Mathematics. Don´t use a calculator.

Mathematics.

Don´t use a calculator.

1. Factorize 25x²81. (Write the expression as a product.) 2. Factorize 14x 4x².

3. Factorize 464x⁴. 4. Factorize ka² 9k6ak. 5. Factorize 12a ³ 108a.

6. Factorize ^{x4 2x3yx2y2.} 7. Factorize 2x²  x2 0,5.

8. Simplify ( xh  x)( xh  x)

9. Simplify ²

3

1

1 a

a

a 





10. Simplify ₂

4 2 2 1

a a

a  



11. Simplify 



 



 

 



 

 1 1

b a

a b a

12. Simplify

1 4 1

1 2

2 



  x x x

13. Simplify b a

b a



1 1

14. Simplify y y 3 3

1 1





15. Simplify 1 3

2 3 3

a a a







16. Factorize x³ x² x1.

17. Simplify

1 1

2 2 3







 x

x x x

18. Solve the equation x²  x2 30.

19. Solve the equation 2xx² 0. 20. Solve the equation 53x2x² 0.

21. Solve the equation 7x² x13 20. 22. Solve the equation 5x²  x6 20. 23. Solve the equation 3x1x1 24. Solve the equation 458x4x² 0

25. Factorize the polynomial p(x)458x4x² 26. Solve the inequality 12 x3 6.

27. Solve the inequality 458x4x² 0. 28. Solve the inequality x²  x5 60. 29. Solve the inequality 82xx² 0. 30. Solve the inequality 0.

1 ) 3 (

2 



 x

x x

31. Solve the inequality x x

x  



 3

1 3

2

32. Solve the inequality 1. 2

1 x x 



33. Solve the inequality . 2 1 1

  x x

34. Solve the inequality .

2 1 1 2

1

 

x  x x That is to find the x-values solving the two inequalities

x x

1 2

1 

 and .

2 1 1

  x x

35. Factorize x³ x² x1.

36. Solve the inequality x³ x² x1.

37. Simplify 2  8. (Remember; do not use a calculator.) 38. Simplify 9^0,5.

39. Simplify 49 2 98 . 40. Simplify 0,25 32.

41. Simplify 32^2/5.

42. Simplify _7/4

4 , 0

16 32

 .

43. Simplify





44. Simplify ³ 10 .

45. Simplify

 

 

2 10 , 1

2 8

2 

.

46. Simplify (3^x 3^x 3^x)².

47. Solve the equation (2^x 2^x12^x1)² 98.

48. Simplify

 

(

5 / 3 1

5 2

3 4 , 0

x x

x 

49. Simplify _{a b}

b a

x x x

2 3

3 3 ) (



 

.

50. Solve the equation x^h 1h.

51. Solve the equation ax(axb)^2/3(axb)^1/3 0

52. Simplify ln 8 ln2

54. Simplify

2 ln 1 2 ln 8 ln 8

ln   

55. Simplify

17 ln18 18 ln17 

56. Simplify

17 ln 9 18

ln17 

57. Simplify lnxlnx² ln2x 58. Find

y

x

59. Solve the equation lnx13ln2 60. Solve the equation 2lnxln2x 61. Solve the equation 2lnxln3ln12 62. Solve the equation lnx² 1lnx

63. Solve the equation ln(1x)ln(1x)ln0,75. 64. Solve the equation 3x1x1

65. Find the equation for the straight line passing through the points (-2; 4) and (6; - 2).

66. Find the equation for the straight line passing through the point (4; 1) and is parallel the line to the line 2y x3 40

67. Solve the system of equations















9 4

3

13 2

7 y x

y x

68. Solve the system of equations













650 150

100

1810 450

230

q p

q p

69. Solve the system of equations



























2 2

5 3

1 3

2

z y x

z y x

z y x

70. Find the derivative of f(x)7x⁹ 71. Find the derivative of f(x) xⁿ 72. Find the derivative of f(x) x 73. Find the derivative of f(x)⁵ x 74. Find the derivative of f(x)x^ 75. Find the derivative of f(x)e^x 76. Find the derivative of f(x)2e^3x 77. Find the derivative of f(x)4x⁵3e^6x 78. Find the derivative of f(x)7e^8x3x3 79. Find the derivative of f(x) x( 5)¹⁷ 80. Find the derivative of f(x) x( 5)^17

