + + + + = ( ) ( )( ) ( )22225 2 8 38 3 5 2x xx x + + + + = ( )( )( 5 4) 8 3(8 9) 5 2x xx x+ + + + = ( )( )( 1) 8 3( 1) 5 2x xx x+ + + + + = 8 35 2xx ++ + 1limxf (x) = 1limx8 35 2xx + + + = 8 1 31 5 2+ + + + = 3 32 2++ = 64 = 32. ) f (x) = 1 2 7 63x x xx + + +. f, 1 02 7 06 03 0xxxx + + 12 763xxxx 17263xxxx 72x1x 3. Df = [72,3) (3,1], 3limxf (x). xDf:f (x) = 1 2 7 63x x xx + + + = 1 2 2 7 1 6 33x x xx + + ++ = 1 23xx + + 2 7 13xx+ + 6 33xx + = ( 1 2)( 1 2)( 3)( 1 2)x xx x ++ + + ( 2 7 1)( 2 7 1)( 3)( 2 7 1)x xx x+ + ++ + + ( 6 3)( 6 3)( 3)( 6 3)x xx x ++ + = ( )221 2( 3)( 1 2)xx x + + + ( )222 7 1( 3)( 2 7 1)xx x+ + + + ( )226 3( 3)( 6 3)xx x + + = = 1 4( 3)( 1 2)xx x+ + + 2 7 1( 3)( 2 7 1)xx x+ + + + 6 9( 3)( 6 3)xx x+ + = ( 3)( 3)( 1 2)xx x ++ + + 2( 3)( 3)( 2 7 1)xx x++ + + ( 3)( 3)( 6 3)xx x ++ + = 11 2 x + + 22 7 1 x + + + 16 3 x +. 3limxf (x) = 3limx(11 2 x + + 22 7 1 x + + + 16 3 x +) = 3limx11 2 x + + 3limx22 7 1 x + + + 3limx16 3 x + = 14 + 1 + 16 = 312 + 1212 + 212 = 1112. - 3, , - 00 114 1:-- ) f (x) = 3 2344 2x xx . f, 34 04 2 0xx 304 2xx 04 8xx 02xx . fDf = [0,2) (2,+) , 2limxf (x). xDf:f (x) = 3 2344 2x xx = 2 2 2 3 32 2 3 3 3( 2)( 2)[( 4 ) 4 2 2 ]( 4 2)[( 4 ) 4 2 2 ]x x x x xx x x ++ + + + +

= 2 2 3 33 3 3( 2)( 2)[( 4 ) 2 4 4]( 4 ) 2x x x x xx ++ + + = 2 2 3 3( 2)( 2)[( 4 ) 2 4 4]4 8x x x x xx ++ + +

= 2 2 3 3( 2)( 2)[( 4 ) 2 4 4]4( 2)x x x x xx ++ + + = 2 2 3 3( 2)[( 4 ) 2 4 4]4x x x x ++ + +. 2limxf (x) = 2limx2 2 3 3( 2)[( 4 ) 2 4 4]4x x x x ++ + + = 2 2 3 3(2 2 2)[( 4 2) 2 4 2 4]4+ + + + = 8 (4 4 4)4 + + = 24. 4.14.i) f (x) = x xx x223 + 95 +6 . , , 3limxf (x). ii) , , -1limxx x xx x323 4 + 1+ 2 . : i) f, x2 5x + 6 0 x 2x 3. Df =\{2,3} = ( ,2) (2,3) (3,+), 3limxf (x). x(2,3)f (x) = 223 95 6x xx x + + + = 2265 6x xx x + = ( 2)( 3)( 2)( 3)x xx x+ = 22xx+ x(3,4)f (x) = 223 95 6x xx x + + = 22125 6x xx x+ + = ( 3)( 4)( 2)( 3)x xx x + = 42xx+. , 3limxf (x) = 3limx22xx+ = 3 23 2+ = 5 3limx+f (x) = 3limx+42xx+ = 3 43 2+ = 7. 3limxf (x). ii)g(x) = 323 4 12x x xx x + . g, x2x2 0 x 1x 2. Dg = ( ,1) (1,2) (2,+), 1limxg(x). 1limx(x33x4) = (1)3 3(1) 4 = 1 + 3 4 = 2 < 0 , , x33x4 < 0 , x(,1) (1, ) g(x) = 323 4 12x x xx x + + + = 324 32x xx x + + = 2( 1)( 3)( 1)( 2)x x xx x+ + ++

