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(1 " . ¯ )1 ¯ – ¯ , ¯ ¯ ¯ ¯ . . ¯, www.GooL.co.il ¯ ¯, ¯ , , ¯ ¯ . . ¯ www.GooL.co.il/hedva1.html : ¯ ¯ , . www.GooL.co.il © ¯ – ¯ 2 ¯ 3 ............................................................................................ 1 5 ........................................................................................... 2 , 12 ................................................................. 3 , 9 .......................................................... 4 15 ................................................................................ 5 18 ................................................................. 6 20 .............................................................................................. 7 22 ................................................................................................... 9 25 ............................................................................................. ¯ 10 30 ..................................( 32 ........................................................ 11 33 ............................................................................ 12 44 ...............( 45 , " ¯8 ") , 13 ) 14 ............................................................................ 50 ................................................................................................' 15 52 ..................................................................................... 16 57 / ........................................................................................................ 18 61 ...................................................................................... 62 ...................................." ¯ " 17 ¯ 19 63 .......................................................... 20 64 ............................................................................ 21 65 ............................( 22 ) 23 66 ........................................ 69 ...................( ¯ ¯) 24 72 .....................................( ) 25 78 ) 26 )" 27 81 ............................................................................... 28 82 ................................................................................................. ...................................( 80 ................................................( www.GooL.co.il © ¯ – ¯ 3 1 – (1) : 4x + 1 (3 x2 + 1 y= y= y = x − 4 (6 y = x 3 − x 2 − 4 x + 1 (1 x2 (5 x2 − x − 2 y= 1 (9 1− | x | 1 (2 x −4 2 1 (4 x −x y= 3 y = 3 x 2 + x − 1 (8 y = x 2 + x − 2 (7 1 (11 log x y = ln ( x 2 + x − 2 ) (10 y = cot ( 4 x ) (15 y = tan (10 x ) (14 y = log x ( x + 4) (13 y = arccos( x + 1) (18 y = arcsin( x − 4) (17 y = arctan( x + 4) (16 y= y = ex 2 + x +1 . h( x ) = h ( h ( x )) (6 y = log x + (12 4 , g ( x) = x 2 , f ( x) = x − 4 : x : ¯ f ( f ( x )) (5 h( f ( x )) (4 (2) f ( g ( x )) (3 h ( g ( f (5))) (2 " (3) ¯ . f ( x) = x 2 − 4 ( x ≥ 0) (4 f ( g (1)) (1 ¯ . f ( x) = 3x − 2 (3 x−2 f ( x) = x +1 (2 x f ( x) = (4) : 1 (4 x y = sin x ⋅ cos x (8 y = 1 (3 y = x 4 + x10 (2 y = 4 x 3 (1 y = ln x + x 2 (7 y = 2 x (6 y = x 2 + sin 2 x (5 ¯ (5) y= : y = sin 2 x (4 ." y = tan x (3 3 * . y= " x −1 (1 3 |x| (4 x " ¯ © " " y = 3 | x + 1| (2 ," ¯ – y = 2sin x (1 ¯ y = x 2 + 2 | x − 1| (3 " www.GooL.co.il y = 5 + 3sin(4 x + 1) (2 ¯ " (6) y =| x − 2 | (1 * 4 1 – (1) x ≠ 2, −1 (5 x < −2 x > 1 (10 x ≠ π 4 ⋅ k (15 x ≠ 0,1, −1 (4 −1 < x < 1 (9 π x ≠ 20 + π 10 k (14 x ≠ ±2 (2 x ≥ 1(7 x ¯ (12 3 < x < 5 (17 x ¯ (3 x ¯ (8 0 < x ≠ 1 (13 −2 < x < 0 (18 x ¯ (1 x ≥ 4 (6 0 < x ≠ 1(11 x ¯ (16 x ≤ −2 (2) x (6 4 (4 x−4 x − 8 (5 x 2 − 4 (3 4 (2 −3 (1 (3) y ≠ 3 , f −1 ( x) = 2x − 2 (3 x−3 y ≠ 1 , f −1 ( x ) = 1 (2 x −1 y ¯ , f −1 ( x) = 3 x + 1 (1 y ≥ −4 , f −1 ( x ) = x + 4 (4 (4) .6,7 – ¯ 1,4 – 2,3,5,8 – (5) π (4 3π (3 π 2 (2 2π (1 (6) 3 x + 3 x ≥ −1 y= (2 −3 x − 3 x < −1 x − 2 x ≥ 2 y= (1 2 − x x < 2 1 x > 0 y= (4 −1 x < 0 x2 + 2 x − 2 x ≥ 1 y= 2 (3 x − 2x + 2 x < 1 www.GooL.co.il © ¯ – ¯ 5 2 – :( x →100 x →1 xn − x (4 x −1 lim / x →1 x7 − x (3 x −1 x2 + x + 2 − 2 3− x + 6 (4 lim (3 2 x →3 x −1 2x − 6 x →1 2 x 2 − 50 (2 x →−5 2 x 2 + 3 x − 35 lim lim x →3 x →4 (2) ) lim x →3 x2 − x − 6 (1 x2 − 9 ¯) :( lim lim x 2 + x + 1 (1 lim x →1 :( lim x +1 (2 x →10 x + 2 lim x + 3 (3 + lim 20 (4 (1) ) x −3 (2 x +1 − 2 (3) lim x →1 1− x (1 1− x 1− 3 x 2 − 3x + 1 2x +1 − x + 5 (7 lim (6 lim (5 x →1 1 − x x →1 1 − 2 x − 1 x →4 x−4 lim :( lim sin x = 1 x lim x →0 x cos x (3 sin 2 x sin(3 x) (2 x →0 sin(4 x ) lim 1 + sin x − cos x (6 x lim x →0 (4) ) x →0 tan x − sin x (5 x →0 x3 lim sin(3 x) (1 x →0 4x lim 1 − cos x (4 x →0 x2 lim 1 − cos x 3sin x − sin 3 x 1 − cos(1 − cos x) (9 lim (8 lim (7 2 3 x →0 x →0 x →0 x x x4 lim :( x2 −1 (4 x → 2 ( x − 2)( x − 5) − x2 (3 x → 2 (2 − x ) 2 lim x →0 ( x − 1)2 (2 x →2 x − 2 lim 1 lim e x (8 lim 1 lim− − ln(2 − x) (6 x →2 2 1 1 lim (11 lim (10 1 1 x →0 x → 0− 1+ 2x 1+ 2x ( ) lim (ln x) 2 + 2 ln x − 3 (7 + x →0 lim ln x ⋅ cot x (12 x →0 + www.GooL.co.il © (5) ) ¯ – ¯ lim x →0 x2 + 4 (1 x ln x (5 x →0 x 1 lim (9 1 x → 0+ 1+ 2x lim + 6 :( 4 x2 + 2 (3 x →∞ x 2 + 1000 x x →−∞ x 2 − 5x + 6 x lim − (6 x →∞ 2 2 x + 10 x4 + 2x2 + 6 (5 x →∞ 3 x 5 + 10 x 9 x6 − 5x (9 x →− ∞ x 3 − 2 x 2 + 1 x →−∞ 16 x + 4 x +1 (12 x →∞ 2 4 x + 2 + 2 x + 3 x →∞ 4 ⋅ 9 x + 3x +1 (15 x →− ∞ 810.5 x + 3 x + 3 lim x →∞ lim x →−∞ ( ( ) lim 5 x →∞ ) x 2 + x + 1 + x (24 lim x →∞ lim x →∞ x →∞ ) ( 16 x + 4 x +1 (13 x →−∞ 2 4 x + 2 + 2 x + 3 ( lim x →∞ ) x 2 + x + 1 − x (23 lim x →∞ ) lim( ( ) x 2 + kx − x (22 x 4 + x 2 + 1 − x 2 ) (25 x →∞ 1 = lim (1 + x ) x = e (7) ) x →0 x x x+2 lim (3 x →∞ x 1 lim 1 + 2 (2 x →∞ x x 2x + 3 lim (5 x →∞ 2 x − 3 1 x lim (1 + sin x ) (6 x→0 10 x x x2 + 4 x + 1 1 lim 1 + tan (9 lim 2 x →∞ x →∞ x + 2 x + 2 x © 4 x2 + 2 (16 x 2 + 1000 x x4 + 2 x2 + 6 lim sin (19 5 x →−∞ 3 x + 10 x ax + 1 (20 bx + 2 x 2 + ax − x 2 + bx (26 www.GooL.co.il x2 + 1 (7 x lim 3x3 − 5 x − 1 lim ln 3 (17 2 x →∞ x − 2x +1 (18 1x x x →∞ lim x 2 + 5 x − x (21 :( lim (1 + lim 4 ⋅ 9 x + 3x +1 (14 x →∞ 810.5 x + 3 x + 3 lim x →∞ x4 + 2x2 + 6 (4 x →− ∞ 3 x 3 + 10 x 34 x + 2 − 3x − 3 x + 2 x 2 + 6 + 27 x 6 (11 lim (10 x →∞ 4 x + 1 − 5x − 1 3 x3 + 10 x + 4 x 4 lim lim (1 lim x2 + 1 (8 x lim ln x x →∞ lim lim lim e lim ( e − x ) lim arctan x + e x (2 lim x4 + 2 x2 +6 3 x4 +10 x (6) x) ¯ – ¯ x 1 lim 1 + (1 x →∞ 2x 1 lim 1 − 2 x →∞ x x2 + x + 1 (8 lim 2 x →∞ x + x + 4 x 2 −1 (4 4 x2 (7 7 :(' lim x →∞ ¯ 3 x + sin x (3 4 x + cos x ( lim x →∞ x →0 cos(2 x + 1) (2 x 1 lim x ⋅ sin (5 x →0 x ) lim x 2 ⋅ cos ln x 2 (6 (8) ") lim x →∞ sin x (1 x 3 x 2 + x + sin 2 x (4 x →∞ x 2 + cos 3 x lim 1 3 x + arctan(2 x − 3) (7 [ x ] (9 lim x 2 x + 3x + 4 x (8 lim x →∞ x x →∞ x →∞ 4 x + arctan( x − ln x ) lim lim x →0 :( ) x2 + x − 2 ( a = 1) f ( x) = x − 1 x −1 x −1 1 [ x ] (10 x (9) lim f ( x ) x→a x >1 ( a = 0) (2 x <1 x>0 (1 x<0 f ( x) = ( a = 0) f ( x) = |x| (3 x ( a = −∞ ) (a = ∞) | x| (4 x sin 4 x f ( x) = x 1 4 + ex f ( x) = | x| (5 x ! . (8 ) ¯ .4 " 3 ,2 ¯ www.GooL.co.il © ¯ – ¯ 8 2 – (1) 40 (4 2 (3 11 12 6 (3 10 8.5 (2 21 (1 (2) n − 1 (4 1 3 (7 3 4 (7 1 2 (6 1 6 (6 1 2 (5 3 8 (5 1 2 (4 −1 12 (4 1 2 (2 5 6 (3 4 (2 1 2 (3 3 4 3 4 (1 (3) (1 (4) 1 (9 4 (8 1 8 (2 (1 (5) φ (8 0 (9 ∞ (7 ∞ (6 −∞ (5 φ (4 φ (2 φ (11 −∞ (3 −∞ (12 −3 (9 −1 (8 1 (7 e (18 ln 3 (17 2 (16 1 3 a −b 2 −5 (6 1 9 (15 0 (5 4 (14 −∞ (4 π − 2 (2 4 (3 1− 3 2− 5 0 (13 0.25 (12 (26 1/ 2 (25 −1/ 2 (24 1/ 2 (23 k / 2 (22 φ (1 1 (10 (6) 0 (1 (11 1.5 (10 2.5 (21 (**) (20 0 (19 (7) 1 2 e (9 e30 (8 e −12 (7 e (6 e3 (5 e −1 (4 e 2 (3 1 (2 e (1 (8) 1 (9 4 (8 0.75 (7 0 (6 0 (5 3 (4 0.75 (3 0 (2 0 (1 0 (10 (9) − 1 (5 1 (4 : φ (3 φ (2 20 6 4 (1 (**) a ⇐ b ≠ 0 (I b lim = ∞ ⇐ a > 0, b = 0 (II lim = 5 lim = −∞ ⇐ a < 0, b = 0 (III www.GooL.co.il © ¯ – ¯ 9 3 – * : " (1) " .( sin x x f ( x) = 2 1 1 + e x 4 x>0 ) sin 4 x f ( x) = x 1 x 4 + e x = 0 (2 x<0 x x ≥ 1 f ( x) = 2 (4 x x < 1 1 x f ( x) = x − 2 1 x−2 3 x>0 (1 x<0 x +1 x ≤ 2 f ( x) = (3 5 − x x > 2 sin x 2 x f ( x) = 2 − x x − 3 x ≤1 1 < x < 2 (6 x=2 x>2 x<0 0 ≤ x <1 1≤ x < 2 (5 x≥2 * . .x=0 :x ¯ 1 , (2) k x2 + 2 x − 3 x ≠1 f ( x) = x − 1 (2 k x =1 kx 2 + x − 2 f ( x) = 5kx − 6 2 x − k f ( x) = 2 x x x2 + 5 − 3 x≠2 f ( x) = x − 2 (3 k x=2 .(8 ) x≤0 x>0 (4 ¯ www.GooL.co.il © ¯ ¯ – ¯ 4 x≤2 x>2 : (1 10 (3) a b : 3 2 x < −1 a x + x 2 − 1 ≤ x ≤ 1 (2 f ( x) = bx + x − 1 4 x − 1 + a − a x >1 a ( x − 1) 1 1 1 + e1− x f ( x) = ax 2 + b 1 ( x − 1) x − 2 ax + b sin x f ( x) = 2x a cos x 1 ≤ x ≤ 2 (4 x>2 ) ¯ ¯ . 0 < x < π (1 x≥π 1 1− x x ( x − 1) ln( x + 1) + b f ( x) = 1 x a 2 1 − 2 x 2 +4 x <1 .(8 x≤0 ¯ 4 x >1 0 ≤ x ≤1 (3 x<0 3 : (1) (4) ¯ : ¯ (5) . ¯ .1 . .2 . ¯ .3 . . .4 ¯ www.GooL.co.il © ? f +g . ¯ – ¯ g f (6) 11 ( ) (7) . : ¯ x 2 = − ln x (2 x − 0.25sin x = 7 (3 x3 + 4 x − 1 = 0 (1 x 3 + bx 2 + cx + d = 0 . ¯ : . f (0) = 1, f (1) = 2 : x 2 = 10 − e x − 5 x = 0 (1 x¯ . . 1 = 0 (2 x (11) f ¯ f ( x ) + sin x = 4 x 1 x (12) ¯ . f ( x) = x 2 + 1 x −1 (13) . f (0), f (2) x2 + . (0, 2) , x = 0,1 :' . a = 2,b = 1 (5 . . (2 . . . – (5 . . k = 4 (2 . k = 1 (1 (2) . 1 =0 x −1 3 (4 . .x=2 ' (3 . © (6 . x = 2 . k = −1 (4 . k = 2 (3 3 . a = e / 3 , b = −e / 3 (4 . a = −2e−1 , b = e −1 (3 . f (0) = −1 , f (2) = 5 . www.GooL.co.il 1 (1 (3) 2 (1 (1) (2 . . x =1 ' a = 1, b = 2 (2 . a = 0, b = (1 (4) (9) (10) ¯ 4 x3 + 5 x − (8) (13) . [ 0.1,1] (12) ¯ – ¯ . (6 . 12 4 . ( ) , – , . (1) ¯ . ¯ .3. . . ¯ ¯ . . x2 − 5x f ( x) = 3 x − 14 x≥2 x<2 ¯ x2 − 4 x f ( x) = 3 x − 14 (2 2 x + 8x f ( x) = 3 x + 12 ln(1 + 2 x) − 0.5 < x < 0 f ( x) = 2 (4 x≥0 x + 2x f ( x) = 3 x 2 + x | x | +1 (6 1 2 x sin f ( x) = x 0 x>0 x≥2 x<2 x≥2 x<2 (1 (3 f ( x) = 2 + 4 | x − 1| (5 1 x sin f ( x) = x 0 (8 x≤0 x>0 (7 x≤0 (2) 3 x + 1 x ≥ −1 . f ( x) = 1 + a x < −1 x . x = −1 . a a . . x = −1 (3) 3 x −1 . f ( x) = 2 −( x + 1) ? . x =1 x≥0 x<0 . . www.GooL.co.il © ¯ – ¯ 13 . b . e x f ( x) = ax + b (4) ¯ a , 0 < x ≤1 x >1 ¯ ln 3 x f ( x) = ax + b ( 0< x≤e x>e ( (5) : 1 (2 x +1 f ( x) = x 2 + 4 x + 1 (1 f ( x) = ln x (5 f ( x) = sin 4 x (3 f ( x) = e x (4 f ( x) = f ( x) = x + 10 (6 . ¯ * : ¯ (6) f '(0) f ( x) = x( x − 1)( x − 2)( x − 3) ( x − 44) (1 f ( x) = 2 x(| x | +1) 1 + x + x 2 (2 f ( x) = sin x ( x − 4)10 (1 + tan x)4 cos( x + sin x) (3 ( x − 1) 2 ( x − 10)10 ( z(0) = 1, lim z( x) = 4 : ) x →0 f ( x) = x ⋅ z ( x) (4 f ( x) = | x 4 − x3 + sin(10 x) − 1| (5 . x=0 (4 :( . . x0 . x0 ¯ f = g +h (7) (1) , ¯ x0 ) g , x0 ¯ (8) h . f = g +h g , x0 h . f = g ⋅h . x0 x0 x0 g , x0 h . h . . x0 f = g ⋅h www.GooL.co.il © x0 g ¯ – ¯ , x0 14 4 – (1) x>2 2 x − 5 f '( x ) = 2 (2 x<2 3 x 2 − 0.5 < x < 0 f '( x) = 1 + 2 x (4 2 x + 2 x≥0 8 x f '( x) = 4 x 1 1 2 x sin − cos f '( x) = x x 0 . . x≠2 x≥0 (6 x<0 x>0 (8 x≤0 2 x − 4 f '( x) = 2 3 x x>2 x<2 2 x + 8 f '( x) = 2 3 x x≥2 x<2 4 f '( x) = −4 1 11 sin − cos f '( x) = x x x 0 , (1 (3 x >1 (5 x <1 x>0 (7 x<0 ! 1 , . . (2 a = 1 (1 (2) . (2 (1 (3) . a = e , b = 0 ( . a = 3 / e , b = −2 ( (4) (5) f '( x) = 4 cos ( 4 x ) (3 f '( x) = −1 (2 ( x + 1) 2 1 f '( x) = (5 x f '( x ) = 1 (6 2 x + 10 −10 (5 10 ( 0.4) 4 (4 (3 © ¯ – ¯ f '( x) = e x (4 2 (2 . www.GooL.co.il f '( x) = 2 x + 4 (1 44! (1 (6) (7) 15 5 :( f ( x) = – 2 x2 (3 ( x + 1) 2 f ( x) = x 2 − 5x + 6 (2 2 x + 10 3 x +1 f ( x) = (6 x −1 f ( x) = f ( x) = x ⋅ ln x (9 f ( x) = ln 2 x + 2 ln x − 3 (12 f ( x) = ( x + 2) ⋅ e 1 x x3 (5 ( x + 1) 2 f ( x) = f ( x) = ln ln x (8 x 1 (11 2−x f ( x) = e (15 (1) ) 27 29 1 x (14 f ( x) = x2 + 2 x + 4 (1 2x f ( x) = x3 (4 x2 − 4 f ( x) = ln x (7 x f ( x) = x 2 ⋅ ln x (10 f ( x) = ln 2 x + 1 (13 ln 2 x 2 f ( x) = 3 x 2 − 1 (18 f ( x ) = 3 x 2 (17 f ( x) = x ⋅ e−2 x (16 f ( x) = cos( x 4 ) (21 f ( x) = sin( x 3 ) (20 f ( x) = 3 x 2 (1 − x) (19 f ( x) = ln(cos x 2 ) (24 f ( x) = tan( x 2 ) (23 f ( x) = sin 3 x (22 sin x f ( x) = ( x + 1) (27 f ( x) = arctan( x 2 ) (26 f ( x) = arcsin(2 x + 3) (25 f ( x) = ( cos x ) www.GooL.co.il © ln x (29 ¯ – ¯ x f ( x) = ( sin x ) (28 16 5 – (2 (1 2 x + 20 x − 62 448 2x − 8 4 f '( x) = , f ''( x) = f '( x) = , f ''( x) = 3 2 3 2 (2 x + 10) (2 x + 10) 4x x (4 (3 2 2 2 4x 4(1 − 2 x) x ( x − 12) 4 x ⋅ (2 x + 24) f '( x) = , f ''( x) = f '( x) = , f ''( x) = 3 2 2 2 3 ( x + 1) ( x + 1) 4 ( x − 4) ( x − 4) (6 (5 2 2 6( x + 1) ( x + 1)( x + 3) x ( x + 3) 6x f '( x) = − , f ''( x) = 12 f '( x ) = , f ''( x) = 3 4 5 ( x − 1) ( x − 1) ( x + 1) ( x + 1) 4 (8 (7 2 − ln x 3ln x − 8 1 − ln x 2 ln x − 3 f '( x) = , f ''( x) = f '( x) = , f ''( x) = 2 x1.5 4 x 2.5 x2 x3 (10 (9 f '( x) = x(2 ln x + 1), f ''( x) = 2 ln x + 3 1 f '( x) = ln x + 1, f ''( x) = x (12 (11 2 −2 ln x 1 1 f '( x) = (ln x + 1), f ''( x) = f '( x) = , f ''( x) = 2 2(2 − x) x x (4 − 2 x) 2 (13 4 5 4 2 ( ln x) − 1 2 (ln x) − (ln x) − (ln x) − 3 f '( x) = , f ''( x) = − 2 3 x (ln x) x (ln x) 4 2 2 (15 x −x−2 5x + 2 f '( x) = e , f ''( x) = e 4 2 x x 1 x 1 x 2 (14 1 1 1 + 2x f '( x) = e ⋅ − 2 , f ''( x) = e x 4 x x (16 −2 x2 2 −2 x2 f '( x) = e (1 − 4 x ), f ''( x) = −4 xe (3 − 4 x 2 ) (17 2 2 f '( x) = 3 , f ''( x) = − 3⋅ x 9 ⋅ 3 x4 (18 1 − x2 − 1 2x 23 f '( x) = , f ''( x) = ⋅ 2 3 ( x − 1)5/3 3 3 ( x 2 − 1) 2 1 x (19 2 − 5x 2 1 + 5x f '( x) = 3 , f ''( x) = − ⋅ 9 3 x4 3x (20 3 2 4 3 f '( x) = cos( x ) ⋅ 3 x , f ''( x) = −9 x sin( x ) + 6 x ⋅ cos( x 3 ) www.GooL.co.il © ¯ – ¯ 17 (21 f '( x) = − sin( x 4 ) ⋅ 4 x 3 , f ''( x) = −16 x 6 cos( x 4 ) − 12 x 2 ⋅ sin( x 4 ) (22 f '( x) = 3sin 2 x ⋅ cos x , f ''( x) = 6sin x cos 2 x − 3sin 3 x (23 2 2 2 2 2x 2 ⋅ cos ( x ) − 8 x cos( x ) sin( x 2 ) f '( x) = , f ''( x) = cos 2 ( x 2 ) cos 4 ( x 2 ) (24 2 −4 x f '( x) = tan( x 2 ) ⋅ ( −2 x ) , f ''( x) = − 2 tan( x 2 ) 2 2 cos ( x ) (25 1 2x + 3 f '( x) = , f ''( x) = 3/2 − x 2 − 3x − 2 2 ( − x 2 − 3x − 2 ) (27 (26 sin x 2x 2 − 6x4 f '( x) = xsin x cos x ⋅ ln( x + 1) + f '( x) = , f ''( x) = 2 x +1 1 + x4 (1 + x 4 ) f '( x) = ( cos x ) ln x (29 ln(cos x) ⋅ − tan x ⋅ ln x x www.GooL.co.il © f '( x) = ( sin x ) ¯ – ¯ x (28 ( ln(sin x) + cot x ⋅ x ) 18 6 – ¯ : ,¯ ( arctan x ) ' = ¯ 1 (3 1 + x2 ( arcsin x ) ' = 1 1− x 2 (2 ,¯ (1) ( x ) ' = 2 1 x (1 , , f ( n ) ( x) , n : . y= x4 (4 x2 −1 y= (2) x 2x + 3 (3 y = 2 (2 ( x − 1)( x − 2) x − 3x + 2 2 , y (10) , : y = x3e x (1 (4) : y' y (3 4 ln x + 10 ln y = y 2 (2 x 2 + y 5 − y = 1 (1 x xy = sinh x y − y x = 1 (5 x + y = xy (6 . xy − y 3 + x 2 − x = 0 y '' 1 (1 x+a (3) y = x 3 sin 5 x (2 . y =1 y= x y − xy = 10 (4 (5) (6) . x(t ) = t cos t (2 2 y (t ) = t − 1 x(t ) = t − sin t (1 y (t ) = t cos t h(x) g(x) : f ( x ) = ( cos x ) www.GooL.co.il © ln x (3 sin x f ( x ) = ( x + 1) (2 ¯ – ¯ (7) x f ( x ) = ( sin x ) (1 19 6 – (2) y ( n ) = ( −1) n n !( x + a ) − n −1 (1 ) ) (2 ( y ( n ) = ( −1) n n ! −5( x − 1) − n −1 + 7( x − 2) − n −1 ( 1 2 y ( n ) = ( −1) n n ! − 1 ( x − 1) − n −1 − 6 ( x + 1) − n −1 + 3 ( x − 2) − n −1 2 ( y ' = 2 x − 1 ( x − 1) −2 − ( x + 1) −2 2 ( y '' = 2 + ( x − 1) −3 − ( x + 1) −3 ) (3 (4 ) ( y ( n ) = 1 ( −1) n n ! ( x − 1) − n −1 − ( x + 1) − n −1 2 ) ( n > 2) (3) (e 3 (10) ( sin 5 x ⋅ x ) x ⋅ x3 ) (10) = e x x 3 + 103x 2 + 456 x + 120 ⋅ 6 (1 = −510 x 3 sin 5 x + 6 ⋅ 510 x 2 cos 5 x + 54 ⋅ 59 x sin 5 x − 24 ⋅ 59 cos 5 x (2 (4) y − y ( x 2 + cosh ) x (3 y'= y 2 x( x − cosh ) x y'= y− y x −x (6 y'= y' = −4 y (2 x(10 − 2 y 2 ) y x ln y − x y −1 y (5 x y ln x − y x −1 x y'= y'= 2x (1 1− 5y4 y (1 − x y −1 ) (4 x y ln x − x − 1 , − 1 (5) 8 (6) cos t − t sin t 1 − cos t (−2sin t − t cos t )(1 − cos t ) − sin t (cos t − t sin t ) y ''( x ) = (1 − cos t )3 2t y '( x) = cos t − t sin t 2(cos t − t sin t ) − 2t (−2sin t − t cos t ) y ''( x) = (cos t − t sin t )3 y '( x) = .6 www.GooL.co.il © ¯ – ¯ 37 39 (1 (2 (7) 20 7 – ( ) . . f ( x) = e x 2 . f ( x) = 2 + 3 x b . b . .x = 0 a .x =e (2) y = 3x 1 y = ax + 2 (3) (4) (5) f ( x) = x 3 + 1 . x=0 (6) x 2 + y 2 = 25 . (3, 4) . (1) y = 4x + b . f ( x) = x x + b 2 g ( x) = x+c f ( x ) = ln x b .c y = x+b 1 y = − x2 + k 2 . k (7) y= 1 x (8) (9) . y= x ( ( −3,1) (2, −3) y = x2 − 2 x + 1 ( (10) : 1 . y = − x2 − 5 4 . y = g ( x) = 1 x y = f ( x) = x2 . y2 = 2x . ¯ © (11) x2 + y 2 = 8 x2 − y2 = 2 www.GooL.co.il y = x2 x2 + 2 y 2 = 8 ¯ – ¯ (12) ¯ (13) 21 7 – . y = x +1 . y = 4x + 9 (1) (2) (3) (0,1) (−1,5) .b=4 (4,12) 1 (0, ) 2 1 .y= x e 1 1 .y =− x+ 8 2 (4) (5) . y =1 (6) 3 25 y =− x+ 4 4 (7) , k = 1.5 (8) . (1,1) y = 6 x − 15 , (4, 9) .y= , y = −2 x + 1 , (0,1) ( (9) 1 3 x + , (9,3) : 6 2 ( y = 2 x − 1 , y = −2 x − 1 (10) 71.57 (11) 71.56 (12) www.GooL.co.il © ¯ – ¯ 22 8 – ¯ (1) : lim x →1 lim x →3 2 x 2 − 50 (2 x →−5 2 x 2 + 3 x − 35 xn − x (3 x −1 lim x2 + 7 − 4 (6 x − 2 −1 x→4 lim 2e x − x 2 − 2 x − 2 (12 x →0 2 x3 lim x →∞ ex − x −1 (11 x →0 x2 lim ln 2 ( x + 1) + x (15 x →0 x sin(ax) lim (18 x → 0 sin(bx ) 3 2x2 −1 − x lim (7 x →1 x −1 a x − bx (a, b > 0) (10 x →0 x lim x2 + 1 ln 2 x −1 lim (14 x →∞ 1 x2 sin(ax 2 ) lim (17 x →0 bx 2 lim 1 + sin x − cos x (21 x sin 2 x − sin( x 2 ) (24 x →0 x4 lim ln x − x + 1 (13 x →1 x 2 − 2 x + 1 lim lim x →0 tan x − sin x (20 x →0 x3 lim e x sin x − x(1 + x) (23 x →0 x3 lim arctan( x 2 + 3 x) (26 x →0 arcsin( x 2 − 4 x ) lim tanh x (27 lim x →∞ x2 + 1 (30 x →∞ 2 x 2 + x + 3 lim x→ (ln x) 2 + 2 ln x − 3 (33 x lim lim lim x→∞ x →3 3 −1 x (8 1 x 0 2 cosh x − 2 (29 1 − cos 2 x ln x + x + 1 (32 x →∞ ex www.GooL.co.il © ¯ – ¯ x2 − x − 6 (1 x2 − 9 x−3 (4 x +1 − 2 lim 1− e −1 lim (9 x →0 x x →0 x →3 2x +1 − x + 5 (5 x−4 lim x lim lim lim x →0 tan x (16 x x − sin x (19 x3 1 − cos(1 − cos x) (22 x →0 x4 lim ln(cos x 2 ) (25 x →0 x4 lim lim x →0 sin x (28 sinh x lim x→∞ ex (31 x 23 1 x 1 x lim ⋅ e x (36 x→∞ lim − x →0 e (35 x lim ln(sin x) x→0+ ln(tan x ) (34 lim x 2e− x (38 x→∞ 1 lim ⋅ ln x (37 x →∞ x lim x ⋅ ln x (41 lim(1 − cos x) cot x (40 1 1 − (45 lim x →0 sin x x 5 lim x ⋅ 1 + − 1 (44 x →∞ x x+3 lim x ⋅ ln (43 x →∞ x −3 lim x 2 + x + 1 − x (48 lim [ ln(3 x) − ln(sin 5 x) ] (47 + 1 1 − lim (46 x →1 ln x x −1 lim tan x ⋅ ln x (39 x →0 + lim ( x 2 − 9) ⋅ ln( x − 3) (42 + x → 0+ x →3 x →∞ x →0 x →0 1 lim (ax) x (a > 0) (51 + lim x x −1 (50 x →1 x →0 x 2 + x + 1 + x (49 lim x → −∞ x2 lim x sin x x → 0+ 1 2 x4 lim(cos x ) x →0 (54 x2 + 1 lim 2 (53 x →∞ x − 1 (57 tan x x2 lim (56 x →0 x lim (2 x − 4) x − 2 (52 x →2+ 1 lim( x + 1)cot x (60 lim x tan x (59 x → 0+ x →0 1 lim(1 + tan 3 x) x (55 x →0 lim (sin x) tan x (58 x →0 + 1 sin x x2 lim (63 x →0 x 2 lim (1 + x 2 )cot x (62 x →0 + www.GooL.co.il © ¯ – ¯ lim ( x + sin x) tan x (61 x → 0+ 24 ∞ . ∞ ¯, ¯ ¯ (2) . 16 x + 4 x +1 (2 x →∞ 2 4 x + 2 + 2 x + 3 3x + sin x (3 x →∞ 4 x + cos x lim 8 , lim lim x →∞ x2 + 1 (1 x – (1) 5 (7 6 3 (6 2 1 (5 6 4 (4 n − 1 (3 20 (2 17 1 (13 2 1 (12 6 1 (11 2 a ln (10 b 1 (9 1 (21 2 1 (20 2 1 (19 6 a (18 b a (17 b 1 (16 1 (15 1 (28 1 (27 1 (24 3 1 (23 3 1 (22 8 0 (35 ∞ (34 0 (33 ∞ (32 1 (31 2 1 (30 2 2 (29 3 0 (42 0 (41 0 (40 0 (39 0 (38 0 (37 1 (36 3 (48 5 0.5 (47 0 (46 2.5 (45 6 (44 0 (43 1 (56 e 2 (55 1 (54 1 (53 1 (52 e (51 1 (63 e (62 1 (61 1 (60 e −1/ 2 (59 e1/3 (58 e3 (57 e −1/6 (65 e (64 2 (14 1 (49 2 − ln − 3 (26 4 − 1 (25 2 − 5 (1 6 − − 3 (8 2 1 (50 2 (2) 0.75 (3 www.GooL.co.il © ¯ – ¯ 0.25 (2 1 (1 25 9 , : ** , – , , . f ( x) = f ( x) = x −1 (3 x2 x2 − 1 f ( x) = (8 ( x − 2)( x − 5) ln x (12 x f ( x) = f ( x) = x( x − 9) 2 (1 f ( x) = 2 x2 (4 ( x + 1) 2 3 x +1 f ( x) = (7 x −1 f ( x) = x3 − x 2 (10 x2 − 1 1 (14 2−x f ( x) = x ⋅ ln x (13 1 (17 ln 2 x f ( x) = 4ln 2 x − 4ln x − 3 (16 1 f ( x) = ( x + 2) ⋅ e x (20 2 f ( x) = 3 x 2 (1 − x) (23 x 2 − 1 (24 f ( x) = e x (19 f ( x) = 1 x2 + 1 (22 | x −3| (26 x−2 f ( x) = 3 x 2 − 1 (25 f ( x) = 2cos 2 x − sin 2 x (29 f ( x) = arcsin(sin x) (28 f ( x) = x − 2arctan x (27 f ( x) = 8cos x + 2cos 2 x − 3 (30 , 1 2 f ( x) = x ⋅ e−2 x (21 ) , f ( x) = ln 2 x + f ( x) = x − e x (18 3 ln x (11 x , f ( x) = ln f ( x) = ln 2 x + 2ln x − 3 (15 ( x3 (5 x2 − 4 f ( x) = x2 − 4 x + 3 f ( x) = (9 x2 − 4 f ( x) = , *** f ( x) = x 4 − 2 x 3 (2 x3 (6 ( x + 1) 2 f ( x) = ¯ , (1) * f ( x) = (0≤ x≤π ) (0≤ x ≤ 2π ) : . 