Post-Algebra II, Pre-Precalculus Summer Packet

Transcription

1 Post-Algebra II, Pre-Precalculus Summer Packet (Concepts epected to be understood upon entering Precalculus course) Name Grade Level School Teacher In order to be successful in a Precalculus course at the high-school level, a student needs to have a firm grasp of the material covered in Algebra I, Algebra II, and Geometry. This packet is an attempt to highlight the skills obtained in those courses that are used most often in a Precalculus course. During the school year, other skills that may not be covered in this packet will be necessary for success in a Precalculus class, but this packet should serve as a guide to the skills considered most vital. Parents are requested to see that this assignment is completed properly and seriously. Your child s success in the first semester depends on the correct completion of this packet and on understanding of the concepts covered. Answer key will be available on or after May 1, 017. If you realize later that you need another packet or a different packet, you can just download a copy from this link. PRE-CALCULUS STUDY PACKET ( ) y z if 1. Evaluate 6, y, z 7.. Solve 5 a Solve 9 5k. 4. Given f ( ) 5 6, find f ( ). 5. Find the slope intercept form of the equation of the line passing through 1,4 and 5,. 6. Find an equation of the line that passes through 5, and is perpendicular to the line with the equation y Solve 4y y Simplify: 1 b c ( c d) b c d 4 9. Simplify: Solve m Simplify: i 4 i 1. Solve: a a 14.

2 1. Solve: Solve by factoring : Solve by completing the square: Solve by the quadratic formula: A rectangular garden 0 meters by 0 meters is increased on all sides by the same amount. The area of the new garden is 875 square meters. Find the amount by which each dimension is increased. 18. Simplify and write with positive eponents: 6 54 y y Find the solution set: n Factor: 4a 19ab 5 b. 1. Factor: Factor: Find the quotient: 1a b 4a b 48a b 6ab 5. Simplify: y. 4. Divide using synthetic division: 5 a a Assume c > 0 - simplify: 18c c 9c 4 6c c Simplify ( > 0): a Solve for : Simplify: i i Solve the system; identify the figures: y y 1 1

3 1. Specify the coordinates of the maimum or minimum point of the graph of y Find the remainder for : Find all of the rational zeros of: 4 f ( ) Find: Find: Assume $500 is deposited in a savings account. The interest rate is 8% compounded continuously. Find when the rt investment will be tripled. A Pe. 7. Graph the following system: y1 y 9. Graph: y Graph: y. y1 40. Graph: The hypotenuse of a right triangle is 7 inches longer than the base and 14 inches longer than the height. The perimeter of the triangle is 84 inches. Find its area. 4. The perimeter of a rectangle is 96 cm. Its area is 4 square centimeters. Find the length and width of the rectangle. 4. Maria can cut a lawn and trim the hedges in 8 hours. Debbie can do the lawn and hedges in 6 hours. When Maria, Debbie, and Bill worked together it took hours to complete the job. How long would it take Bill if he worked alone? 44. Two trucks leave from the same truck stop. One travels east on interstate 10 and the other travels west on interstate 10. The eastbound truck averages 40 miles per hour, and the westbound truck averages 50 miles per hour. In how many hours will they be 00 miles apart?

4 45. Sketch graphs of the following functions: a. f b. f c. f 4 d. f log e. f log f. f g. f h. f Evaluate the following logarithmic epressions: a. log 16 log 16 b. 5 log c d. e. f. g. log log e log 4 e log 1 e log 6 log 7 log 8... log 5 h Rewrite each epression using a single logarithm: a. log log 4 log 1 log log 4 5log 7 log log 4 log b. c Solve the following equations: log 7 a.

5 b. log 7 log c. e 5e What are the domains of the following functions? a. f ln b. f ln 7 c. f ln 5 4 d. f ln 7 ln( ) e. f ln 7 f. f g. f 50. Let f ln( 17) ln( ) 6 ln and g log a. What is the domain of f()? b. What is the domain of g()? c. What is the domain of the function h() = f(g())? d. What is the domain of the function j() = g(f())? e. What is the domain of the function k() = f(f())? f. What is the domain of the function l() = g(g())? 51. Find the inverse of each of the following functions: a. f ln b. f 4ln 7 f 4e 7 c. d. f e. f 5 5. You deposit $1,000 in a bank account which yields % interest per year, compounded yearly. a. Write a function A(t) which gives the amount of money in the account after t years. b. How much money will you have in the account after months? c. How many years will it take for your money to triple? d. How much interest do you earn each month (ie, what is the effective monthly interest rate)? Each day? Each decade? 5. You wish to have $1,000,000 in 0 years. An investment account promises 7% returns per year. How much do you need to invest now? 54. You have $100,000 now, and wish to have $1,000,000 in 0 years. What rate of return (in %/year) do you need to get on your money?

