AP Calculus AB Summer Assignment 018 Name: Date Submitted:
Welcome to AP Calculus AB. This will be the toughest class yet in your mathematical careers, but the benefit you will receive by having this eperience in high school is immense. Because of the unique nature of this class, it is very important that you are ready to start working on the first day. We will NOT be reviewing the material in this packet. The reason for this is to make sure we have adequate time for all of the new material that you did not see this year, and more importantly, to make sure we have completed all of the required material well before the AP eam. As the instructor of AP Calculus, I have etremely high epectations of students taking this course. Epected is a certain level of independence by anyone taking AP Calculus. Your first opportunity to demonstrate your capabilities and resourcefulness is through this summer work packet which will help you maintain/improve your skills. This packet is a requirement for those entering AP Calculus AB and is due the first week of class. There are certain math skills that have been taught to you over the previous years that are necessary to be successful in calculus. If you do not fully understand the topics in this packet, it is possible that you will get calculus problems wrong in the future not because you do not understand the calculus concept, but because you do not understand the algebra or trigonometry behind it. Don t fake your way through any of these problems because you will need to understand everything in this very well. You may work with someone else while you do this, but copying will not be tolerated. Also, don t wait until the last minute to do everything in the packet because you may run out of time and rush through them. Likewise, do not do all the problems right at the beginning of summer and completely forget how to do all of them by the time school starts again. If you feel that you need help with some of these topics, the best resource may be the internet. There are great sites where you can just type in the topics to get help. Using youtube.com, khanacademy.org, and ck1.org not only will help you on this packet this summer, but will be referenced often throughout the upcoming school year. For this packet you must show all of your work on a separate sheet of paper and circle your answer. If I cannot find each question and answer easily I will assume they are not there or incorrect, so make sure your work is neat. Do not rely on a calculator to do all of the work for you. Half of the AP eam does not allow any calculator at all. This packet will be worth a test grade. Additional Requirements: In addition to this assignment, you will be given a calculator assignment to complete which will help you learn the TI nspire CAS calculator and become familiar with some of its features. Tetbooks: Required: Finney, Demana, Waits and Kennedy, Calculus: Graphical, Numerical, Algebraic, 4 th AP Edition This tet will be issued to you by the school before you leave for summer break. Recommended: 5 Steps to a 5: AP Calculus AB (paperback, approimately 15.00). It is important to purchase the 017 edition or later. It can be pre-ordered from amazon or your favorite bookstore. ISBN-13: 978-159583360 ISBN-10: 159583368 Archbishop Curley Mathematics Department 018 P a g e 1
This packet contains the topics that you have learned in your previous courses that are most important to calculus. Make sure that you are familiar with all of the topics listed below when you enter Calculus. Algebra Basics You must be able to factor polynomials (GCF, by grouping, trinomials, difference of squares, and sum & difference of cubes) You must know how to add, subtract, multiply, and divide rational epressions. You must know how to work with comple fractions Function Basics We use functions almost every day in calculus. It is critical that you are able to identify and do the following: Identify functions from equations, graphs, sets Use function notation Find the domain and range of a function Identify functions as even, odd or neither. You should also be able to identify what lines of symmetry go with even and odd functions. Perform operations on functions. Compose functions. Graph and write the equations of piecewise functions. Identify discontinuities. Find inverse functions algebraically and graphically. Use long and synthetic division Find the left hand and right hand behavior of a graph. Find the zeros of a polynomial function algebraically and graphically. Find vertical, horizontal, and slant asymptotes and any holes in the graph of a function. Logarithmic