18 .( 8 . . .x 27 ) 1,2,28,29,30 ¯ 9,17 ¯ www.GooL.co.il © ¯ – ¯ * ** *** 26 9 – (1) (1 (2 y y x x (4 (3 y y x x (6 (5 y y x x (8 (7 y y x x www.GooL.co.il © ¯ – ¯ 27 (10 (9 y y x x (12 (11 y y x x (14 (13 y y x x (16 (15 y y x x www.GooL.co.il © ¯ – ¯ 28 (18 (17 y y x x (20 (19 y y x x (22 (21 y y x x (24 (23 y y x x www.GooL.co.il © ¯ – ¯ 29 (26 (25 y y x x (28 (27 y y x x (30 (29 y y x x www.GooL.co.il © ¯ – ¯ 30 10 " – "– (1) .a . . . f ( x) = ax3 + x 2 (1, 2) ( . f ( x) = ax3 + bx 2 x =1 ( . a, b .a . . . f ( x) = ax3 + x 2 (1, 2) ( . f ( x) = ax3 + bx 2 x =1 ( . a, b .33 f ( x) = ax3 + x 2 x=3 ( .a .12 . f ( x) = ax 3 + bx 2 (3,9) ( . a, b . ax3 + x 2 . f ( x) = 3 2x + x + 6 y=4 ( .a . f ( x) = y = 0.5 x + 1 ax 2 + bx + 4 x .b . x =1 f ( x) = a ( . x2 + 2 x + 4 x 2 + ax + 6 ( .a www.GooL.co.il © ¯ – ¯ 31 f ( x) = x 3 − 3 x . f ( x) = 5 . f ( x) = 2 . f ( x ) = 0.5 f ( x) = k f ( x) = k f ( x) = k f ( x) = k ." . . . . (2) . . . . . . . . k k k k y (-1,2) x (1,-2) : ¯ ( 0 < x < π3 ) ( x ≥ 0) x < 2sin x (2 ( −∞ < x < ∞ ) ( x > 0) ln( x + 1) ≤ x (4 10 (3) 8 x 3 ≤ 3 x 4 + 6 x 2 (1 x +1 < 1+ x (3 2 – (1) a=−2 ( 3 a=−1 ( 3 a= 2 3 . b = 6, a = −4 ( , b = −1 ( .a =1 ( . b = 3, a = −1 ( a = −7 ( a = 0.5 ( a =8 ( 3( 2( −2 < k < 2 ( . k = ±2 ( −1 < x < 1 www.GooL.co.il © x < −1 ( x >1 ¯ – ¯ (2) 1( . k < −2 k >2 ( ( 32 11 – (1) :( 4 x − 2 f ( x) = ( x − 2)( x − 3) ( −5 < x < −1) x <1 x ≥1 ) ( −1 ≤ x ≤ 3 ) f ( x) = x 3 − 3 x 2 + 3 x (1 ( −1 ≤ x ≤ 20 ) f ( x) = x 2/3 (20 − x) (3 f ( x) = − x 2 + 4 x + 5 (2 (1 ≤ x ≤ 7) 2 2 ¯ (4 x2 (6 x +1 ( −5 ≤ x ≤ 1) f ( x) = 1+ | 9 − x 2 | (5 ( −∞ < x < ∞ ) f ( x) = f ( x) = x 3 − 9 x + 1 (7 . ¯ ( x ≥ 0 ) xe− ( x ≤ 1) 0 ≤ x 2 e x −1 ≤ 1 (3 11 x ( x ¯ ) x 3e − x ≤ ≤ 1 (2 (2) 27 (1 e3 – (1) . . . (3, 9) , ( −1, −7) (1 (2,3) , (5, 0) , (−1, 0) (2 (8, 48) , (20, 0) , (0, 0) (3 . (1, 2) , . (2.5, −0.25) (4 (−5,17) , . ( −3,1) (5 . ( −2, −4) (6 . (7 : [a, b) ⇔ a ≤ x < b , www.GooL.co.il © ( a, b ) ⇔ a< x<b ¯ – ¯ , [ a, b ] ⇔ a≤ x≤b 33 12 – * ¯¯ , : ( 1) (AB||CD) ABCD D C ." 6 "4 .( )D DE ¯ DE A E B ? .( A B x ( 2) . ABCD x ) " 60 x ( . ¯ C D ( p ¯ ¯ .( p ¯ , AD = BC = A P " 5 .( Q ) AP = AQ = CS = CR = x : S ¯x D R ? C www.GooL.co.il © – ¯ PQRS ¯ ¯ ( 3) ¯ ABCD . AB = CD = B ) " 10 34 ( 4) E ¯ ( C = 90° ) ∆ABC ." 8 AB ¯ A , ¯ . D C ¯ .ABDE ¯ AEDBC B ( 5) "8 ¯ , ABCD AB C D D 8 , C .( A ) B ? ¯ AB (6) ¯ , ( B = 90° ) ∆ABC A ¯ AD . " 30 ¯ .( B D , C ¯ ¯ . ¯ 600 ." ) BC ¯ ( 7) , 8 ," 8 ." 3 , 3 ¯ ) ¯ .( www.GooL.co.il ¯ 35 G,F,E CF = CG , BE = BF ( 8) ABCD ¯, DC , BC , AB .( A E B ) ." 6 ¯ , BE x F ) FCG BF . x ¯ EBF .( D G C ¯ .1. x . . ¯ E . " 10 E A M N " B ¯, ¯ ¯ .(ED = EC) .( )N ¯ M ¯ AM . C (*9) ABCD DEC AB D .2. BNC , AMD , EMN ," (*10) .R ) ¯ , ¯ .( R . www.GooL.co.il © ¯ – ¯ 36 O . " 10 (*11) O¯ ¯ ,ABCD DC B A B , A .( ) ¯ ABCD D C . ¯ ¯ , A * ABCDE ( 12) ¯ ABE .( E ) EBCD . AB = AE = " 4 , BC = " 2 : B . D C (*13) ¯ ABC A . ¯R ¯ ? B C ¯ R 72 . 2π 5 . www.GooL.co.il ¯ ¯ (*14) 37 ) " " (15) ." 8 ¯) ( ¯ ? , ( (16) ," y ¯ ¯ ,( )"x . " 12 ? ¯ (17) , ¯ )" 75 ¯ .( ) , . , ( )" ." ( " ¯ (18) 1000 ? (19) " a¯ . , : ¯y . . . www.GooL.co.il © x . ¯ – ¯ 38 , ¯ ¯ (*20) ¯ , (*21) ,a . , ," 200 . ) " 12 ¯ ¯ 12 .( . ¯ . " 64 ¯ ¯ . . ¯ 4 3 ¯ ¯ ? www.GooL.co.il (23) ¯ π 10 ,( 10 ¯ (22) )" (24) 39 (25) ,A y ¯ A C , y = − x2 + 5x ¯ .( ) ABOC ¯A ¯ ¯ . ? x O . B ¯A ? (26) y = 9 − x2 ¯ , ABCD y .( )x ¯ CD C D AB ? x A B y = 9 − x2 y D (27) ABCD .( A ¯A )x ¯ ? . ABCD x C B .ABCD www.GooL.co.il © . ¯ – ¯ 40 (28) . y = − x 2 + 12 y .( A B )B A x .O , B A ¯ AB x . ? 0 AOB ? AOB . (29) y = ex y . y = e⋅ x − 2 y A .( )B A B x . x¯ AB . x AB . ? (30) : y 1 1 . y = − x 2 + 3x , y = x 2 + 7 4 2 P y Q x .( , © ¯ – P ¯ .PQ www.GooL.co.il )Q ¯ 41 (31) .y= x x y .( ) A( 4.5, 0) M ¯ ,M . . AM x A(4 5,0) (32) f ( x) = 3x − 4 . (0,1) (*33) : y . g ( x ) = 36 − 6 x , f ( x ) = 3 x ,x . x ¯ . (*34) y = − x2 + 2 x y , ) y=0 ¯, x = 0 , x =1 : ? x www.GooL.co.il © ( ¯ – ¯ 42 y = x2 ¯ y ¯ (0, a ) A. ) a > 0.5 .( ,B A (*35) B . a . AB B x . a .2 (*36) , y = x2 (t , t 2 ) y . y = 6x − 9 . .( )M ¯ .t (t,t2) . x , M . t . M ¯ A(2, 2) y M .O A(2,2) M O ¯ x . B ( 2, −2) . x>0 OM + MA + MB : ¯ B(2,-2) x ¯ ,M ? www.GooL.co.il © ¯ – ¯ (*37) 43 12 . x = 3.75cm (3) (7) " 40 : . AM = 5 / 2 (9) . 0.25 p . ." .32 . (17) .x = y = 403.1 (24) ." . A(1,8) . (27) . PQ = 4 (30) 12 3 (12) ." 1 a. 9 .x= ." ." . CD = 2 3 (26) . . (0.5, 0.75) (34) . t = −3 / 37 . . y = 2t ⋅ x − t 2 . (36) 48 : (10) .R = . 3π 3π 2π (14) , , 10 10 5 . " 120 (18) (22) ." ." 500 (21) 3 . A(3, 6) . (25) . S∆AOB = 16 . (29) .8 (33) . 4.25 . (19) . " 15 : A(0, 0) . . AB = 4 . (28) . A(5, 5) . x =1 . (8) . " 4 (15) 1 1 a, y= a . 12 6 24 : . . AC = BC = 4cm (4) . 4 5 cm (11) . " 4 (16) . AE = 1.7 cm (1) (2 ) . (30 − x) . . S = x 2 − 6 x + 18 . . x = 3 .1. 9 .2. . " 5: 433 a (20) 27 ¯. . B = 6cm , BC = 24cm (6) . AB = 2 32 cm (5) . 45° , 45° , 90° (13) 2.5 : – . (1.5, 0.5) (32) . M ( 4, 2) (31) . B ( (2a − 1) / 2, (2a − 1) / 2) . (35) . M (0.845, 0) (37) www.GooL.co.il © ¯ – ¯ 44 13 ( ,( – ) , ) : −4 x 3 + 21x 2 − 48 x + 28 = 0 (4 (1) ¯ x 2 = − ln x (2 x 3 + 4 x − 1 = 0 (1 ax 3 + bx 2 + cx + d = 0 (2) x − 0.25sin x = 7 (3 . b 2 < 3ac ¯ . ¯ ? . (3) ¯ x 2 + x sin x = 1 − cos x (4 ln( x + 5) − 4 = x (3 e x −1 = x (1 arctan x − x = 0 (2 . f '( x ) ≤ 1 , f (0) = 1, f (1) = 2 : x¯ . (4) f ¯ f ( x ) + sin x = 4 x : (5) ¯ 1 + 4 x 4 = 8 x 3 (3 4 x3 + 5x − 1 = 0 (2 x e x − 5 x = 0 (1 ¯ (6) .( ¯ ) ax 3 + bx 2 + cx + d = 0 (2 ax 2 + bx + c = 0 (1 (n > 4, odd ) ax n + bx n − 2 + cx n − 4 − d = 0 (4 x + a cos(bx) = 1 (3 :( 2,3 −4 x3 + 21x 2 − 48 x + 28 = 0 (3 13 (7) ) 1 + 4 x 4 = 8 x 3 (2 7 x3 − 33 x 2 + 21x + 61 = 0 (1 – . x = 0 (4 x = −4 (3 x = 0 (2 x = 1 (1 (3) 1 < −1 ab (2) . 