81. Find the derivative of f(x) x( ²5)¹⁷ 82. Find the derivative of f(x) x( ²5)^17

83. Find the derivative of f(x) x^{17 2} 5 84. Find the derivative of f(x)lnx

85. Find the derivative of f(x)x⁷ ln(x1) 86. Find the derivative of f(x)ln(1x²) 87. Find the derivative of ^f(x)ln





88. Find the derivative of f(x)xxlnx 89. Find the derivative of f(x)³12x⁴ 90. Find the derivative of f(x)ln³12x⁴

91. Find the derivative of f(x)e^3x2 ln(1x²)

92. Find the derivative of ₂ ) 1

( x

x x

f  

93. Find the derivative of

1 ) 1

( ₂

2



  x x x f

94. Find the derivative of _{2 2} ) 1 ( ) 4

(  

x x x f

95. Consider the function f(x)x² x6 having the domain 3x2 a) Give the global maximum and minimum of the function.

b) Give the range of the function.

96. Consider the function f(x) xe^x having the domain 1x2 a) Give the global maximum and minimum of the function.

b) Give the range of the function.

97. Consider the function f(x)x² x6

a) Give the largest possible domain to the function.

b) Find any zeros to the function, i.e. x-values where f(x)0.

c) Find the functions stationary points, i.e. x-values where whewre the derivative is zero.

d) Find the intervals where the function is increasing and where it is decreasing.

e) Classify the stationary points, i,e find any local minimum or maximum.

f) Find the intervals where the function is concave and convex.

g) Give any points of inflection.

h) Give the range of the function.

i) Sketch the graph.

98. Consider the function f(x)x³6x²9x and answer questions 97. a) – i).

99. Consider the function f(x) xe^x and answer questions 97. a) – i).

100. Consider the function ₂

2

1 ) 1 ) (

( x

x x

f 

  and answer questions 97. a) – i).

101. Consider the function ( ) ^ln₂ x x x

f  and answer questions 97. a) – i).

Answers.

1. (5x9)(5x9) 2. (12x)² 3. 4(14x²)(12x)(12x) 4. _{k( a 3)}² 5. 12a(a3)(a3) 6. x²(xy)²

7. 0,5(2x1)² 8. h 9. a1

10.

2 1

  a

11. _1 12.

) 1 (

1



 x x

x

13.

ab

 1 14.

3

1 15.

a

a 3

16. (x x1)( ²1) 17. x1 18. _{x1 1} x₂ 3

19. x₁ 1 x₂ 2 20. x₁ 1 x₂ 2,5 21. x₁ 1/7 x₂ 2

22. 5

19 3

1

 

x 5

19 3

2

 

x 23. x 5 24. x₁ 4,5 x₂ 2,5

25. p(x)4(x4,5)(x2,5) 26. x2 27. 2,5x4,5 28. 2 x3 29. x2 x4 30. x1 0 x3 31. 0 x1 x2 32. 0 x1 x2 33. x0 x2

34. x2 35. (x x1)( ² 1) 36. x1

37. 4 38. 1/3 39. 21

40. 2 41. 4 42. 2 ⁹

43. 5 44. 10^1/6 45. 1

46. 3^2x2 47. x 1,5 48. x

49. x^{6 a 3b} 50. x( 1 h)^1/h 51. xb/2a

52. 4ln2 53. 2ln2 54. 0

55. 0 56. ln2 57. ln2

58. e 59. e/8 60. x2

61. x6 62. x e 63. x₁ 0,5 x₂ 0,5

64. x5 65. y 0,75x2,5 66. y1,5x7

67. 



 3 1 y

x 68.







 3 2 q

p 69.