= 232x xx + +. 11042 224 1120 10431 113 1130 4:1151limxg(x) = 1limx232x xx + + = 2( 1) ( 1) 31 2 + + = 1 1 33 + = 13 = 13. 4.15. f 1limxf (x) = 2f (x) 2,f (x) 2 1. o o: 1limxf x f xf x22 ( ) 5 ( ) 2( ) 2 . : g(x) = 22 ( ) 5 ( ) 2( ) 2f x f xf x 1. 1limxf (x) = 2 > 0 1 f (x) > 0 1limx(2f 2(x) 5f (x)) = 2.22 5.2 = 8 10 = 2 < 0 1 2f 2(x) 5f (x) < 0 , g(x) = 22 ( ) 5 ( ) 2( ) 2f x f xf x + = ( ( ) 2)(2 ( ) 1)( ) 2f x f xf x = (2f (x) 1) = 2f (x) + 1. 1limxg(x) = 1limx[2f (x) + 1] = 2.2 + 1 = 4 + 1 = 3. -. . . - , - . . 4.16. f : \ \ 1limx[2f (x) x2 + 24+1xx + 5] = 6, 1limxf (x). : g(x) = 2f (x) x2 + 241xx + + 5 ,x\ .1limxg(x) = 6. x\g(x) = 2f (x) x2 + 241xx + + 5g(x) + x2 241xx + 5 = 2f (x) f (x) = 12 g(x) + 22x 221xx + 52. 1limxf (x) = 1limx(12 g(x) + 22x 221xx + 52) = 121limxg(x) + 1limx22x 1limx221xx + 52 = 12 6 + 12 21 1+ 52 = 3 + 12 + 1 52 = 4 2 = 2. 4.17. f \ limx [8f (x) f 2(x)] = 16. limx f (x). : g(x) = 8f (x) f 2(x) ,x\ . limx g(x) = 16. 116 1:-- x\g(x) = 8f (x) f 2(x)f 2(x) 8f (x) = g(x) f 2(x) 8f (x) + 16 = 16 g(x) [f (x) 4]2 = 16 g(x). limx [f (x) 4]2 =limx [16 g(x)] = 16 limx g(x) = 16 16 = 0. limx [ ]2( ) 4 f x = 0limx ( ) 4 f x = 0limx [f (x) 4] = 0limx f (x) = 4. 4.18. f, g \ . : 2limxf xx2( )4 = 32limx[g(x)(x32x4)] = 5. 2limx[f (x).g(x)]. : h(x) = 2( )4f xx x\ {2,2} 2limxh(x) = 3. h(x) = 2( )4f xx f (x) = (x24).h(x) = (x2)(x+2).h(x). (x) = g(x).(x32x4)x\2limx (x) = 5. x32x4 = 0 (x2)(x2+2x+2) = 0 x2 = 0x2+2x+2 = 0x = 2. (x) = g(x).(x32x4) g(x) = 3( )2 4 xx x = 2( )( 2)( 2 2) xx x x + +x\ {2}. x(1,2) (2,3):f (x).g(x) = (x2)(x+2).h(x)2( )( 2)( 2 2) xx x x + + = 2( 2) ( )2 2x h(x) xx x+ + + 2limx[f (x).g(x)] = 2limx2( 2) ( )2 2x h(x) xx x+ + + = 2 2 222lim( 2) lim ( ) lim ( )lim( 2 2)x x xxx h x xx x + + + = 4 3 510 = 6. 4.19. x 1lim3 2+1x +x +x = 5 , , . : f (x) = 3 21x +x x++x 11limxf (x) = 5. x 1f (x) = 3 21x +x x++ x3+x2+ = (x+1).f (x). 1limx(x3+x2+) = 1limx[(x+1).f (x)] 1limx(x3+x2+) = 1limx(x+1).1limxf (x) (1)3 + .(1)2 + = (1+1).5 1 + + = 0 = 1 (1). f (x) = 3 21x +x x++ (1)=3 211x +x x+ + = 3 2( 1) ( 1)1x xx+ + + = 2( 1)( 1) ( 1)( 1)1x x x x xx+ + + + +

