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0, x = −y . 5 g 1 (x, y) = 1, g 2 (x, y) = ⎧ ⎨ ⎩ 0, x 2 = y 2 1, x 2 = y 2 &% 5 A ε = 1 50 , π − 1 50 ∪ π + 1 50 , 2π − 1 50 . 5 A ε = 0, 1 − 1 101 !5 A ε = 1 1001 , 1 . 5 A ε = 10 −4 2 , 1 5 A ε = 1 5 · 10 5 − 1, − 1 5 · 10 5 ∪ 1 5 · 10 5 , 1 − 1 5 · 10 5 5 A ε = 0, 2 − 1 2 · 10 6 . && 2 μ (R) = ∞. &'&. f (x) ≡ 0 '/ f (x) ≡ 0 '$ _ &9 "0 # * " "4 - "4 > # 8#" A 1 = {x ∈ E : χ A (x) = 0} = E\A, A 2 = {x ∈ E : χ A (x) = 1} = A & ! χ A ! ! A 1 = {x ∈ E : sign x = −1} = [−1, 0), A 2 = {x ∈ E : sign x = 0} = {0} A 3 = {x ∈ E : sign x = 1} = (0, 3] & ! " y = sign x E ! % K : [0, 1] → R ! 7%`" K [0, 1] \K ! n ∈ N ! K n 38%" ! 5 K ! 2 ! K ∞ ∪ n =1 K n = [0, 1] \K ' 5 ∞ n =1 n · 2 n −1 3 n . b ) 1 4 . f n (x) = [nK(x)] n . % ' y 1 = 0, y 2 = 1, y 3 = 2, y 4 = 3, y 5 = 4, A 1 = [0, 1), A 2 = [1, 2), A 3 = [2, 3), A 4 = [3, 4), A 5 = [4, 5) & , 8%" ( 10 ' 4π 3 . \ ! (0 1) , [0, 3] & ! 0 / Z ! 0, 1, 1 2 , . . . , 1 n , . . . " , [0, 1] & ! 0 3 2 1 1 $ 4 % 3 2 . & e − 2 $% 8a" $& 5 − √ 2 − √ 3. $' . $. 0 %/ 12 % 0 % 2 % −6 % 4 ln 2 + 5 %$ e + 1 2 . %% 5 > 5 ln 4 − 1, !5 7 5 5 + 3 √ 2 + √ 3 2 . %& 5 1 2 , 5 6, !5 10, 5 2 ln 2 + 1, 5 ln 3 + e, 5 1 2 , 5 1 2 , 5 1 2 , 5 1, +5 5 16 , 5 0. '9 " "4 - "4 "4 4 5 _ 5 _ !5 _ 5 _ 5 5 5 0, 5 1, !5 0, 5 π. % 5 α > 1, 5 α > 1, !5 α > 2. & α > 1. ' / 5 π, 5 π 2 . .9 -- * +4 # "4-4- ,-0) / = 0, 1, 2, 3 '! 1 = x = 0 x = 0 ! & x = 0 2 = x = 0 x = 0 ! & 1 [−4, 0) [0, 5] ' x = −1, x = 0. x = −1 3 x = 0 ! & x = 0 1 & f d (x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 0, x ∈ [−2, −1) 1, x ∈ [−1, 0] 2, x ∈ (0, 2], f c (x) = x − 1. ' f d (x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 0, x ∈ [−10, −2) 1, x ∈ [−2, 0) 5, x ∈ [0, 4], f c (x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ x 3 , x ∈ [−10, −2) −8, x ∈ [−2, 0) x − 8, x ∈ [0, 4]. . f d (x) = ⎧ ⎨ ⎩ 0, x ∈ , − π 2 , 0 1, x ∈ , 0, π 2 - , f c (x) = ⎧ ⎨ ⎩ sin 3 (x) − 1, x ∈ , − π 2 , 0 sin 2 −1, x ∈ , 0, π 2 - . / f (x) = f c (x) + f d (x), f d (x) = [x] + 2, f c (x) = 2x − 2 . / & $ `7 V [f] = 6, `8 V [f] = 20, `W V [f] = 2, `$ V [f] = 8, b> V [f] = 2, b# V [f] = ln(1 + e), b% V [f] = 32 3 4 , b` V [f] = e 2 + 1, bb V [f] = 6. $$ 2 ∨ 0 [f] = 3, 2 ∨ 0 [g] = 7. $% 2 ∨ 0 [f] = 4+|a|+|1 − a| ! a ∈ [0, 1] 2 ∨ 0 [f] %/ f (x) = v(x) − ϕ(x), v(x) = ∨ x a [f] v f (x) = ⎧ ⎨ ⎩ sin x, x ∈ , 0, π 2 - 2 − sin x, x ∈ π 2 , π - ϕ f (x) = ⎧ ⎨ ⎩ 2 sin x − 1, x ∈ , 0, π 2 - 1, x ∈ π 2 , π - v g (x) = x ∨ 0 [g] = 1 − cos x, x ∈ [0, 2π] ϕ g (x) = |cos x| + cos x v φ (x) = = ⎧ ⎨ ⎩ 4 − (x − 2) 2 , x ∈ [0, 2] 4 + (x − 2) 2 , x ∈ (2, π] , ϕ φ (x) = ⎧ ⎨ ⎩ 4 − 2(x − 2) 2 , x ∈ [0, 2] 4, x ∈ (2, π]. v ψ (x) = ⎧ ⎨ ⎩ sin 2 x, x ∈ , 0, π 2 - 2 − sin 2 x, x ∈ π 2 , π - , ϕ ψ (x) = ⎧ ⎨ ⎩ 0, x ∈ , 0, π 2 - 2 − 2 sin 2 x, x ∈ π 2 , π - % {36. v(x) = 3x, ϕ(x) = −1. 