Math Analysis Summer Packet

Transcription

1 Math Analysis Summer Packet Name: This packet is to be completed correctly and turned in at the beginning of the third class in the fall of 2017; you may print it out or copy the problems and complete them on a separate sheet of paper. It will be worth 50 points and will NOT be accepted late. The expectation is that each student enter Math Analysis with extensive knowledge of the content of this packet. You will be tested over this material on the fourth class. The test will also be worth 50 points. Significant class time will NOT be spent reviewing skills that should have been honed previously. If you don t understand a concept, look it up and practice it. These skills will prove vital to your success in this course. It would also be in your best interest to purchase a TI-84+ graphing calculator early in the summer if you do not already own one. You should also learn to use the calculator to its full potential; it will prove invaluable in your home studies in the upcoming school year. DOMAIN AND RANGE Identify the domain and range of each given function. Write both the domain and range in interval notation. 1. f x = 2x 3 2. f x = x2!3x!2 x 2!2x!8 Determine all holes and asymptotes of each given function. Be sure to describe each hole as an ordered pair. 3. f x = x2!4x!3 x 2!4x!5 4. f x = x2!25 x 2!2x!15 LINEAR AND ABSOLUTE VALUE EQUATIONS AND INEQUALITIES Solve each equation or inequality x + 3 < x 5 = 19

2 7. 3x x 2 > 4 SYSTEMS OF LINEAR EQUATIONS Solve each system of equations by using whatever means is convenient. Be sure to note if the system has infinite solutions or no solutions, if necessary. 9. 3x + 2y = 9 5x y = x + 2y = 4 3x + 6y = x 3y = 2 2x 5y = x y = 1 3x 3y = 5

3 SIMPLIFYING EXPRESSIONS OF VARYING TYPE Fully simplify each of the given expressions. Be sure to express all fractions in lowest terms, reduce radicals fully, and remove any common factors from both the numerator and denominator. 13. Simplify: a!2 b 2 a 2 b!1 14. Rationalize: 8 4! Simplify: 81x 4 y 9 z 16. Simplify: 8x 6 y ln Simplify: e 18. Simplify: 2xy 2 x y 2!3 19. Simplify: 40x 7 y Simplify: 3x 2 8x 3 2x 72x Simplify: (3 6 2) 22. Multiply: x + 2 (x 2 3x + 7) 23. Condense: 1 2 log x + 3 log y 24. Expand: log 5x 6y 2

4 25. Multiply: 2x 2!x!6 x2!x 2x 2!5x!3 x 2!4 26. Divide: 6x2!x!1 4x 3!4x 2 6x2!2x x 2!4x!5 27. Subtract: 2x 5 3x!1 x!4 28. Add: x!6 x!6 SOLVING EQUATIONS OF VARYING TYPE Solve each equation. Be sure to check for extraneous solutions. 29. log 3 x log 3 x 1 = x = 5x x 4/3 5 1 = x + 2 = = 14 3x!7 x! x = 3 x 2

5 35. log 4 x = x 2!x!6 2x 2!5x!5 = x!4 = x!1 = 81 x!2 GRAPHING FUNCTIONS AND SYSTEMS OF VARYING TYPE Graph each equation or system of equations/inequalities on the coordinate plane provided. Be sure to accurately transform the parent graphs for each equation. 39. y = 2(x 3) y < 2x y = x y = e x

6 OPERATIONS ON COMPLEX NUMBERS Fully simplify each of the following expressions. 43. Simplify: Simplify: 9 7i 6 + 5i 45. Multiply: 3 + 4i 5 8i 46. Rationalize: 8!6i!2!3i 47. Simplify: i Simplify:!4± 16!48 2 PROPERTIES OF FUNCTIONS Given f x = 2x 4 and g x = 1, determine the expression that describes each of the following functions. x 49. f x + g(x) 50. f x g(x) 51. f g x 52. f f x PROPERTIES OF POLYNOMIAL AND RATIONAL FUNCTIONS Fully factor each polynomial. If it does not factor, write prime. 53. f x = 27x f x = 8x

7 55. f x = 2x f x = x f x = 4x 2 + 3x f x = 3x 2 x 10 Determine the roots of each polynomial. 59. f x = 2 x 1 x 3 x f x = x 4 + x f x = 2x 4 6x 3 8x f x = x 2 4x 63. f x = x 2 + 3x f x = 2x 2 + x 3 Complete each operation using either synthetic or polynomial long division x 3 2x 2 5 (x + 3) 66. 7x x 2 + 7x + 5 (x 2 + 1)

8 67. 4x 3 + x 2 3x + 7 x x 3 + 2x 2 3x + 1 (x 2 + x + 1) Determine the equation(s) of the horizontal and vertical asymptote(s) for each function described. 69. f x = 2x2!6x!8 x 2!5x!6 70. f x = x2!2x!1 x 2!4x!3 PROBABILITY AND STATISTICS Determine the total number of possible outcomes for each given situation. 71. Positions that can be assigned to 12 members of a committee. The positions are president, vice president, secretary, and treasurer. 72. Three-person committees that can be formed from a pool of 16 possible members. 73. Meal combinations given that each of one of 2 salads, 4 entrees, 3 sides, 5 drinks, and 3 desserts is chosen.

9 SEQUENCES AND SERIES Identify the next three terms in each sequence , 1, 2, 3, , 5, 8, , 9, 27, , 1, 1, Write out the first five terms of each sequence. 78. t n = 2n n g n = 1 2 n!1 80. s n = 1 n n!3 n!2 81. b n = en n! Determine the value of each infinite geometric series, if it exists. If not, write infinite TRIGONOMETRY Convert each expression to radians or degrees π π

10 Determine the exact value of each of the following expressions. 88. sin π cos tan 4π sec 7π 4 Graph each sinusoidal function. Be sure to note amplitude, period, and phase shift. 92. y = 3 sin 4x π y = 2 cos 2x π 4