Dear Calculus BC student, I hope that ou re all enjoing our first few das of summer! Here s something that will make it a little more fun! Enclosed ou will find a packet of review questions that ou should complete before the first da of classes. This packet covers mainl Algebra I, Algebra II, and Precalculus skills. These are skills that we will work on having down pat so that ou won t even have to think about, how do I solve this problem? The packet will be graded upon our return to school and will be worth 4 homework assignments (a week s worth of assignments). Solutions are also attached, but I epect to see full work for each problem. There will be a quiz covering this material on the second or third da of classes. Man of ou ma want to complete this as soon as possible and have it over and done with. If this is the case, please be sure to review our solutions in the das before school starts. The aim of this summer work is to keep our mathematical mind from rusting in the months that ou ll spend awa from school. We ll be hitting the ground running in August and ou don t want to be out of breath. Enjo! E-mail an questions: jtonkin@keschool.org See ou all in August! Ms. Tonkin If ou come across a topic with which ou are uncomfortable, ou ma find the following websites useful: http://www.khanacadem.org http://www.coolmath.com/ http://www.purplemath.com/ Help is also available in our Precalculus notebook or tetbook. Did ou save our notes? The following problems involve the skills that ou will be using throughout the ear in Calculus BC. Let s see what ou remember! We will answer some questions on the first da of class, but our goal is to answer our own questions before class starts. You can phone a friend, google, use khanacadem.org, use our Precalculus notes/tetbook etc. Be honest with ourself and how comfortable ou are with these skills. We ll discuss in August!
Part 1: Simplifing Simplif each epression completel. 1. 1 1 4 4. Rationalize the numerator: 1. 56 1 4. ( ) ( 1) 5. sin cos cos 6. sin cos 1 sin 1 7. Epand: log 1 ln 1 ln5 ln 8. Condense: 10. 1 t 9. 4 t
11. 1 1 1 1 1 1. 1 1. 8! 6! 14. 10!! 8! n 15. n! 1! n 16. n 1!! 17. Factor the following epressions completel. a.) 4 81 b.) 8 15 c.) 4 8 5 50 d.) 1 4 4 0
18. Write the sum using sigma notation. a.) 17 1 b.) 4 4 4 4 4 4 6 40 19. Rewrite the trigonometric epression: tan arccos as an algebraic epression. Part II: Evaluating 1. Fill in and memorize the values in the following table. You should be able to come up with these values without having to draw the unit circle each time. Angle (degree) Angle (radians) sin() cos() tan() 0 0 45 60 90
. Find eact values for each of the following trigonometric epressions. a.) cos 11 b.) sin 6 1 c.) sin 4 d.) cos 1 1 1 e.) sin 1 f.) tan g.) sin arctan 4 h.) 5 cot arcsin 1. Perform the following polnomial division: a.) 5 4 4 7 1 using snthetic division b.) 5 4 4 1 using long division.
4. Calculate the volume of each: a.) Outer radius: 5 cm, Inner radius: cm, height: 10 cm b.) Cone with height: 8 in, Base Circumference of 10 in 5. Evaluate the following sums: 5 a.) n n1 n0 b.) If n0 n 1, find n 1 n. c.) If n5 n 18 4, find n 4 81 n.
Part III: Graphing 1. Sketch each of the following functions and state their domain and range. Sketch an asmptotes. You should have these functions and their characteristics memorized. 1 1
e ln sin cos tan arctan
. Sketch the following graphs on the aes provided. a.) b.) 9 c.) f 5 5 ( ) 4 5 d.) 4
. Given the graph of f(), sketch the following: a.) f 1 b.) f c.) f( ) 4. Identif an asmptotes (vertical, horizontal, or slant) of each of the following functions: a.) f( ) b.) 1 f( ) 1 5. Sketch the following polar functions. a.) r 5 b.)
c.) r cos d.) r 1 sin Part IV: Solving 1. Solve for the indicated variable. a.) A P nrp for P b.) d d for d. Solve for. a.) 4 1 10 e b.) ln 1 5 4
100 log log 6 d.).5 1 e c.) 50 e.) sin 1 0 on 0, f.) cos1 sin 0 on 0, g.) tan h.) 5 1 Part V: Applications 1. Calculate the average rate of change of f on the given interval. a.) the graph of f is given. Find the average rate of change of f on the interval 4,
b.) on,6 f ( ) 5 c.) on a, a h f ( ) 5..