1 > 1 (3 4b 2 − 12ac < 0 (2 b 2 − 4ac = 0 (1 (6) ab b 2 (n − 2) 2 − 4anc(n − 4) < 0 (4 . x = −1 (2 x = 0.5576 , x = 1.9672 x = 0.8459 www.GooL.co.il © ¯ – ¯ (1 (7) (3 45 14 ¯ ' 0.5 – . ¯ ? ' 20 2 ¯ ¯ ,¯ ¯ . ¯ . (1) ,' 2.5 (2) 1 ' ? .( ) ¯ ' 900 .' 1200 (3) ' 260 ¯, ¯ ¯ . ? ? " 40 " . . (4) 30 , " 20 ¯ . " 10 ? . . . 1200 " 480 (5) . α = 30° , . α ? . ? . . 1200 α www.GooL.co.il © ¯ – ¯ 46 . ?"3 ¯ ¯ (6) . " 2 ." 3 . " (7) L 9π L . ¯ .* ¯¯ .t . h(t ) ? h(t ) = 1.5cm . xy 3 8 =: 2 5 1+ y 6 (8) x . (1, 2) . . y ? . ¯ ." 3 ¯ .t ¯ . 1/64 ¯ . ¯ (*10) V (t ) ¯ ln 2 " www.GooL.co.il © ¯. R .t . . ¯ ? −3V (t ) ¯ (*9) "4 ¯ (*) ¯ – ¯ ¯ ¯¯ : 47 8 6 .( . 120 (11) )' 2 ¯ '5 , 5.5 '. 12 (12) . . . ' 25 ' 25 ' ¯ ' . ' ¯ ¯ 8 ¯ 2 . 6 '. 20 , ¯ (13) . ? www.GooL.co.il © 2.5 ? ¯ – ¯ 48 . ¯ 7. 350 (14) 5 . ¯ 3 ¯ . 25 : . ( Ω) R2 (15) R1 . . ¯ 0.7 11 1 =+ R R1 R2 R 0.4 R2 ? R1 = 80Ω , R2 = 105Ω : www.GooL.co.il © ¯ – ¯ ¯ R1 ¯R 49 14 . 208 m / sec . 0.104 rad / sec . (3) 4 m / sec (2) 3 5 π 100 Rad / hour . (5) − .. 60 . 7 (8) . 0.25m / sec (11) . 7.9958m / sec (14) .− − . 115.4 m / sec . . . – . 4L . (7 ) 9π . t = 4.5hours . . 20π m 2 / sec (1) . −0.38 cm / min (4) . . R (t ) = " 0.75 (6) 12 . 2t + 3 . 2.0833 m / sec (13) . 1.6923 m / sec . 3.6923 m / sec . . 0.002045 Ω / min www.GooL.co.il © ¯ – ¯ (9) (12) (15) 50 15 – ' ¯ (1) b−a b b−a < ln < b a a (1 b−a b−a < b− a< 2b 2a (2 π b−a b−a < tan b − tan a < 0 < a < b < 2 2 cos a cos 2 b (3 : (0 < a < b) (0 < a < b) ( a < b) (4 b−a b−a < arctan b − arctan a < 2 1+ b 1+ a2 (0 < a < b) ( a − b )e − a < e − b − e − a < ( a − b ) e − b (5 b−a ( 0 < a < b < 1) 2 < arcsin b − arcsin a < b−a 1− a 1 − b2 b−a a rc sinh(b) − a rc sinh(a ) b−a < < (0 < a < b) 2 b−a 1+ b 1 + a2 b−a b−a < a rc tanh(b) − arc tanh(b) < ( 0 < a < b < 1) 2 1− a 1 − b2 (0 < a < b) (1 < a < b ) n b⋅ b−a n b−a < b−n a < n a⋅ n⋅b n⋅a ( x > 0) x 1 + x2 ( x > 0) < a rc sinh( x) < x (4 x < ln(1 + x) < x (6 1+ x ( x > 0) ( 0 < x < 1) ¯ x < arctan x < x (2 1 + x2 © ( x > 0 ) 1 + x < e x < 1 + xe x sin x ≤ x (8 (9 (2) (7 π 0 < x < tan x < 4 x (9 3 ¯ – (8 π x (1 0 < x < x < tan x < 2 cos 2 x x (3 ( 0 < x < 1) x < arcsin x < 1 − x2 x (5 ( 0 < x < 1) x < a rc tanh( x) < 1 − x2 arctan x > ln(1 + x) (*10 www.GooL.co.il (7 b 2 + 1 2a (b − a ) 2b(b − a ) < ln 2 < (10 b2 + 1 a2 +1 a +1 : ( x > 0) (6 ¯ 51 : (3) ¯ cos x2 − cos x1 ≤ x2 − x1 (2 sin x2 − sin x1 ≤ x2 − x1 (1 | tan y − tan x |≤ 8 | sin x − sin y | (* 4 arctan y − arctan x < y − x (3 : (4) ¯ + 1 < 2 < 1.5 (2 1 3 1 < ln < (1 3 2 2 3π 1π + < arcsin ( 0.6 ) < + (4 15 6 86 3π 4 1 π + < arctan < + (3 25 4 3 6 4 1 22 . | f '( x ) | ≤ 5 x¯ . f ( 2) = 8 ¯ ¯ . | f '( x ) | ≤ 7 ,' . f (1) = 3, f (4) = 18 ¯ x¯ . 4 ≤ f (2) ≤ 10 ¯ ¯ 4 . © 3 10 2 * ¯ ¯ – . f ( x) . f (1) = 3, f (4) = 18 ¯ ¯ www.GooL.co.il . (5) f ( x) ¯ 52 16 – / : )x=0 ( (1) . ¯ * f ( x ) = sinh x (3 f ( x ) = x 2e −4 x (2 f ( x ) = sin 2 x (1 f ( x) = 2 x (6 f ( x) = cos 2 x (5 f ( x) = sin 2 x (4 f ( x) = arcsin x (9 f ( x) = ln(2 − 3 x + x 2 ) (8 f ( x) = x cos(4 x 2 ) (7 f ( x) = 1 (12 1 + 9 x2 f ( x) = 3 (11 1 − x4 f ( x) = 1 (10 1+ x x (15 9 + x2 f ( x) = x (14 4x +1 f ( x) = 1 (13 x−5 f ( x) = f ( x) = 1 (18 (1 + x) 2 f ( x) = ln f ( x) = 1+ x (21 1− x 7x −1 (17 3x + 2 x − 1 f ( x) = 2 3 (16 x + x−2 2 f ( x) = ln(1 − x) (20 f ( x) = arctan( x / 3) (24 f ( x) = ln(1 + x) (19 x2 (23 (1 − 2 x) 2 f ( x) = ln(5 − x) (22 f ( x) = . ¯ ¯ 16,17 : 18,19,23,24 . x = x0 : (2) ( x0 = π2 ) f ( x) = sin x (3 ( x0 = 2) f ( x) = , 1 (2 x , (3) :( f ( x) = www.GooL.co.il © ( x0 = 1) f ( x) = ln x (1 sin x (3 ex ¯ f ( x) = tan x (2 ¯ – ¯ ) 2 f ( x) = e− x cos x (1 53 : ∞ ∑2 n =0 ¯ (4) (−1)n 2 n (2 n! n =0 ∞ 1 (3 n ⋅ n! ∞ ∑ (−1) n ∑ (2n + 1)! (6 n=0 (−1)n ∑ 2n+1 (n + 1) (9 n =0 n =0 (−1)n ∑ 2n + 1 (5 n =0 (−1)n ∑ n + 1 (8 n=0 ∞ ∞ ∞ 1 ∑ n ! (1 ∞ n +1 (4 n=0 n ! ∑ (−1)n ∑ (2n)! (7 n =0 ∞ ∞ : e x sin x − x (1 + x ) (3 x →0 x3 lim lim x→0 (5) sin x − x + 1 x 3 6 (1 x→0 x5 x − arctan x (2 x3 lim : 0.001 (6) arctan 0.25 (3 ( ) 1 (1 e sin 3° (2 (7) n : ( n = 4 ) ln1.5 (3 ( n = 1) cos 4° (2 ( n = 3) 1 e (1 (8) π 6 . | x |< 0.01 . |x| ≤ x3 3! ln(1 + x ) ≅ x sin x ≅ x − cos x ≅ 1 − . | x | ≤ 0.2 . . x2 x4 + 2! 4! . (9) 3 . 0.001 . 0.01 www.GooL.co.il © x ,x ¯ 3! x3 x5 x 7 arctan x ≅ x − + − ,x ¯ 357 sin x ≅ x − ¯ – ¯ . . 54 .ε (10) 0.1 ( ε = 0.001) ∫ 0 0.2 ln(1 + x) dx (2 x ( ε = 0.0001) ∫ 0 0.5 (ε = 0.0001) ∫ 0 sin x dx (1 x 1 − cos x dx (3 x2 ' , x0 = 0 f ( x ) = 3 64 + x 3 . , x0 = 0 . (11) .' 66 (12) f ( x ) = tan x .' tan 0.1 , x0 = 0 f ( x) = . , x0 = 16 (13) .' f ( x) = 4 ¯ x+4 5 . ¯ 4 ¯ (14) x .' 15 ¯ : , , n . 0.5 × 10 − n . 0.5 × 10 −3 = 0.0005 . www.GooL.co.il © , ¯ – ¯ 55 16 – (1) (3 (2 x 2n +1 ∑ (2n + 1)! n =0 −∞ < x < ∞ ) ( (1 22 n +1 x 2 n +1 (2n + 1)! n=0 ( −∞ < x < ∞ ) 4n x n+ 2 n! n =0 ( −∞ < x < ∞ ) ∞ ∞ ∞ ∑ (−1)n ∑ (−1)n (6 (5 (4 1∞ 22 n −1 x 2 n + ∑ ( −1) n 2 n=0 (2n)! ( −∞ < x < ∞ ) (ln 2) n x n ∑ n! n=0 ( −∞ < x < ∞ ) ∞ (9 ∑ (−1)n+1 (8 1 ⋅ 3 ⋅ ... ⋅ (2n − 1) x 2 n +1 ⋅ 2 ⋅ 4 ⋅ ... ⋅ 2n 2n + 1 n =1 ( −1 < x < 1) 22 n −1 x 2 n (2n)! n =1 ( −∞ < x < ∞ ) ∞ ∞ 1 x n +1 ln 2 − ∑ 1 + n +1 2 n +1 n =0 ( −1 ≤ x < 1) ∞ x+∑ ∞ (| x |< 1) ∑ 3x 4 n (7 42 n x 4 n +1 (2n)! n=0 ( −∞ < x < ∞ ) ∞ ∑ (−1) ∞ (| x |< 1) ∑ (−1) n x n (11 n =0 ∞ (10 n=0 −1 (| x |< 5 ) ∑ n +1 x n 5 ∞ (| x |< 1 ) ∑ (−1) n 9n x 2 n 3 (13 n =0 x 2 n +1 (| x |< 3) ∑ (−1) n+1 (15 9 n =0 (| x |< n ∞ 1 4 ) ∑ (−1)n 4n x n +1 (−1) n+1 n − 1 x (16 n +1 n =0 2 ∞ (| x |< 1) ∑ (17 n =0 ∞ (−1)n x n +1 ∑ n + 1 (19 n =0 ∞ (| x |< 1) ∑ (−1) n +1 ⋅ n ⋅ x n −1 2 x 2 n +1 (21 (| x |< 1) ∑ n = 0 2n + 1 x n+1 (20 ( −1 ≤ x < 1) ∑ − n +1 n =0 ∞ (22 ∞ n=0 (18 n =1 ∞ (| x |< 1 ) ∑ 2n (n + 1) x n + 2 2 (14 n=0 ∞ (| x |< 1 ) ∑ ( 2(−1) n − 3n ) x n 3 (12 n =0 ∞ ( −1 < x ≤ 1) n (23 n +1 ∞ x (n + 1) n =0 5 ln 5 − ∑ ( −5 ≤ x < 5 ) n +1 (24 ∞ x 2 n +1 (| x | ≤ 3) ∑ (−1)n 2 n+1 3 (2n + 1) n =0 www.GooL.co.il © ¯ – ¯ 56 (2) ( −1) n ( x − π ) 2 n 2 (3 2n ! n =0 ( −∞ < x < ∞ ) ( −1) n ( x − 1) n +1 (1 n +1 n =0 ( 0 < x ≤ 2) ( −1) n ( x − 2) n (2 2n +1 n=0 ( 0 < x < 4) ∞ ∞ ∑ ∞ ∑ ∑ (3) 3 1 x − x 2 + 1 x 3 − 30 x 5 + .. (3 3 5 7 2 4 331 6 x x + x3 + 215 + 17 x + ... (2 1 − 3 x + 25 x − 720 x + .. (1 2 24 315 (4) ln 3 (9 2 ln 2 (8 cos1 (7 sin1 (6 π / 4 (5 2e (4 e (3 e (1 e −2 (2 (5) 1/3 (3 1/3 (2 1/120 (1 (6) 47/192 (3 53/144 (1 π / 60 (2 (7) 77 192 1 160 (3 π ⋅π 1 (2 4050 5 8 1 48 (1 (8) 6 2 (0.2) / 6! (2 (0.01) / 2 (2 5 (π / 6) / 5! (1 (9) | x |< 9 9 / 100 (2 | x |< 5 3 / 25 (1 (10) 143 / 576 (3 39 / 400 (2 www.GooL.co.il © ¯ – ¯ 449 / 2250 (1 57 17 – (1) : n 4 + 2n 2 + 6 (3 n →∞ 3n 3 + 10 n 4n 2 + 2 (2 n →∞ n 2 + 1000 n lim lim n →∞ 16n + 4n +1 (9 n →∞ 2 4 n + 2 + 2 n + 3 lim 3n 3 − 5n − 1 lim ln 3 (12 2 n →∞ n − 2n + 1 lim n →∞ ( 4 n + 1 − 5n − 1 n →∞ n 4 + 2n 2 + 6 (4 n →∞ 3n 5 + 10 n (8 lim n →∞ 3n3 + 10n + 4n 4 lim( n 4 + n 2 + 1 − n 2 ) (18 lim n →∞ n →∞ ( n n4 + 2 n2 + 6 4 lim e 3n +10 n n →∞ ) n 2 + n + 1 − n (17 lim n →∞ n 1 lim 1 + 2 (21 n →∞ n 1 lim 1 + (20 lim n →∞ n →∞ 2n n 2n + 3 lim (24 n →∞ 2n − 3 1 lim 1 − 2 n →∞ n ) n 2 + kn − n (16 ) n n+ 2 lim (22 n →∞ n n2 + n + 1 lim 2 n →∞ n + n + 4 (26 cos(2n + 1) (29 n →∞ n lim 4 n2 (25 sin n (28 n →∞ n lim lim 3n 2 + n + sin 2n (31 n →∞ n 2 + cos 3n 3n + arctan(2n − 3) (32 n →∞ 4n + arctan( n − ln n ) lim n 2n + 3n + 4 n (33 lim lim ! ! ¯x .x ,n www.GooL.co.il © , ) ¯( ¯ . ¯ – (13 n 2 + an − n 2 + bn (19 (23 n 2 + 4n + 1 lim 2 n →∞ n + 2 n + 2 3n + sin n (30 n →∞ 4 n + cos n ( ( n 2 −1 10 n n 1 lim 1 + tan (27 n →∞ n (7 4 ⋅ 9n + 3n +1 lim 0.5 n (10 n →∞ 81 + 3n + 3 an + 1 lim 5 (14 n →∞ bn + 2 ) x →∞ n 4 + 2n 2 + 6 + 27 n 6 4n 2 + 2 lim 2 (11 n →∞ n + 1000n n + 5n − n (15 (1 lim 3 n + 2 − 3n − 3 lim ln n n →∞ n 2 − 5n + 6 n − (5 lim n →∞ 2 2n + 10 n2 + 1 (6 n 2 lim ( e − n ) lim ¯ , 58 (2) : lim n n →∞ 2n (2 n→∞ n ! (2n)! (3 (n !)2 n lim 1 + 2 4n+ 1 n n →∞ lim n (6 lim n→∞ n! (4 4n lim n→∞ 4 lim n ⋅ sin (7 n →∞ n lim n n + (−1)n lim (11 n →∞ n 1 (12 n n! (1 nn n (5 1 + 2 + ... + n (8 n →∞ n 2 + 4 n + 1 lim 4n sin 2n n→∞ 12 + 22 + ... + n 2 (9 n →∞ n3 + n 2 + 1 ( 2n ) ! lim lim sin πn 2 n →∞ (3) : 1 1 1 lim + + ... (1 n →∞ 1 ⋅ 2 n( n + 1) 2⋅3 1 ⋅ 3 ⋅ 5 ⋅ ⋅ ⋅ (2n − 1) (2 n →∞ 2 ⋅ 4 ⋅ 6 ⋅ ⋅ ⋅ 2n lim 1 1 1 + + ... + lim (4 2 n →∞ n2 + 2 n2 + n n +1 . an < 1 ¯¯ -2 1 + 2 + 3 3 + ... n n (3 n →∞ n lim . 2n + 1 .( 1 n( n + 1) = 1 n − 1 n +1 1 : * (4) ) ¯ . 1 1 . an +1 = an + 2 an (10 ¯ , a1 = 2 (3 an +1 = 2an − 1, a1 = 2 (2 an +1 = 2 + an , a1 = 2 (1 (5) . an +1 = 2an + 3an −1 , a1 = 1, a2 = 1 . ¯ lim bn n →∞ . . bn = an : an +1 .1 . bn ¯ an .( .2 ¯) a n .1 . . .1. .¯ ¯ .2 ¯ www.GooL.co.il © ¯ – ¯ .3 59 (6) :¯ ¯ n 2 + sin n 1 =. n →∞ 2 n 2 + 3 2 n2 − 1 =1 . n →∞ n 2 + 1 lim 2n + 1 1 =. n →∞ 4n + 3 2 lim n ⋅ cos 2 n =0 . n →∞ n 2 + 2 lim 4n 2 − 2n + 1 =2 . n →∞ 2 n 2 + n + 3 lim n 2 + ( −1) n =1 . n →∞ n2 + 1 lim lim n3 − n2 + 5n + 6 = ∞ . lim 2 n + 4 = ∞ . 1 = −∞ . n lim lim e 2 n +1 = ∞ . n →∞ lim log n →∞ lim n →∞ n →∞ ( ) n 2 + 4n − n = 2 . lim log(2 n + 5) = ∞ . n →∞ n →∞ : ¯ . . lim bn = −∞ an (1 bn (2 lim | cn |= k (3 dn (4 lim bn = ∞ n →∞ n→∞ . lim cn = − k lim cn = k n →∞ (7) n→∞ n →∞ . ( an ⋅ bn ) ( an + bn ) bn an (5 . . ( an / bn ) bn an (6 ( an ⋅ bn ) . . ¯ , bn ¯ an (7 ( an ⋅ bn ) , bn ¯ an (8 lim an2 = L (9 . lim an = L n→∞ . lim an < lim bn n →∞ n →∞ . lim an bn = ∞ bn n →∞ .k <1 n ¯ an < 1 n . lim ( an ) = 1 n →∞ www.GooL.co.il © ¯ – ¯ n→∞ n ¯ an < bn (10 lim an = ∞ (11 lim an = k (12 lim an = 1 (13 n→∞ n →∞ n →∞ 60 17 .2 (11 .4 (10 .0.25 (9 . 1− 3 2− 5 – (8 .1.5 (7 .1 (6 . 5 (5 .0 (4 . ∞ (3 .4 (2 .0 (1 (1) ( ) , ( lim an = ∞ ) ⇐ ( a > 0, b = 0 ) , lim an = 5 a / b ⇐ ( b ≠ 0 ) (14 . e1/3 (13 . ln 3 (12 . e 0.5 (20 . a −b (19 .0.5 (18 .0.5 (17 . k (16 .2.5 (15 . ( lim an = −∞ ) ⇐ ( a < 0, b = 0 ) 2 2 3 (31 .0.75 (30 .0 (29 .0 (28 . e (27 . e30 (26 . e −12 (25 . e3 (24 . e−1 (23 .1 (21 . 1 1 3 (9 .0.5 (8 .4 (7 .1 (6 . ∞ (5 . (4 .4 (3 .0 (2 .0 (1 (2) .4 (33 . (32 3 4e 4 .2 (1 (4) .1 (4 .1 (3 .0 (2 .1 (1 (3) . ∞ (12 . (11 . (10 1 1 . an = ⋅ 3n − ⋅ ( −1) n (.1. 6 2 www.GooL.co.il © . 1 3 (.1. – ¯ ¯ (5) .1 (3 .1 (2 61 18 ( – ) : 1 ∫x ∫ 4x 2 10 ∫ (x 2 ∫ x dx (2 ∫x dx (6 3 ∫(x + 1) 2 dx (9 4 5 x ∫ ( x − 1) 4 ∫ (x dx (15 2 xdx (20 x +1 +1 ∫ (2 x ∫ (x 2 2 xdx (4 − x + 1) dx (7 + 1)( x + 2) dx (10 ∫ (4 x + 1) 10 ∫ ∫ 3 dx (13 4 x − 10dx (16 dx (19 x −1 − x 12 ∫ (1 + x ) dx (23 1 ∫ 4 x − 1 dx (24 1 + x + x2 ∫ x dx (22 4x +1 dx (26 x+2 ∫ x + 2 dx (25 ∫ + e− x ) dx (27 1 x ∫ 4 e + 3 e4 x + 2 3 x )dx (8 ∫ ∫ dx (5 10 dx (17 2x + 4 ∫ dx (18 1 4x x − 2 x + 1)10 dx (14 ∫ 4 x dx (21 ∫ (e 4 1 1 + 2 x2 + x4 ∫ x 2 dx (11 x +1 ∫ x dx (12 ∫ ( x − 2) ∫ 4dx (1 4 dx (3 dx (30 x+3 2 x + 4 2 x + 103 x dx (29 ∫ 5x x2 ∫ 1 − x 2 dx (33 ∫ (e ) 1 ∫ 1+ 4x ∫ 4− x 2 dx (32 x ∫ 2 sin 4 x + cos xdx (36 ∫ sin 2 dx (35 ! www.GooL.co.il © – ¯ x +1 2 1 2 dx (28 dx (31 ∫ cos 4 xdx (34 * 62 19 ( – ¯ ) : x2 ∫ x3 + 1 dx (3 ∫ cot xdx (2 e x+ 2 ∫ e x + 1 dx (6 ∫ x ln x dx (5 ∫ tan xdx (4 −2 x ∫ e xdx (9 e tan x ∫ cos2 x dx (8 ∫e cos(ln x) dx (12 x ∫ cos(sin x) ⋅ cos xdx (11 1 2 ∫ ∫ sin x dx (15 x ∫ sin( x 2 + 1) xdx (14 ln(tan x) dx (18 cos 2 x ∫ x 2 + 1 ⋅ 2 xdx (21 ∫ ∫ cos(2 x 2 ∫ cos(10 x arctan x dx (17 1 + x2 cos x dx (20 2sin x ∫ ∫ ∫x ∫ arctan x dx 1 + x2 ∫ ln x dx (23 x . ∫ x2 2 xdx (7 + 1) ⋅ 4 xdx (10 4 + 1) x 3dx (13 ∫ ∫ 2x dx (1 +1 2 ln x dx (16 x 2x x2 + 1 x 3 + 4 ⋅ x 2 dx (22 : ! www.GooL.co.il © ¯ – ¯ dx (19 * * 63 20 – ) ( : ∫ x sin xdx (3 ∫x 2 5 1 4 ∫ xe dx (1 x ln xdx (2 ∫ x cos 2 xdx (4 sin 4 xdx (5 ∫ ln ∫ x ⋅ ln ∫x (1) dx (8 2 + 2 x + 3) ln xdx (4 ∫ ln xdx (7 x − 2dx (11 ∫ (x ∫ arcsin xdx (10 3 x ∫ x arctan xdx (14 ∫ ∫x e 2 −4 x dx (6 ∫ arctan xdx (9 ln x dx (13 x2 x ∫ cos 2 x dx (12 2 ln x ∫ x dx (17 ∫ ∫ ln ∫e 1 − x 2 dx (20 4 ∫ ( x + 1) ⋅ x + 2dx 2x 2 . ∫ cos 4 xdx . . n ∫ cos n . ∫ sin 4 xdx . . n ∫ sin n 1 (1 + x 24 ) dx . . . ∫ x e dx nx n ∫ n © xdx . (3) xdx . (4) dx . (5) n ¯ – cos xdx (18 (2) ! www.GooL.co.il x . 1 (1 + x2 ) ln( x 2 + 1)dx (15 xe x ∫ ( x + 1)2 dx (21 2 ∫ x tan xdx (22 . 2 ∫e sin 4 xdx (19 . ∫ x 4 e x dx .∫ ∫x xdx (16 ¯ * 64 21 ( – ) :( 2 x3 ∫ ∫x ∫ x2 + 1 1 1 − ln 2 x 1 x(1 + x) ∫ dx (3 ∫e ∫ x (3x 3 x 3 dx ∫ x8 + 2 (15 ∫x 5 2 3 x 3 ∫ cos( x ∫ 1 x (1 + 3 x ) . www.GooL.co.il © ¯ – ¯ + 1) 2 dx (1 x2 x 3 dx (7 + 1) ⋅ 2 x 3 dx (10 1+ ∫ arctan 1 dx (13 x2 xdx (19 ∫ cos(ln x)dx (22 , ! 