= 2( 1)[( 1) ( 1)]1x x x xx+ + + + = x2 x + 1 + (x1). 1limxf (x) = 5 1limx[x2 x + 1 + (x1)] = 5 (1)2 (1) +1+(11) = 5 1 + 1 + 1 2 = 5 2 = 2 = 1 (1) = 1 (1) = 2. 10242 244 1220 4:117 . - . . - . , - .. . 4.20. f : \ \ : x(x22f (x))f 2(x) x3 + x(x2f (x)) x\ . 0limxf(x). : x\ :x(x22f (x)f 2(x)x3 + x(x2f (x)) x3 2x.f (x)f 2(x)x3 + x2 2x.f (x) x3f 2(x) + 2x.f (x)x3 + x2 x3 + x2f 2(x) + 2x.f (x) + x2x3 + 2x2 x3 + x2[f (x) + x]2x3 + 2x2 . 0limx(x3 + x2) = 00limx(x3 + 2x2) = 0 0limx[f (x) + x]2 = 0 0limx[ ]2( ) f x x += 0 0limx( ) f x x += 0 0limx[f (x) + x] = 0 g(x) = f (x) + x ,x\ .f (x) = g(x) x0limxg(x) = 0. 0limxf (x) = 0limx[g(x) x] = 0limxg(x) 0limxx = 0 0 = 0. 4.21. f \

x 1lim f x f x f x3 2[ ( ) 4 ( )+6 ( ) 4] = 0. : x 1lim f (x) = 2. : 1limxf (x) = 2 1limx[f (x) 2] = 0. x\ :f 3(x) 4f 2 (x) + 6f (x) 4 = f 3(x) 2f 2(x) 2f 2(x) + 4f (x) + 2f (x) 4 = f 2(x).[f (x) 2] 2f (x).[f (x) 2] + 2[f (x) 2] = [f (x) 2].[f 2(x) 2f (x) +2]. f 2(x) 2f (x) + 2 = f 2(x) 2f (x) + 1 + 1 = [f (x) 1]2 + 11 ( ) 2 f x 2( ) 2 ( ) 2 ( ) 2 f x f x f x + = 3 2( ) 4 ( ) 6 ( ) 4 f x f x f x + . -f (x) 2, - - , - . 118 1:-- 3 2( ) 4 ( ) 6 ( ) 4 f x f x f x + f (x) 23 2( ) 4 ( ) 6 ( ) 4 f x f x f x + ,1limx3 2( ) 4 ( ) 6 ( ) 4 f x f x f x + =0= 0.,1limx[f (x) 2] = 0. -. , limx f (x). x, y . f (x+y) - 0limx x f (x) = 0limhf (x0+h). , f (x+y) , xx0 = h x = x0 + h. x x0h 0. 0limx x f (x) = 0limhf (x0+h). f (x.y) h = 0xx x = x0.h , x0 0, x x0 h 1. 0limx x f (x) = 1limhf (x0.h). 4.22. f : \ \ : f (x+y) = f (x) + f (y) + 2xy 1 x, y\ , f (1,1). 0limxf (x) = f (0) limx f (x). ) f (0) f (2). )0limxf (x) = f (0) 2limxf (x). : )x = y = 0f (0+0) = f (0) + f (0) + 2.0.0 1f (0) = f (0) + f (0) 1 f (0) = 1. Cf (1,1)f (1) = 1. x = y = 1f (1+1) = f (1) + f (1) + 2.1.1 1f (2) = 2f (1) + 2 1 f (2) = 2(1) + 1 f (2) = 2 + 1 = 1. )0limxf (x) = 0limhf (h) = f (0) = 1. , 2limxf (x) = 0limhf (2+h) = 0limh[f (2) + f (h) + 2.2.h 1] = f (2) + 0limhf (h) + 4.0limhh 1 = 1 + 1 + 4.0 1 = 1. 