4. Using a graphing calculator, find an relative etrema (min s and ma s) of the polnomial: f 4 ( ) 5 6 10. 5. Determine from the graph whether f has a minimum on the interval. Write es or no for each graph. 6. Calculate the area of the shaded region.
7. A curve is traced b a point P, which moves such that its distance from the point A1,1 three times its distance from the point B, 1. Determine the equation of the curve. is 8. Find an eponential model of the form the model to predict the -value when 6. b ae that passes through the points, and 5,7. Use 9. For each function, find the difference quotient: f ( h) f ( ). Simplif as much as ou can. h a.) f ( ) 4 b.) f ( ) 1 c.) f( ) 1 d.) f ( ) sin (HINT: Use addition identit for sine)
10. Write the partial fraction decomposition of: 1 a.) f( ) 1 b.) f( ) 11 4 11. Complete the square for the following quadratic epressions. Write in the form: a h k a.) 6 b.) 18 c.) 10 1. Find the n th term of the sequence starting at n 0 a.),7,11,15, b.),6,18,54, c.) 5 7 1,,,, 6 4
1. 14. A particle is moving along a diagonal line. Its speed in both the horizontal and vertical directions are given. Calculate the speed of the particle in the direction in which it is traveling. 15. Write a formula for the area of a sector of a circle of radius r with given angle : A = Find the area of the sector shown here: 16. Convert the following coordinates to either rectangular or polar form. a.) 5,9 polar coordinates b.) 7, rectangular coordinates 6
ANSWER KEY Part I: Simplifing 1. 1 4 4. 1 1. 6 4. 4 5. sec 6. cos 7. log log 1 log 8. ln 5 1 9. 4 16 10. t 6 11. 1 1 1. 1 1. 56 14. 45 15. 1 n 16. n n 1 17. a.) 9 9 b.) 54 10 5 c.) 5 5 1 d.) 4 10 18. a.) 17 n 0 b.) 4 n n0 n0 19. 9 Part II: Evaluating 1.. a.) 0 b.) 1 c.) d.) e.) f.) 4 g.) 5 h.) 1 5. a.) 6 1 6 45 b.) 4 89 1 1 4. a.) 10 cm b.) 00 in 5. a.) 79 b.) 1 c.) 16
Part III: Graphing 1.
. a.) b.) c.) d.)
. 4. a.) v. a.: h. a.: 0 b.) v. a.: 1 h. a.: D: 1,1 5.
Part IV: Solving 1. a.) A P b.) 1 nr d. a.).811 b.) e 0 1 c.) 600 d.) 1.86 e.) 99 g.) n, n Z h.) 4 Part IV: Applications 7 11 5 1 5 17, f.),,,,,, 6 6 6 6 6 6 1a.) 1 b.) 4 c.) 10a5h. a.) rt t b.) h r rt h. a.) 1 b.) P 4r r c.) 4 9 4 m d.) 100 5km e.) 0 4. rel min: f( 0.87) 4.69, f(1.094) 10.949, rel. ma: f (0.56) 11.566 5. a.) es, b.) no, a.) no, b.) es 6. 9 4 7. 8 8 8 0 4 0 8. 0.8 1.705e ; (6) 9.84 9. a.) h h 4 b.) h h c.) h 1 1 d.) sin cos( h ) sin( h )cos sin h 10. a.) 1 1 1 1 1 b.) 5 1 11. a.) 9 b.) 9 81 5 5 c.) 1. a.) a 4n n b.) a n c.) n a n n 1 n 1! 1. a.) 6 b.) 1 1 c) 1 14. 10 m s 15. A 1 r, A 5 16. a.) 106,1.064 b.) 7 7,