 4 , 1 8 , 0

0

z y x

70. _{f(x)63x}⁸ 71. f(x)nx^n1 72.

x x

x f

2 5 1

, 0 )

(  ^0,5 

 ^

73. f(x)0,2x^0,8 74. f(x)x^1 75. f )(x e^x

76. f(x)6e^3x 77. f(x)20x⁴18e^6x 78. _{f(x)7(89x}²_)e^8x3x3

79. f(x)17(x5)¹⁶ 80. f(x)17(x5)^18 81. f(x)172x(x² 5)¹⁶ 82. f(x)172x(x²5)^18 83.

17 / 16 2 17

/ 16 2

) 5 ( 17 ) 2

5 ( 17 2 ) 1

(     

 ^

x x x

x x

f

84.

x x

f 1

) ( 

 85.

1 7 1

)

( ⁶

 

 

x x x

f 86.

1 2

) 2

( x

x x

f   87.

4 2

3

1 4 ) 2

( x x

x x x

f  

 

 88. f(x)lnx 89. _{3 4 2/3}

) 2 1 ( 3 8 ) 1

(    ^

 x x x

f

90. 3(1 2 )

8 )

2 1 ( ) 1

2 1 ( 3 8 ) 1

( ₄

3

3 4

3 / 2 4 3

x x x

x x

x

f  

 





  ^

91. ^{3 2 3} ₂

1 ) 2

1 ln(

6 )

( ^{2 2}

x e x

x xe

x

f ^{x x}

 









  ^{ }

92. 2 2

2

) 1 ( ) 1

( x

x x

f 

 

 93. ₍₁ ²₎²

) 4

( x

x x

f  

94. 2 3

2

) 1 (

12 ) 4

( x

x x

f 

 



95. Maximum: f(0,5)6,25 Minimum: f(3)6 Range: 6 f(x)6,25

96. Maximum:

f e1 ) 1

(  Minimum: f( )1 e Ramge:

x e f

e 1

) ( 





97. a) Largest possible domain to the function is the real line.

b) Zeros: x₁3 x₂ 2 c) Stationary point: x0,5 d) x0,5 gives a maximum.

e) f(x) is increasing in the interval x0,5 f(x) is decreasing in the interval x0,5 f) f(x) is cancave in the domain.

g) No inflection point.

h) Range: f(x)6,25

98. a) Largest possible domain to the function is the real line.

b) Zeros: x₁ 3 x₂ 0

c) Stationary points: x₁ 3 x₂ 1

d) x3 gives a local maximum. x1 gives a local minimum.

e) f(x) is increasing in the intervals x3 x1 f(x) is decreasing in the interval 3x1

f) f(x) is concave in the interval x2 f(x) convex in the interval x2 g) Inflection point: x2

h) Range: the real line.

99. a) Largest possible domain to the function is the real line.

b) Zero: x0

c) Stationary point: x1 d) x1 gives a maximum.

e) f(x) is increasing in the interval x1 f(x)is decreasing in the interval x1 f) f(x) is concave interval x2 f(x) is convex in the interval x2

g) Inflection point: x2 h) Range:

x e

f 1

) ( 

100. a) Largest possible domain to the function is the real line.

b) Zeros: x1

c) Stationary points: x1 och x1

d) x1 gives a maximum. x1 gives a minimum.

e) f(x) is increasing in the intervals x1 x1 f(x) is decreasing in the interval 1x1

f) f(x) concave in the intervals  3x0 x 3 f(x) convex in the intervals x 3 0 x 3

g) Inflection points: x 3 x0 x 3 h) Range: 0 f(x)2

101. a) Largest possible domain to the function: x 0 b) Zeros: x1

c) Stationary point: x  e d) x  e gives a maximum.

e) f(x) is increasing in the intervall 0x  e f(x) is decreasing in the intervall x  e

f) f(x) concave in the interval 0x e^5/6 f(x) convex in the interval x e^5/6 g) Inflection point: x e^5/6

h) Range:

x e

f 2

) 1 ( 