4.23. f : \ \ : f (x+2y) = f (x) 2f (y) + 2xy x, y\ . 0limxf (x) = f (0) limx f (x). 4:119: x = y = 0f (0+2.0) = f (0) 2f (0) + 2.0.0 f (0) = f (0) 2f (0) = 0 f (0) = 0. 0limxf (x) = 0. x = 2h x = + 2h. x h 0. limx f (x) = 0limhf (+2h) = 0limh[f () 2f (h) + 2h] = f () 20limhf (h) + 20limhh = f () 2.0 + 2.0 = f (). 4.24. f : \ \ : f (x.y) = (x).f (y) (y).f (x) x, y\ . 1limxf (x) = f (1) 0limx xf (x) = f (x0). : x = y = 1f (1.1) = .f (1) .f (1) f (1) = f (1) + f (1) f (1) = 0. 1limxf (x) = f (1) = 0. h = 0xx x = x0.h. x x0h 1. 0limx x f (x) = 1limhf (x0.h) = 1limh[(x0).f (h) (h).f (x0)]= (x0).1limhf (h) 1limh (h).f (x0) = (x0).0 .f (x0) = (1).f (x0) = f (x0). -. 0limx x f (g(x)) , u = g(x) , u0 = 0limx x g(x) g(x) u0 x0 , 0limx x f (g(x)) = 0limu u f (u). 4.25. :1limx341x xx. : f (x) = 341x xx. Hf 401 0xx 401xx 01xx . fDf = [0,1) (1,+ ), 1limxf (x). u = 12x, u6 =x,u4 = 3x, u3 = 4x . x 1u 1u = 12x 1 x0 = 1. 1limx341x xx = 1limu6 431u uu = 1limu4 22( 1)( 1)( 1)u uu u u + + = 1limu42( 1)( 1)( 1)( 1)u u uu u u + + +

= 1limu42( 1)1u uu u++ + = 421 (1 1)1 1 1++ + = 23. 120 1:-- 4.26. : 0limx2 241+ 2x xx . : f (x) = 2 2 241 x xx+ . fDf = *\ , - 0limxf (x). x*\:f (x) = 2 2 241 x xx+ = ( )( )( )2 2 2 2 2 24 2 2 21 1 1 x x x xx x x+ + ++ + = ( )( )22 2 2 24 2 2 21 1 x xx x x+ + + = ( )2 2 2 24 2 2 21 1 x xx x x+ + + = ( )2 2 2 24 2 2 2 1 x xx x x++ + = 2.2 24 xx2 2 211 x x+ + = 2.222xx 2 2 211 x x+ +. u = x2 , x 0u 0u = x2 0 x0 = 0. 0limx222xx = 0limu2uu = 12 = 10limx2 2 211 x x+ + = 11 0 1 + + = 12. 0limxf (x) = 2.0limx222xx 0limx2 2 211 x x + + = 2.112= 1. 4.27. f \ 0limx( ) f xx = 3. , , 0limx( ) f xx. : x(,0) (0, ) ( ) f xx = ( ) f x xx x . u = x. x 0u 0 = 0 , 0limx( )f xx = 0limu( ) f uu = 3. 