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Guia1 ua-2010
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Guia1 ua-2010

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primera guía de cálculo

primera guía de cálculo

Published in: Education
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  • 1. UNIVERSIDAD AUTONOMA FACULTAD CS. EMPRESARIALES CALCULO I Profesor: Gustavo Benavente K. Ayudante: Darío Guerrero Guía 1. Inecuaciones y Valor Absoluto 1) Inecuaciones de primer grado a) ( x - 2 )2 > (x + 2)⋅ ( x - 2) + 8 R. ] - ∞ , 0 [ x2 – 4x + 4 > x2 – 4 + 8 x2 – 4x + 4 > x2 + 4 -4x > 0 /*(-1) x<0 b) ( x - 1 )2 < x ( x - 4) + 8 x2 – 2x + 1 < x2 – 4x + 8 x2 – 2x + 4x - x2 < 8-1 2x < 7  x <7/2 R. ] - ∞ , 7/2 [ c) 3x - 5 - x - 6 < 1 /*(12) 4 12 3(3x-5)-(x-6)<12 9x-15-x+6<12 8x<12+9 R. ] - ∞ , 21/8 [ x<21/8 d) 3 - ( x - 6) ≤ 4x – 5 R. [ 14/5 , + ∞ [ e) 1 - x - 5 < 9 + x R. ] -67/10 , + ∞ [ 9 f) Determine en cada uno de los siguientes ejercicios el intervalo real para x, tal que cada expresión represente un número real. i) x + 5  x + 5 > 0 2 x2 − 1 ii) iii) x+6 x −1 R. [ -5 , +∞ [ R. ] - 6 , +∞ [ R. [ - 1 , 1 [ ∪ ] 1, + ∞ [ 2) Inecuaciones de segundo grado a) x2 ≥ 16 R. IR - ] -4 , 4[ b) 9x2 < 25 R. ] - 5/3 , 5/3 [ c) 36 > ( x - 1) 2 R. ]-5,7[ d) (x + 5)2 ≤ ( x + 4 ) 2 + ( x - 3 )2 R. IR - ] 0 , 8 [ e) x ( x - 2 ) < 2 ( x + 6) R. ]-2,6[ f) x2 - 3x > 3x - 9 R. IR - 3 h) 2x2 + 25 ≤ x ( x + 10 ) R. 5 2x2 + 25 ≤ x2 + 10x 2x2 + 25 - x2 - 10x ≤ 0 x2 – 10x +25 ≤ 0 (x – 5)2 ≤ 0
  • 2. i) 1 - 2x ≤ (x + 5)2 - 2(x + 1) R. IR j) 3 > x ( 2x + 1) R. ] -3/2 , 1 [ k) x ( x + 1) ≥ 15(1 - x2 ) R. IR - ] -1 , 15/16 [ l) ( x - 2 ) 2 > 0 R. IR - 2 m) ( x - 2)2 < 0 R. ∅ 3) Inecuaciones fraccionarias x R. IR - [ 0 , 1 ] a) >0 x −1 x+6 R. IR - [ -6 , 3 ] b) <0 3− x x R. [ 5 , 10 ] c) −2≥ 0 x−5 2x − 1 R. ] - ∞ , -5 [ d) >2 x+5 x −1 R. ] -11 , -5 [ e) >2 x+5 1 R. ] - ∞ , 3 [ f) ≤0 x−3 x −1 R. IR - [ -1 , 1 [ g) ≥0 x +1 −1 R. ] - 1/2 , 0 [ h) >2 x x x R. ] - ∞ , -1 [ ∪ [ 0, 3 [ i) ≤ x − 3 x +1 x2 + 2 R. ] - 3 , 2/3 [ j) >x x+3 x2 R. IR - ]-3/2 , 3 ] k) ≥ x +1 x−3 x2 − 4 R. ] - 6, -2 ] ∪ [ 2 , +∞ [ l) ≥0 x+6 ( x + 1)( x − 7) R. ] -3, -1 [ ∪ ] 1 , 6 [ ∪ ] 7 , + ∞ [ m) >0 ( x − 1)( x − 6)( x + 3) El Valor Absoluto  x, si a ≥ 0  x= − x si a < 0  Propiedades útiles: 1) x < a ↔ −a < x < a 1) x > a ↔ x > a ∨ x < − a
  • 3. Ejercicios resueltos 1) 2 x + 3 < 9 − 9 < 2x + 3 < 9 − 12 < 2 x < 6 −6< x <3 R : ] − 6, 3[ 2x 2) + 3 > 11 3 2x − 11 > + 3 > 11 3 2x − 11 − 3 > > 11 − 3 3 2x − 14 > >8 3 − 42 > 2 x > 24 − 21 > x > 12 R : ] − 21, 12[ Ejercicios Propuestos 1. 5x - 7 ≤ 3 2. 3 - x < 4 3. 5x - 2x ≤ 7 4. 2x - 3 ≤ −5 5. 5x + 2 > 7 6. 2 3 - x − 10 ≥ 0 7. x + 3 ≥ 2 8. 2x - 1 > 3 R. IR - [ -1 , 2 ] x R. [ 2 , 10 ] 9. 3 − ≤ 2 2 x 1 R. IR - ] -45/2 , 55/2 [ 10. − ≥ 5 5 2 x R. ] 0 , 6 [ 11. 1 − < 1 3 12. x - 3 > -1 R. ] - ∞ , +∞ [ 2x − 1 R. [ - 2/3 , 4 ] 14. ≤1 x+3 15. 3 - 2x < x + 4 R. ] - 1/3 , 7 [

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