and Eponential Functions Know the properties of eponents. Know the properties of logarithms. Natural logs will be used a lot in Calculus, so be familiar with them. Solve eponential and logarithmic equations Trigonometry We use the trig functions and a few of the identities regularly in Calculus. The following are the most important: Calculus is always in radians, never degrees. Know the special right triangles! You need to know, without a calculator, the eact values of sin, cos, tan, csc, sec, cot for the following values: π, π, π 3, π 6, π 4, π Be able to identify the graph of sine, cosine, and tangent and the important characteristics The Pythagorean Identities and Double Angle Identities are the most used. Be able to work with arcsin, arccos, and arctan. Archbishop Curley Mathematics Department 018 P a g e
There are three parts to your summer work. First, you must sign out your tetbook and calculator from Ms. Jenkins before final eams begin. You must complete all three parts of this packet. PART 1: Review Question Packet You must complete all of the problems in the Review packet. You may consult one another or other resources for material you may have forgotten. This packet will be your first major grade of the first quarter. PART : MATH XL registration, introductory assignment and pre-test From the information provided to you in your folder, use the Math XL code to get registered in our class. This is a resource directly connected to our tet and one we will use for section quizzes throughout the course. You MUST get registered in the course and complete the introductory assignment which will guide you through the site. Once you have completed the Intro assignment, you will take the Getting Ready for Calculus Pretest. If you have completed your packet, this pretest should give you a good idea how well you have mastered the prerequisite material. From the results of the summer assignment and the pretest, we will review key concepts the first week of classes and then test on this material. PART 3: Calculator Assignment The calculator assignment is to introduce you to the TI nspire CAS calculator you will be using throughout this course. You will use this calculator in class and at home for calculus assignments. You will also use this calculator for the calculator sections of the AP test in May. Sign out your calculator before school ends so that you may use it over the summer and to do your calculator assignment. I will collect the calculator assignments from your calculators on the first day of class. Archbishop Curley Mathematics Department 018 P a g e 3
PART 1: Review Question Packet Answer every question. Show all work in clear, legible and organized manner. Clearly mark all answers. Label all diagrams and graphs appropriately. Eponents and Radicals: Simplify: 1) g 5 g 11 ) (b 6 ) 3 3) w -7 4) y y 1 8 5) (3 7 )(-5-3 ) 6) (-4a -5 b 0 c) 7) -15y 7 - -9 5 5 y 8) 4 1 9 4 3 Epress the following in simplest radical form. 9) 50 10) 4 11) 19 1) 169 13) Simplify 7 9 5 6. Epress your answer using a single radical. Factoring Polynomials Factor completely: 14) 5 3 1 15) 4 3 0 9 16) 15 5 75 15 17) 15 56 18) 8 3 7 19) 16 3 1 Archbishop Curley Mathematics Department 018 P a g e 4
Comple Fractions and Rational Epressions Simplify: 0) 1 1 1 1 1) 1 1 3 1 3 = ) 4 10 5 = 3) 1 6 5 n n n n = Function Composition and Inverses 4) Let f ( ) 3 and g ( ) 1. Compute ( g f )( ), 5) Find f(g()) if f() = and g() = + 4 6) Let 3 7 f( ). Find f 1 ( ), the inverse of f( ). 7) Find f 1 () if f() = + 16 Archbishop Curley Mathematics Department 018 P a g e 5
Linear and Quadratic Functions 8) Write an equation of a line with slope 3 and y-intercept 5. 9) Use point-slope form of a linear equation to find an equation of the line passing through the point (6, 5) with a slope of /3. 30) Find an equation of a line passing through the points (-3, 6) and (1, ). Graph it below. 31) Find an equation of a line with an - intercept of (, 0) and a y-intercept of (0, 3). Graph it below. 3) Find an equation of a line with an -intercept of (, 0) and a y-intercept of (0, 3). 33) Solve the inequality: 1 0. 34) Find an equation for the parabola whose verte is (, 5) and passes through (4,7). Epress your answer in the verte form for a quadratic. 35) Transform y = -3 4 + 11 to verte form by completing the square. Archbishop Curley Mathematics Department 018 P a g e 6