0limx( ) f xx = 0limx( ) f x xx x = 0limx( )f xx.0limxxx = 3.1 = 3. ___________________________________________________________________ 4.28. f (x) =4 x + 23 25 4 x x + . : i) 4limxf (x)ii) 1limxf (x) 4.29. : ) 1limx232 3 51x xx +) 2limx222 5 25 6x xx x + +) 1limx327 61x xx 4:121) 1limx423 2 14 5x xx x + ) 12limx328 12 7 4xx x+ ) 12limx242 14 1xx [.)-73 )-3 )2 )109 )23 )12] 4.30. : ) 4limx21 147 12xx x + ) 2limx312 128xx ) 3limx 1limx2 32 31 1 x x [.)1 )-12 )12] 4.31. : ) 2limx3 17 3xx + )limx x x x , > 0 ) 2limx 25 2 32 3 4xx x + +

[.)-3 )3 2 )-23 )532] 4.32. :) 3limx41 3 73x xx+ +) 1limx3211xx ) 2limx32 x xx [.)532 )16 )-13] 4.33. : )2limx+24 22x xx + )1limx+21 11x xx + )2limx+24 22x xx + [.)3 ) 2)2] 4.34. f (x) =21 1 1 03 0x xxxx xxx+ . 0limxf (x). [.=-13] 4.35. f (x) = 32 115 12 2 11x xxxx xxx+ + . 1limxf (x) f yy 4. [.=3, =-1] 4.36. : ) 0limx2233x xx x+) 2limx24 3 12x xx + )2limx+222 24x x xx+ + +

122 1:-- )3limx+22 33x x x xx + [.)-1 )0 )-34 )0] 4.37. 1limx4 3 211x x xx = 6. 4.38. :) 1limx321x xx+ ) 1limx37 31x xx+ + [.)56 )-16] 4.39. : ) 1limx22 3 1x xx x +) 1limx25 43 2x xx x x x + + [.)12 )6] 4.40. : ) 0limx3xx ) 0limx( ) ( ) x x x+ + ) 0limx221 xx) 0limx1+ 1 x xx [.)13 )2 ) )1 )1] 4.41. : i)0limx22 1+xxii)0limx1+ x xx [.i)28 ii)12] 4.42. :) 0limx2 1 x xx+ ) 0limx21 2x x xx+ [.)1 )3] 4.43. : ) 0limxxx ) 0limxxx ) 0limxxx ) 0limx xx) 0limx xx 4.44. : ) 0limx 2 ... ( ) x x xx+ + + ) 0limx2 212x xx xx [.) (+1)2 )12] 4.45. : i) 0limx 2 ... ( ) x x xx+ + + = 36 ii) 0limx 2 ... ( ) x x xx = 120 [.i)8 ii)5] 4:123 4.46. f, g : )0limxf (x) = 0 , 0limxg(x) = 0 0limx( )( )f xg x = 3. )2limxf (x) = 0 ,2limxg(x) = 0 2limx( )( )f xg x =2 . 4.47. f \ , 1limx[x.f (x) x2 + 5x 3] = 8. , , 1limxf (x). [.7] 4.48. f, g 2. 2limx[5f (x) + 2g(x)] = 4 2limx [3g(x) 4f (x)] = 17, 2limxf (x)2limxg(x). [.-2, 3] 4.49. f \ , 5limx[2f (x) + f 2(x)] = 1. 5limxf (x). [.-1] 4.50. 0limx( ) f xx = l \ , 0limxf (x) = 0. 4.51. f, g (4, 2) (2,0), :2limx2( )4f xx = 18 2limx[g(x).(x3+8)] = 6. 2limx[f (x) g(x)] [.14] 4.52. f \ 1limx( ) 11f xx = 72. 1limx( ) 11x f xx . [.4] 4.53. f \ 0limx[( 3 1 x + 1).f (x)] = 6. 0limx[f (x).x]. [.4] 4.54. f \ 2limx( ) 12f xx = 72. 2