2 ln 4 x ∫ x dx (16 dx (23 , 2 ∫ xdx (14 x7 ∫ (1 − x 4 )2 dx (20 ⋅ 3 x 3 + 1dx (24 (x ∫e arctan 2 x ∫ 1 + x 2 dx (17 (21 2x ex ∫ e2 x + 1 dx (4 dx (8 − 1)14 dx (11 ∫ ln dx ∫ x ⋅ ln x ⋅ ln(ln x) (18 1 + e2 x ∫ 1 ∫ x ln 4 x dx (5 dx (9 dx (1) x3 + 4 ⋅ x5 dx (2 dx (6 cos 2 (ln x) ∫ x dx (12 ∫ ) : * 65 22 – ( ) : ∫x dx (3 −4 ∫ 2 2x + 5 (x x2 + x − 1 ∫ x3 − x dx (6 2 − 2 x + 1) ∫x 8x ∫ ( x − 2)2 ( x + 2) dx (9 ∫ (x 2 2 4 (1) x +1 ∫ ( x − 4) dx (2 ∫x ∫x 6 x2 + 4 x − 6 ∫ x3 − 7 x − 6 dx (7 9 x + 36 dx (11 + 6 x2 + 9 x 3 2x2 + x − 1 ∫ ( x 2 + 1)( x − 3) dx (15 ∫x 2 dx (1 2− x dx (4 2 + 5x x dx (5 + 5x + 6 10 x ∫ x 4 − 13x 2 + 36 dx (8 dx (12 − 2 x + 1)( x 2 − 4 x + 4) 2 5− x dx (10 3 + x2 ∫x 1 dx (14 + x +1 ∫x 2 1 dx (13 + 2x + 3 1 ∫ x( x 2 + 1)2 dx (18 3 ∫ ( x 2 + 1)( x 2 + 4) dx (17 2x2 + 2x + 1 ∫ ( x 2 + 1)( x + 2) dx (16 x 4 + 2 x3 − 10 x 2 − 8 x dx (21 ∫ x+4 3 x3 − 5 x 2 + 4 x − 2 dx (20 ∫ x −1 25 x 2 ∫ ( x − 1)( x 2 + 4)2 dx (19 x4 − 4 x2 + x + 1 ∫ x 2 − 4 dx (24 x4 − 2 x3 + x2 + x ∫ ( x − 1)2 dx (23 12 x3 − 11x 2 + 6 x − 1 dx (22 ∫ 4x − 1 : ∫1+ ∫ 1 4 x −1 dx (3 ∫ dx 3 © ¯ – x+ x (2 1 ∫ 1 + e x dx (5 1 + e x dx (6 ! www.GooL.co.il (2) ¯ ∫ dx 3 x−x (1 3 x2 ∫ x + 1 dx (4 * 66 23 – ( ) ( ) : ∫ sin 1 (3 10 x ∫ (sin x + cos x) 2 1 ∫ (sin x cos x) ∫ (sin 4 2 ∫ sin 4 3 ∫ ( cos 2 4x dx (2 x ∫ (sin 2 x − 4 cos 3 )dx (1 ) ∫ ( cos xdx (8 ∫ sin x cos x cos 2 xdx (7 ∫ (cos x cos 2 x + sin x sin 2 x)dx (11 ∫ sin 7 x cos 5 xdx (10 dx (6 4 x − sin 4 x dx (5 ∫ tan dx (9 x + cos 4 x)dx (12 ∫ cos 1 ∫ cos 2 (1) ∫ sin xdx (15 2 ∫ cos 2 xdx (18 sin 2 x − cos 2 x + 1 2 4 xdx (14 4 xdx (17 sin 5 x − sin x 2 ) x − sin 2 x dx (4 ∫ cos ∫ sin 3 2 xdx (13 4 xdx (16 1 + cos 2 x ∫ sin 2 x + cos 2 x + 1 dx (21 ∫ sin 4 x − sin 2 x dx (20 ∫ 1 − cos 2 x dx (19 2 4 ∫ sin x cos xdx (24 1 + cos3 x ∫ cos2 x dx (23 2 sin 3 x ∫ 1 − cos x dx (22 www.GooL.co.il © ¯ – ¯ 67 ( ) :¯ sin x = t ∫ f (sin x) ⋅ cos xdx = ( x = arcsin t ) = ∫ f (t )dt cos x = t ∫ f (cos x) ⋅ sin xdx = ( x = arccos t ) = ∫ f (t ) ( −dt ) : ∫ cos ∫ sin 5 3 xdx (3 ∫ (cos 3 ∫ (sin x + cos x − 2)sin xdx (2 ∫ sin x cos 4 xdx (6 4 1 ∫ cos x dx (9 2sin x 2 x + sin x + 2)cos xdx (1 ∫ sin x cos5 xdx (5 ∫ tan ∫ cos 2 x + 4cos x + 7 dx (12 (2) ∫ sin 2 x ⋅ e 5 2 xdx (4 ∫ cos xdx (8 cos x 3 5 xdx (7 dx ∫ sin x (10 dx (11 ( ) :¯ ∫ f ( sin x, cos x ) dx = x =∫ 2 ( x = 2arctan t ) t = tan 2t 1 − t 2 2 f , dt 2 2 2 1+ t 1 + t 1 + t : cos x dx ∫ 2 − cos x dx (3 www.GooL.co.il © ∫ 1 + sin x + cos x (2 ¯ – ¯ (3) 1 ∫ 1 + sin x (1 68 ( ) ) x = a sin t ( a 2 − x 2 dx = ∫f( a 2 + x 2 dx = ∫f( a −a sin t x − a dx = dt cos t = ∫ f ( a tan t ) ⋅ 2 cos t (t = arccos a ) x ∫ f ) 2 2 x (t = arcsin a ) = ∫ f ( a cos t ) ⋅ ( a cos tdt ) x = a tan t a a =∫f dt ⋅ 2 (t = arctan ) cos t cos t x a x= ) : ∫ ∫ 2 dx (9 + 2 x + 5)3/ 2 dx (2 2 ∫x dx (5 ∫x x +4 x2 ∫ x 2 + 2 x − 3dx (6 ∫ (x 1 ∫ 4 x 2 − 1dx (3 4 − x2 ∫ dx 22 (4 + x ) ∫ (8 ! www.GooL.co.il © (4) ¯ – ¯ dx 2 4 − x2 dx 2 x2 − 1 (1 (4 −6 x − x 2 dx (7 * 69 24 – : 1 −x ∫ xe dx (3 0 π ∫ cos (10 x ) dx (6 2 0 (1) 2 4 4x + 1 ∫ 2 x2 + x + 5 dx (2 1 ∫ (x 2 − 4 x + 1)dx (1 1 4 e ln 4 x ∫ x dx (4 1 1 ∫ x 2 + 4 x + 5 dx (5 1 x 0 ≤ x <1 . f ( x) = 1 2 x ≥1 x 4 . ∫ 4+ | x − 1|dx (8 −1 4 ¯ ∫ f ( x)dx : π /2 ∫ 0 4 4 (2) π sin x dx (2 4 sin x + cos x :¯ a x sin x ∫ 1 + cos 0 2 x dx (1 .f (3) a −a (7 0 0 . ∫ f ( x)dx = 2∫ f ( x)dx f . f . a . ∫ f ( x)dx = 0 −a : 4 sin x + 1 ∫4 x 2 + 1 dx (2 − www.GooL.co.il © ¯ – ¯ (4) 1 ∫ (x −1 3 + x 5 ) cos x dx (1 70 : 2 2 2 ≤ ∫ e x dx ≤ 2e 4 (3 2 6≤ 14 π π /2 ≤ ∫ 0 ∫ dx π dx ≤ (6 6 3 + 4sin 2 x 1 3 ln 4 3 4 ≤∫ dx 3 3 ln x ≤ 1 3 ln 3 1 −10 e≤ 2 (5 1 10 ∫ 0 e− x dx ≤ 1 (4 x + 10 1 π4 1 1 sin x ln( x + 1) x 2 arctan (9 − ≤ ∫ x ⋅ sin dx ≤ dx ≤ (8 ∫ 6 20 2 x+4 x +1 0 dx 2 2 ≤∫ ≤ (7 3 9 −1 8 + x 7 : 14 + 24 + 34 + ... + n 4 (1 n →∞ n5 lim sin lim n →∞ n 1 2 + sin + ... + sin n n n (2 n 1 1 1 lim + + ... + (3 n + n n +1 n + 2 n →∞ n n n lim 2 2 + 2 + ... + 2 (4 2 n →∞ n + 1 n +2 n + n2 1 1 1 lim + + ... + (5 2 2 2 2 2 2 n +2 n +n n +1 n →∞ n + 1 + n + 2 + ... + 2n lim (6 3/2 n →∞ n www.GooL.co.il © (5) 2 dx ≤∫ ≤ 4 (1 41 −1 1 + x 4 1 + x 2 dx ≤ 6 17 (2 −4 0 π ¯ ¯ – ¯ (6) 71 :( ) (7) π 1 1 1 ∫ sin xdx (4 3 ∫ x dx (3 2 ∫ x dx (2 ∫ xdx (1 0 0 0 0 : ¯ 1 + 2 + 3 + ... + n = 0.5n( n + 1) 1 n( n + 1)(2n + 1) 6 1 13 + 23 + 33 + ... + n3 = n 2 ( n + 1) 2 4 n + sin ( 2 α ) sin ( n2 1 α ) sin α + sin 2α + ... + sin nα = sin ( α ) 2 12 + 22 + 32 + ... + n 2 = www.GooL.co.il © ¯ – ¯ * 72 25 ( ) : f ( x) = x 2 + 4 x + 6 y (1) g ( x) = x 2 − 4 x + 14 . . . x -2 , x 2 .( )x= 2 x=2 (2) ) y = − x2 + 6x − 5 .( y . . . ? x . , .( ) y (3) f ( x) = ( x − 2) 2 .( , x .( x www.GooL.co.il © ) y = 0.5 x + 0.5 ¯ – ¯ ) 73 : f ( x) = x y (4) 2 g ( x) = − x 2 + 18 A ¯ B .( A B .B A ) . x x . 4 , x .x=4 : (5) y = − x 2 + 3x + 2 y y = x 3 − 3x + 2 x . x . . . , (6) . f ( x) = − x 2 + ax y ) A( 2,8) A .( .a O(0,0) x O x .B B . ¯ . .B . AB , .x www.GooL.co.il © ¯ – ¯ 74 (7) : y f ( x) = e − x + 2 g ( x) = e x . S1 .y S2 x . . S )1 S2 .( . (8) . f ( x) = e −2 x y ) x = −1 .( . . x , . ) .( y 0≤ x≤4 (9) y = cos 2 x .( .x= 4 . x , © . . .y www.GooL.co.il ) π ¯ – ¯ 75 (10) y x=3 y =1 y= 1 2x − 1 .( ) x (11) . f ( x) = e 2 x − e x . ¯ y . x . x . , ¯ .x ,x , ¯ , 3e 2 a − e a , x=a . a < ln 0.5 .a ) f ( x) = e .( y ,A (12) x +1 2 , . .A e2 2 . . x .A , . .y (13) www.GooL.co.il © ¯ – ¯ 76 .x>0 y f ( x) = 8 −2 x A .( ) A( 2, 2) . . x , . )x .( : (14) f ( x) = sin x ; 0 ≤ x ≤ π g ( x) = cos 2 x ; 0 ≤ x ≤ π . ¯ . . . .− .x =− π 2 (15) f ( x) = tg 2 x <x≤0 π . 4 ∫ tg , 2 xdx = tgx − x + c ¯ . .x . y = x 2 − 10 x + 25 . . . www.GooL.co.il © (16) A(8,0) ¯ ¯ – ¯ . 77 (17) f ( x) = x x + 4 y .x≥0 ( ) . . (0,0) x . .y , . f ( x) = cos3 x y = cos 2 x ⋅ sin x (18) . . x 1 3 . π ≤x≤ π 2 2 . , * y2 = −x ¯ (19) x = y2 + 2 ¯ (20) .y = x+6 . y = x −8 a . ∫ a 2 − y 2 dy . −a a . ∫ x 2 − a 2 dx . : (21) 0 ( ) : (1 ≤ x ≤ 2 ) y = x5 1 + 3 (3 15 4 x (1 ≤ x ≤ 8 ) y = x 2 / 3 (2 (1 ≤ x ≤ 8) x 2 / 3 + y 2 / 3 = 4 (6 ( 0 ≤ x ≤ 3) y = (1 ≤ x ≤ 2 ) y = x 2 (9 (22) 1 x (3 − x) (5 3 (1 ≤ x ≤ 2 ) y = ln x (8 www.GooL.co.il © ¯ – ¯ (1 ≤ x ≤ 2 ) y = ( 0 ≤ x ≤ 3) y = x4 1 + 2 (1 8 4x 2 (1 + x 2 )3/ 2 (4 3 ( 0 ≤ y ≤ 4 ) x = 3 y 3/ 2 − 1 (7 78 26 ( , ,y ) , x (1) . .x (cavalieri) y = x2 y = 2x ¯ (2) :¯ . (cavalieri) . . .y . y = x2 y = 2x ¯ (3) :¯ . (cavalieri) . . y x = -1 . f ( x) = 1 − x3 x=2 ¯ (4) : .y=2 x . . y = −1 . .x . .x=2 y=2 . . x = −1 . .y . ? y = -1 . ¯ (5) ¯ (6) ¯ (7) ¯ (8) . . ¯ y y = sin ( x 2 ) y = 0,x = . x www.GooL.co.il © 6 ,x= π 3 .y ¯ – π ¯ : 79 y y = ex 2 ¯ y = 0 , x = 0, x = 1 : . (9) .y x y , f ( x ) = x ln x ¯ x (10) ( e, e ) ? .x x .y (11) x −1 ≤ x ≤ 1 .x y = 4 − x2 (12) ? . . ¯ ¯ ¯ −2 ≤ y ≤ 2 y , x = 9 − y2 (13) (14) (15) ? .a , h .c b (16) (17) a www.GooL.co.il © ¯ – ¯ 80 27 ." : , a ( x ), b( x ) ( ) (1) ¯¯ f ( x) (2) b( x) I ( x) = ∫ f (t ) dt ⇒ I '( x) = f (b( x))b '( x ) (1 a b( x) I ( x) = ∫ f (t )dt ⇒ I '( x) = f (b( x))b '( x) − f ( a ( x))a '( x) (2 a( x) : x2 I ( x) = ∫ x3 dt 1 + t4 x3 + x (4 I ( x) = ∫ (3) x3 t ln tdt (3 