Polynomial and Rational Functions 36) Which of the following could represent a complete graph of number? f ( ) a A. B. C. D. 3, where a is a real 37) Find a degree 3 polynomial with zeros -, 1, and 5 and going through the point (0, 3). 38) Use polynomial long division to rewrite the epression 3 7 14 8 4 39) Use a graphing calculator to approimate all of the function s real zeros. Round your results to 6 5 3 four decimal places. f ( ) 3 5 4 1 5 40) Give that f( ). Find the asymptotes and the domain of the function. Domain: Vertical Asymptote(s): Horizontal Asymptote(s): 41) Factor to solve the inequality. Write your answer in interval notation. 0 3 64 3 Eponential and Logarithmic Functions 3 4) The graph of y a for a 1 is best represented by which graph? A. B. C. D. 43) Rewrite the epression 4 log 5(3 ) y into an equivalent epression by epanding. Archbishop Curley Mathematics Department 018 P a g e 7
44) Solve: log 6 ( + 3) + log 6 ( + 4) = 1 45) Simplify: log 5 + log ( 1) log( 1) 46) Solve: log - log 100 = log 1 47) 116 The number of elk after t years in a state park is modeled by the function Pt () 0.03t 1 75e. a. What was the initial population of elk? b. When will the number of elk be 750? Trigonometric Functions Sketch the graphs using the intercepts, amplitude, period, frequency and midline. 48) f( ) sin +1 Intercepts: Amplitude: Period: Frequency: Midline: 49) f( ) cos( Intercepts: Amplitude: Period: Frequency: Midline: Archbishop Curley Mathematics Department 018 P a g e 8
50) Given and, determine the eact value of A) B) C) D) E) undefined 51) If, find the value of in degrees without the aid of a calculator. 5) Using the figure below, if, determine the value of. Trigonometric Identities csc( ) tan( ) sin( )cos( ) 53) Simplify A. C. sin( ) cos ( ) B. sin ( ) cos( ) D. cos( ) sin ( ) cos ( ) sin( ) 54) Solve the equation sin ( )cos( ) cos( ) algebraically. 55) Find all the eact solutions to sin ( ) 3sin( ) 0 0,. on the interval 56) Solve for : 3sin = cos for interval 0 π 57) Solve for : cos sin = sin for interval -π π Archbishop Curley Mathematics Department 018 P a g e 9
Miscellaneous Problems: 58) Solve the equation both algebraically and graphically. 43 5 4 Algebraic Solution: Graph Solution: 59) Three sides of a fence and an eisting wall form a rectangular enclosure. The total length of a fence used for the three sides is 40 ft. Let be the length of two sides perpendicular to the wall as shown. Write an equation of area A of the enclosure as a function of the length of the rectangular area as shown in the above figure. Then find value(s) of for which the area is 5500? ft Eisting wall 60) Describe the transformations that can be used to transform the graph of f ( ) ( ) 3. Graph each function in the designated area below. f ( ) f ( ) to a graph of f ( ) ( ) 3 61) Graph the piecewise function. 1 f ( ) 1 3 5 1 3 Archbishop Curley Mathematics Department 018 P a g e 10
6) For the function graphed answer the following A. B. C. f (0) D. f ( ) 1 f (3) f( ) f ( ) 0 1 4 1 5 5 63) Simplify the epression as much as possible. 10 (3 4) 6 (3 4) 0 5 64) (a) Find the ratio of the area inside the square but outside the circle to the area of the square in the picture A below. A B (b) Find a formula for the perimeter of a window of the shape in the picture B above. 65) A water tank has the shape of a cone (like an ice cream cone without ice cream). The tank is 10m high and has a radius of 3m at the top. If the water is 5m deep (in the middle) what is the surface area of the top of the water? 66) Two cars start moving from the same point. One travels south at 100km/hour, the other west at 50 km/hour. How far apart are they two hours later? 67) A kite is 100m above the ground. If there are 00m of string out, what is the angle between the string and the horizontal? (Assume that the string is perfectly straight.) Archbishop Curley Mathematics Department 018 P a g e 11
68) Water is poured into a conical container with a diameter of 0 cm and a height of 38 cm. If the volume of a cone is one-third the volume of a cylinder, write an equation for the volume water as a function of the height of the water in the container. 69) If a projectile is fired into the air with an initial velocity v at an angle of elevation θ, then the height h of the projectile at time t is given by h = 16t + vt sinθ a. Find the angle of elevation θ of a rifle barrel, to the nearest tenth of a degree, if a bullet fired at 1500 ft/s takes s to reach a height of 750 ft. b. Find the angle of elevation of a rifle, to the nearest tenth of a degree, if a bullet fired at 1500 ft/s takes 3 s to reach a height of 750 ft. 70) Suppose that f () is a function of and that the graph of y = f () has intercepts (a) y-intercept ; (b) -intercepts 3, 1, 0, and. What are the intercepts of y = f ( 3)? Archbishop Curley Mathematics Department 018 P a g e 1