6 55. Consider the function f ln. a. What is the average rate of change in the function over the interval b. The slope function m f 1 7 e, e? gives the slope of the line which is tangent to the graph of f() at. Write an equation for the line which is tangent to the graph of f() at = e. c. What is the area of the triangle which is bounded by the -ais, the y-ais and the line from part b? d. Approimate the integral as a single logarithm ln( ) d using trapezoids of equal width. Write your answer 56. If you are reading this, do NOT tell anybody else about this. On the first day of Precal class net year, 10 minutes after class starts, stand up, stand on your seat and put your thumbs in your ears. Everybody who does this will receive some bonus points on the first quiz. The more people who do it, the smaller the reward will be, so remember, DO NOT TELL anybody else about this. Don t tell them to check out problem 56 or ask them if they solved problem 56. Just tell them to make sure that they do all the problems in the summer packet. 57. The first few terms in an arithmetic sequence, A n are 17, 1, 7,, a. Write a recursive definition for the sequence. b. Write an eplicit definition for the sequence. c. What is the 100 th term in the sequence? d. What is the sum of the first 100 terms in the sequence? e. Let S n formula for n A. What are the first 5 terms in the sequence n k1 S n k 58. The first few terms in a geometric sequence Gn are 10, -5,.5, -1.5, a. Write a recursive definition for the sequence. b. Write an eplicit definition for the sequence c. What is the 7 th term in the sequence. d. What is the sum of the first 0 terms in the sequence? e. What is the sum of ALL the terms in the sequence? S? Write an eplicit 59. For what values of does do each of the series below converge? Assuming that is in that set, evaluate the sum in terms of. a. k1 1 0 b. 4 k 1 k1 k1 60. For each series, does it converge or diverge? Give evidence to support your answer. a. k k b. k k

7 c k d. e. k1 k 1 k 1 k 1 1 k1 k k (Hint: you might want to write out some partial sums for this one) 61. You play the following game: Starting at a known location, you walk a distance of 1 mile due east. Then, you walk a distance of / of a mile either continuing east or west. Then, you walk a distance of 4/9 of a mile, again either to the east or the west. And you repeat this process indefinitely, each leg is / of the length of the previous leg. a. If, on every leg, you always head east (never west), how far will you be from your starting location at the end of your journey? b. If you reverse direction every leg (that is, you head east 1 mile, west / mile, east 4/9 of a mile, west 8/7 of a mile etc), how far from your starting location will you end up? c. What if you travel EWWEWWEWW? d. What if you travel ENWSENWSENWS? 6. The velocity (in m/s) of a particle moving along the -ais is shown below. At time t = 0s, the particle is 15 m to the left of the origin. v(t) (m/s) time(s) a. At what times was the particle moving to the left? To the right? Stationary (ie not moving)? b. At what times was the particle speeding up? Slowing down? c. Fill in the table with the particle s position at each value of t. T(s) Position T(s) Position d. Write a piece-wise function V(t) for the velocity e. Write a piece-wise function A(t) which gives the acceleration of the particle at time t. f. Write a piece-wise function P(t) which gives the position of the particle at time t. (Hard) g. Assuming that the velocity function continues as indicated by the arrow at the far right, at what time does the particle reach the origin?

8 6. There are 5 afterschool activities that a student can participate in: Math Tutorials, Science Tutorials, Basketball, Boardgames, and Socializing. a. If a student does one activity each day for a week (5 days), how many different weeks can he have (assuming he may repeat activities)? What if he doesn t repeat any activities? b. A student is struggling and has to go to tutorials on Monday, Wednesday and Friday, but is free to do anything on Tuesday and Thursday. How many different weeks can they have? c. A student is free to do any activity, but doesn t want to do the same activity twice in a row. How many different weeks can they have? d. One day, 50 people played boardgames and 7 people socialized. If there were a total of 60 people doing these two activities, how many did both? e. On another day, 50 people played boardgames and 7 people socialized and did both. How many people did either? 64. A group of 100 students get together. a. They run a race and give medals to the top 10 finishers. In how many different ways can the medals be awarded? b. They select a group of 0 to perform a play. In how many different ways can this group be selected? c. They split into small groups of 5. How many possible different groups are there? In how many different ways can the entire group be split into 0 groups of 5? (note: these are two different questions). 65. In how many ways can the letters in the word TENNESSEE be arranged? 66. What is the coefficient on the 8 y term in the epansion of y 11?