I ( x) = 2 x ln t −t 2 ∫ t 2 dt (2 I ( x) = ∫ e dt (1 1 2 : (4) x x lim x→4 x t2 ∫ e dt (3 x−4 4 lim + x →0 1 x3 x 2 ∫ sin tdt (2 0 lim x →0 tdt ∫ cos t 0 sin 2 x (1 x F ( x) = ∫ (t + 1)4 (t − 1)10 dt : (5) 0 . , www.GooL.co.il © , ¯ – ¯ 81 28 ( ¯) : 1 ∫x (4 2 x +1 0 ∞ 1 dx ∞ 1 ∫ x 2 (8 −∞ (1) 1 dx ∫ sin x ⋅ x 2 (3 0 ∞ sin x ⋅ ln x ∫x 2 x2 − 4 3 arctan x ∫ 1 + x 4 dx (3 1 e3 x ∫ 1 + x 2 dx (8 −∞ . x=5 1 ∫ 1 + x 4 dx (7 0 ,y x www.GooL.co.il © − x2 ∞ ∫ 2 x3 + 1 dx (6 x (5 1 (2) x2 + 2 x + 1 ∫ x 4 + 4 x 2 + 5 dx (1 1 ∞ ∫( 1 ) x 2 + 1 − x dx (5 , y = e2 x x =1 x ∫ xe ∞ x2 + 2 x + 1 ∫ x3 + 4 x 2 + 5 dx (2 1 ∞ 2 . x ≤1 ∞ (1 ∞ ¯ ∞ dx (4 22 1 x ∫ x2 + 5 (6 1 1 xdx ∫ (1 + x ) ∞ 2 −2 x ∫ x e dx (7 : ∞ ∞ dx ∫ (1 + x) x (2 1 (3) 1 x (4) , y= ¯ – ¯ 82 – x → −∞ x→0 x →∞ 1 1 1 1 1 =0 = ∞, − = −∞ =0 x −∞ 0+ 0 ∞ ______________________________________________________________________ y= e −∞ = 0 y = ex e0 = 1 e∞ = ∞ ______________________________________________________________________ y = ln x ln(0+ ) = −∞ −−− ln(∞) = ∞ ______________________________________________________________________ y = arctan x atan(−∞) = − π π atan(0) = 0 atan(∞) = 2 2 _____________________________________________________________________ a −∞ = 0 y = ax , a > 1 a0 = 1 a∞ = ∞ y = ax , 0 < a < 1 a −∞ = ∞ a0 = 1 a∞ = 0 _____________________________________________________________________ y = sin x −−− sin 0 = 0 −−− _____________________________________________________________________ −−− cos 0 = 1 −−− y = cos x _____________________________________________________________________ sin x y= 0 1 0 x _____________________________________________________________________ tan x −−− 1 −−− x _____________________________________________________________________ y= 1 y = 1 + x x (from right) 1 e e e 1 1 y = (1 + x) x −−− _____________________________________________________________________ y= x 0+ = 0 −−− ∞ =∞ 3 3 y=3 x −∞ 0 =0 ∞ =∞ _____________________________________________________________________ Defined Limits: ∞ ⋅ ∞ = ∞ , ∞(−∞) = −∞ , Undefined Limits : ∞+∞ =∞ , ∞ ± a = ∞ , ∞ ⋅ (± a ) = ±∞ , 0∞ , , ∞ − ∞, 0 ⋅ ∞, 1∞ , 00 , ∞ 0 0∞ www.GooL.co.il © ¯ – ¯ ∞ / (± a) = ±∞ 83 – 1. y = a → y'= 0 → y ' = n ⋅ f n −1 ⋅ f ' y'= e f ⋅ f ' 4. y = a f → y ' = a f ⋅ f '⋅ ln a 5. y = ln f → y'= 6. y = sin f → f y ' = cos f ⋅ f ' 7. y = cos f → y ' = − sin f ⋅ f ' 8. y = tan f → y'= 2. y = f n 3. y = e f → 1 ⋅f' 1 ⋅f' 2 cos f 9. y = cot f → 1 y'= − ⋅f' 2 sin f 10. y = arcsin f → 1 y'= 1− f 11. y = ar cos f → 2 1 y'= − 1− f 12. y = arctan f → 1 y'= 1+ f 13. y = ar cot f ⋅f' 2 ⋅f' 2 ⋅f' 1 → y'= − 14. y = sinh f → 1+ f y ' = cosh f ⋅ f ' 15. y = cosh f → y ' = sinh f ⋅ f ' 16. y = tanh f → y'= 2 1 ⋅f' ⋅f' 2 cosh f 17. y = coth f → 1 y'= − ⋅f' 2 sinh f 18. y = f ( x) g ( x) → y ' = f ( x) g ( x ) ⋅ ( g ( x) ⋅ ln( f ( x)) ' www.GooL.co.il © ¯ – ¯ 84 – ∫ adx = ax + c x n +1 + c n ≠ −1 n +1 n ∫ x dx = 1 n ∫ (ax + b) dx = 1 1 (ax + b) n +1 + c n ≠ −1 a n +1 1 ∫ x dx = ln | x | +c ∫ ax + b dx = a ln | ax + b | +c ∫ e dx = e 1 ax + b e +c a 1 k ax + b k ax + b dx = +c ∫ a ln k 1 ∫ cos(ax + b)dx = a sin(ax + b) + c 1 ∫ sin(ax + b)dx = − a cos(ax + b) + c 1 ∫ tan(ax + b)dx = − a ln | cos(ax + b) | +c 1 ∫ cot(ax + b)dx = a ln | sin(ax + b) | +c 1 1 ∫ cos2 (ax + b) dx = a tan(ax + b) + c 1 1 ∫ sin 2 (ax + b) dx = − a cot(ax + b) + c −−−−−−−−−−−−−−−−−−−−− x x ∫ k dx = x ∫e +c kx +c ln k ∫ cos xdx = sin x + c ∫ sin xdx = − cos x + c ∫ tan xdx = − ln | cos x | +c ∫ cot xdx = ln | sin x | +c 1 ∫ cos 2 x dx = tan x + c 1 ∫ sin dx = − cot x + c x −−−−−−−−−−−−−−−−− 2 1 ax + b 1 dx = 1 1 ∫ cos x dx = ln | cos x + tan x | +c ∫ sin x dx = ln | sin x − cot x | +c ∫x 1 1 x dx = arctan + c 2 +a a a 1 x dx = arcsin + c 2 2 a a −x 2 ∫ −−−−−−−−−−−−−−−−− ∫ f' dx = ln | f | +c f ∫e f 1 1 x−a dx = +c ln 2 −a 2a x + a 1 2 2 dx = ln | x + x 2 ± a 2 | +c x ±a −−−−−−−−−−−−−−−−−−−−− 1 2 +c ∫ cos f ⋅ f ' dx = sin( f ) + c ∫ sin f ⋅ f ' dx = − cos( f ) + c ∫ ∫ 2 ∫ f ⋅ f ' dx = 2 f ⋅ f ' dx = e f + c 3 f ⋅ f ' dx = ∫x 22 f +c 3 ∫ f' dx = 2 f + c f ∫ u ⋅ v ' dx = u ⋅ v − ∫ u '⋅ vdx www.GooL.co.il © ¯ – ¯ 85 – sin 2 α + cos 2 α = 1 sin α tan α = cos α cot α = cos α sin α sin 2α = 2sin α cos α 2 2 2 2 cos 2α = cos α − sin α = 1 − 2sin α = 2cos α − 1 1 2 1 + tan α = cos 2 α 1 + cot 2 α = 1 sin 2 α 1 2 sin α = 2 (1 − cos 2α ) cos 2 α = 1 (1 + cos 2α ) 2 1 sin α cos β = 2 ( sin( a + β ) + sin(α − β ) ) 1 sin α sin β = ( cos( a − β ) − cos( + β ) ) 2 1 cos α cos β = 2 ( cos( a + β ) + cos(α − β ) ) x = α + 2π k sin x = sin α ⇒ x = (π − α ) + 2π k x = α + 2π k cos x = cos α ⇒ x = −α + 2π k tan x = tan α ⇒ x = α + π k cot x = cot α ⇒ x = α + π k sin x = 0 ⇒ x = π k π cos x = 0 ⇒ x = 2 + π k www.GooL.co.il © ¯ – ¯ 86 – (a + b)2 = a 2 + 2ab + b 2 2 2 2 (a − b) = a − 2ab + b (a + b)3 = a 3 + 3a 2b + 3ab 2 + b3 3 3 2 2 3 (a − b) = a − 3a b + 3ab − b (a + b)4 = a 4 + 4a3b + 6a 2b 2 + 4ab3 + b 4 (a − b) 4 = a 4 − 4a3b + 6a 2b 2 + 4ab3 + b 4 a 2 + b 2 = (a + b) 2 − 2ab 2 2 a − b = (a − b)(a + b) a 3 + b3 = (a + b)(a 2 + b2 − ab) 3 3 2 2 a − b = (a − b)(a + b + ab) a 4 + b 4 = ( a 2 + b 2 ) 2 − 2a 2b 2 a 4 − b 4 = (a 2 − b 2 )(a 2 + b 2 ) a m a n = a m + n m a = a m−n an n a m = a mn (ab)n = a nb n n an a = n b b 0 a = 1 a − n = 1 an m 1 a = a 2 , n am = a n a x = b ⇒ x = ln b a > 0, b > 0 ln a + ln b = ln ab a ln a − ln b = ln b ln1 = 0 , ln e = 1 n ln e = n ln x n = n ln x ( x > 0) eln x = x b b ln a a = e k ln x = k ⇒ x = e () a b = a ⋅d −b⋅c c d a b c ef d d e f =a −b hi g g h i f i +c www.GooL.co.il © d e g a if a ≥ 0 2 | a |= a = − a if a < 0 | a ⋅ b |=| a | ⋅ | b | a |a| = b |b| | x |< a ⇔ − a < x < a | x |> a ⇔ x < − a or x > a h ¯ – ¯ 87 ¯ ∞ xn x1 x 2 x3 = 1 + + + + ... 1! 2! 3! n =0 n ! ex = ∑ ∞ sin x = ∑ (−1) n n =0 ∞ cos x = ∑ (−1) n n =0 x 2 n+1 x3 x 5 x 7 = x − + − + ... (2n + 1)! 3! 5! 7! −∞ < x < ∞ x2n x 2 x 4 x6 = 1 − + − + ... (2n)! 2! 4! 6! −∞ < x < ∞ ∞ ln(1 + x) = ∑ (−1)n n =0 ∞ arctan x = ∑ (−1) n n =0 −∞ < x < ∞ x n+1 x 2 x3 x 4 = x − + − + ... 234 n +1 −1 < x ≤ 1 x 2 n+1 x 3 x5 x 7 = x − + − + ... 2n + 1 357 −1 ≤ x ≤ 1 ∞ 1 = ∑ x n = 1 + x1 + x 2 + x3 + ... 1 − x n =0 −1 < x < 1 ∞ m(m − 1) ⋅ ... ⋅ (m − n + 1) n x n! n =1 m(m − 1) 2 m(m − 1)(m − 2) 3 = 1 + mx + x+ x + ... 2! 3! (1 + x)m = 1 + ∑ www.GooL.co.il © ¯ – ¯ −1 ≤ x ≤ 1 (m > 0) −1 < x ≤ 1 (−1 < m < 0) −1 < x < 1 (m ≤ −1) m ≠ 0,1, 2,3,... ...
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