37. v(x) = ⎧ ⎨ ⎩ 2 − 2x 2 , x ∈ [−1, 0] 2 + 2x 2 , x ∈ (0, 3], ϕ (x) = ⎧ ⎨ ⎩ −4x 3 − 3, x ∈ [−1, 0] −3, x ∈ (0, 3]. 38. v(x) = ⎧ ⎨ ⎩ sin x, x ∈ , 0, π 2 - 2 − sin x, x ∈ π 2 , π - , ϕ (x) = ⎧ ⎨ ⎩ 0, x ∈ , 0, π 2 - 2 − 2 sin 2 x, x ∈ π 2 , π - . 39. v(x) = ⎧ ⎨ ⎩ 2 cos x + 2, x ∈ [−π, 0] 6 − 2 cos x, x ∈ [0, π], ϕ (x) = ⎧ ⎨ ⎩ 2, x ∈ [−π, 0] 6 − 4 cos x, x ∈ [0, π]. 40. v(x) = tgx + 1, ϕ(x) = 1. 41. v(x) = ln(1 + x), ϕ(x) = 0. 42. v(x) = 2 x + 5x + 9 3 4 , ϕ (x) = 9 3 4 . 43. v(x) = xe x +1 + 1, ϕ(x) = −4. 44. v(x) = 3 + 3(x − 1), ϕ(x) = 3x − 4 − 3 |x − 1| } . % λ l = −β, ∧ l = −α, λ r = a, ∧ r = b % _ 3$7a f 5 % * f [a, b] ! f [a, b] ! %. A & A = [a, b] b ∨ a [χ A ] /9 ) +0+ ,-0) " "48" - "4 ' n k =1 |f(b k ) −f(a k )| = n k =1 |3b k + 1 −3a k −1| = 3 n k =1 (b k −a k ) < 3δ δ = ε 3 n k =1 |f(b k ) − f(a k )| < ε + \ f (x) = 3x + 5 [0, 2] . a 2 − b 2 = (a − b)(a + b) #>#" ! / |sin x − sin y| ≤ |x − y| || cos x| − | cos y|| ≤ |cos x − cos y| ≤ |x − y| % ' x k = 1 {8. v(x) = 3x, ϕ(x) = −1. 9. v(x) = x 2 , ϕ (x) = −3. 10. v(x) = ⎧ ⎨ ⎩ sin x, x ∈ , 0, π 2 - 2 − sin x, x ∈ π 2 , π - , ϕ (x) = ⎧ ⎨ ⎩ 0, x ∈ , 0, π 2 - 2 − 2 sin 2 x, x ∈ π 2 , π - . 11. v(x) = ⎧ ⎨ ⎩ 2 + 2 cos x, x ∈ [−π, 0] 6 − 2 cos x, x ∈ (0, π], ϕ (x) = ⎧ ⎨ ⎩ 2 + 2 cos x − 2 |cos x| , x ∈ [−π, 0] 6 − 2 cos x − 2 |cos x| , x ∈ (0, π]. 12. v(x) = tg x 4 + 1, ϕ(x) = 1. 13. v(x) = x ln(1 + x), ϕ(x) = 0. 14. v(x) = 2 x + 5x + 4, 5, ϕ(x) = 4, 5. 15. v(x) = ⎧ ⎨ ⎩ 1 − √ −x x ∈ [−1, 0] 1 + √x x ∈ (0, 1], ϕ (x) = ⎧ ⎨ ⎩ 1 − 2 √ −x x ∈ [−1, 0] 1 + x ∈ (0, 1], #7 v (x) = 3x, ϕ(x) = 3x − 3|1 − x| − 4.} / f (x) = K(x), A = [0, 1] \K. f (x) = K(x), A = K. % = & $/ # $ ` $ #5 −π, %5 b `5 25 1 3 , b5 n (n + 1) 2 , a5 n (n + 1)(2n + 1) 6 , 75 2e −π/2 − 2e π /2 − (1 + π)e π + (1 − π)e −π , 85 2 − π, W5 3 8 , $5 % #>5 # 0 "4- " 5 * #"\ %"\ `"' b"' a"' 7"' 8"* W"* $"' #>"d ##"\ #%"' #`"* #b"' #a"' #7"\ #8"' #W"d #$"' %>"d %#"\ %%"d %`"\ %b"' %a"' %7"' %8"' %W"d %$"' `>"d `#"* `%"d ``"* `b"d `a"' `7"d `8"' `W"\ `$"\ b>"d 6 1 > ' 8 & 3##"#85 ' ##"&; " ! ! ! + #%"& ! " #`"& ! & " ! & & " #b"& ' ' ! ! ! & #a"& + #7"& " && = & & & #8"& + " 9 "0 ,+ ' & ! (: X & Download 1.57 Mb. Do'stlaringiz bilan baham: |
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