Summer Review Packet for students entering IB Math SL The problems in this packet are designed to help you review topics that are important to your success in IB Math SL. Please attempt the problems on your own without any notes and show all work! In addition, do not use your calculator for these problems. When you come across topics that require a little review, feel free to look at your old notes, search a website or ask a friend for help. If you want to check your work with a calculator, that is fine also. You are epected to get each problem correct. It is recommended that you work with one or more people, but each person must submit his/her own work. Before you leave school, write down the names, phone numbers, and/or email addresses for at least two people who are also taking IB Math SL in the fall. Name Phone Email Name Phone Email Bring the finished packet with you to your IB Math SL class on the first day of school. After you have an opportunity to ask questions, you will be assessed on these skills during the first week of school as part of your 1st quarter grade. Enjoy your summer! I am looking forward to seeing you in September. If you have any questions, please contact Mrs. Atamas: Faina_Atamas@mcpsmd.org.
I. Simplify. Show the work that leads to your answer. 1. 4 4. 8. 5 5 4. 4 16 II. Complete the following identities. 1. sin + cos =. 1 + tan =. cot + 1 = 4. cos = 5. sin = III. Simplify each epression. 1 1 1. h. 10 5. 1 1 1 8 6 9 1 4. IB Math SL Summer Review Packet Page of 1
IV. Solve for z: 1. 4 + 10yz = 0. y + yz 8z 4 = 0 V. If f() = {(,5), (,4), (1,7)} g() = h() = {(,), (4,), (1,6)} k() = + 5 determine each of the following: 1. (f + h)(1) =. (k g)(5) =. (f h)() = 4. (g k)(7) = 5. f -1 () = 6. k -1 () = 7. 1 f( ) = 8. (kg)() = VI. 1. Evaluate f ( h) f ( ) h and simplify if f() =.. Epand ( + y). Simplify: ( 5 + ) = 4. Simplify: + = 5. Find sin if sin. How many answers do you epect? 5 IB Math SL Summer Review Packet Page of 1
VII. Epand and simplify. 1. 4 n. n0 n1 1 n. 1 k k=0 = 4. 1 = n! k=0 VIII. Simplify 1.. ln e. e (1 ln ) 4. ln 1 5. ln e 7 1 6. log 7. log 1/ 8 8. 1 ln 9. ln e 10. 4y 1 1 y 5 11. 7 / 1. (5a / )(4a / ) 1. (4a 5/ ) / 14. ln81 ln IB Math SL Summer Review Packet Page 4 of 1
IX. Using the point-slope form y y1 = m( 1), write an equation for the line 1. with slope, containing the point (, 4) 1.. containing the points (1, -) and (-5, ).. with slope 0, containing the point (4, ). 4. perpendicular to the line in problem #1, containing the point (, 4) 4. X. Given the vectors v = i + 5j and w = i + 4j, determine 1. 1 v. w v. length of w 4. the unit vector for v XI. Without a calculator, determine the eact value of each epression. 1. sin 0. sin. sin 4 4. cos 5. cos 7 6 6. cos 7. tan 7 4 8. tan 6 9. tan 10. cos(sin -1 1 ) 11. Sin-1 (sin 7 ) 6 IB Math SL Summer Review Packet Page 5 of 1
XII. For each function, determine its domain and range. Function Domain Range 1. y 4.. 4. y y 4 y 4 4 XIII. Determine all points of intersection. 1. parabola y = + 4 and line y = 5 + 11. y = cos and y = sin in the first quadrant XIV. Solve for, where is a real number. Show the work that leads to your solution. 1. + 4 = 14. 4 1 0. ( 5) = 9 4. + 5 = 8 IB Math SL Summer Review Packet Page 6 of 1
Solve for, where is a real number. Show the work that leads to your solution. 5. ( + )( ) > 0 6. 15 0 7. 1 = 8. sin = sin, 0 9. < 7 10. ( + 1) ( ) + ( + 1)( ) = 0 11. 7 = 9 1. log + log( ) = 1 IB Math SL Summer Review Packet Page 7 of 1
XV. Graph each function. Give its domain and range. 1. y = sin. y = e. y = 4. y = IB Math SL Summer Review Packet Page 8 of 1
Graph each function. Give its domain and range. 5. y = ln 6. y = + 7. 1 y 8. if 0 y if 0 4 if IB Math SL Summer Review Packet Page 9 of 1
XVI. Compute each of the following limits: cos 1. lim. 1 lim 1 1 XVII. Let f 1, if 4, if, if Compute the following limits: a) lim f b) lim f c) lim f d) lim f e) lim f f) lim f XVIII. Write each sum using summation notation, assuming the suggested pattern continues. 1. + 5 + 8 + 11 +... + 9 =. 1 + + 6 + 4 + 10 + 70 =. 6 1 + 4 48 +... = 4. 1 1 + 1 1 +... = 5. 1 + 1 4 + 1 9 + 1 5 + = 6. 0.1 + 0.01 + 0.001 + 0.0001 +... = XIX. Remember you are not using a calculator. 1. In a triangle ABC, angles A and C measure 45 and 0 degrees respectively. Side BC is 14 centimeters long. Sketch a diagram and find. Find the area of the shaded region. a) AB b) Area of the triangle ABC IB Math SL Summer Review Packet Page 10 of 1
XX. The Binomial Theorem. 1. Find the coefficient of 5 in the epansion of ( ) 8.. Use the binomial theorem to complete this epansion. ( + y) 4 = 81 4 + 16 y +.... Determine the constant term in the epansion of ( )9. XXI. Vectors 1. ABCD is a rectangle and O is the midpoint of [AB]. D C Epress each of the following vectors in terms of OC and OD (a) CD (b) OA A O B (c) AD. The quadrilateral OABC has vertices with coordinates O(0, 0), A(5, 1), B(10, 5) and C(, 7). (a) Find the vectors OB and AC. (b) Find the cosine of the angle between the diagonals of the quadrilateral OABC. IB Math SL Summer Review Packet Page 11 of 1
. The vectors i and j are unit vectors along the -ais and y-ais respectively. The vectors u = i + j and v = i + 5j are given. (a) Find u + v in terms of i and j. A vector w has the same direction as u + v, and has a magnitude of 6. (b) Find w in terms of i and j. 4. Find a vector equation of the line passing through ( 1, 4) and (, 1). Give your answer in the form r = p + td, where t. 5. The triangle ABC is defined by the following information OA = ( ), AB = ( 4 ), AB BC = 0, AC is parallel to ( 0 1 ) (a) On the grid below, draw an accurate diagram of triangle ABC. y 4 1 1 O 1 4 1 4 5 6 (b) Write down the vector OC. IB Math SL Summer Review Packet Page 1 of 1
XXII. The following vector problem could be challenging. I hope you will figure it out. Points P and Q have position vectors 5i +11j 8k and 4i + 9 j 5k respectively, and both lie on a line L 1. (a) (i) Find.PQ. (ii) Hence show that the equation of L 1 can be written as r = ( 5 + s) i + (11 s) j + ( 8 + s) k. The point R (, y 1, z 1 ) also lies on L 1. (b) Find the value of y 1 and of z 1. The line L has equation r = i + 9 j +1k + t (i + j + k). (c) The lines L 1 and L intersect at a point T. Find the position vector of T. (d) Find the cosine of the angle between the lines L 1 and L. IB Math SL Summer Review Packet Page 1 of 1