Summer AP Assignment Coversheet Falls Church High School Course: AP Calculus AB Teacher Name/s: Veronica Moldoveanu, Ethan Batterman Assignment Title: AP Calculus AB Summer Packet Assignment Summary/Purpose: The material in this packet should be a review of past material that will be necessary in the first few weeks of the Calculus program Completion of this packet will set each student up for continued success in the AP Calculus program Due date (must be completed before school starts) Assigned during: Summer break Due Date: First day of class Estimated time needed to complete the assignment: Approimately _6-8 hrs (not to eceed 8 hrs) Description of how the assignment will be assessed: The material from this packet will be assessed through quizzes given during the first month of school No quizzes will be given during the first week of classes Students will have an opportunity to get help on the packet before the quiz Grade impact to overall course grade: Content will be integrated into the coursework Tools/resources needed to complete the assignment: TI-8, 8, or 89 graphing calculator Contacts: Name Veronica Moldoveanu Ethan Batterman E-mail vmoldoveanu@fcpsedu ebatterman@fcpsedu
Falls Church High School Summer Review Packet For students entering AP Calculus AB Name: This packet is to be handed in to your Calculus teacher on the first day of the school year All work must be shown in the packet OR on separate paper attached to the packet Completion of this packet will help you to be prepared for the topics that will begin the year in AP Calculus If you have questions over the summer, you may email Mrs Moldoveanu at vmoldoveanu@fcpsedu or Mr Batterman at ebatterman@fcpsedu SHORT ANSWER SECTION Objectives: For objectives # - #5 you should be able to do all without a calculator Identify functions as even or odd Algebraically Graphically Know key points and basic shapes of essential graphs f, f, f ( ) ( ) ( ) f ( ) e f ( ) ln ( ) ( ) sin, ( ) f f cos Review Trigonometry Evaluate trig values Evaluate inverse trig values Solve trig equations Understand characteristics of rational epressions (be able to sketch without a calculator) Vertical asymptotes Horizontal asymptotes Zeros Holes 5 Be able to use properties of natural logarithms to solve equations 6 Know your calculator
I Symmetry Even and Odd Functions Quick Review Eample: y Even Function Symmetric about the y ais f( ) f( ) for all Eample: y Odd Function Symmetric about the origin (equivalent to a rotation of 80 degrees) f( ) f( ) for all To determine algebraically if a function is even, odd, or neither, find f ( ) f ( ), f ( ), or neither and determine if it is equal to Eample: Determine if f ( ) + is even or odd ( ) ( ) + f ( ) f + + ( ) Therefore, f ( ) is an odd function Determine if the following functions are even, odd, or neither f ( ) + f ( ) + 5 f ( ) + + f ( ) + +
II Essential Graphs For each graph, show two key points (label coordinates) and basic shape of the graph f ( ) f ( ) ( ) sin f - - - - - - -π -π π π - - f ( ) e 5 f ( ) ln ( ) 6 ( ) cos f - - - - - - -π -π π π - - III Trigonometry Review On each right triangle with a hypotenuse of, label the lengths of the other sides: 5 radians 90 radians 0 radians 60 radians In calculus we always use radians! Never degrees
Evaluate the sine, cosine, and tangent of each angle without using a calculator It is not necessary (ever again!) to rationalize the denominator (ie, an answer of does not need to be written ) π b) 5 π π c) 6 d) π e) 5 π f) π 6 Find two solutions to the inverse trig equations ( 0 θ π) Draw a triangle, if necessary Label θ and sides Determine reference angle Notice restrictions 5 Place in correct quadrant cosθ b) cosθ c) secθ d) tanθ e) cotθ f) sinθ Solve for y Notice the only difference between # and # is the domain restrictions # y sin b) y cos c) y tan ( ) d) y csc ( ) e) y cos ( 0) f) y cot ( )
IV Rational Functions Rational functions are ratios of polynomials: h( ) f ( ) ( ) g h( ) has a zero when h( ) 0 (which occurs when f ( ) 0 and the factor does not cancel) E h( ) + ( ) h ( + )( ) ( + )( ) ( ) h ( + )( ) ( + )( ) Therefore, h( ) 0 when h( ) has a vertical asymptote when g( ) 0 and the factor that causes g( ) 0 does not cancel E h( ) + ( ) h ( + )( ) ( + )( ) ( ) h ( + )( ) ( + )( ) Therefore, h( ) has a vertical asymptote when h( ) has a hole (is undefined but the limit eists) when g( ) 0 and the factor that causes g( ) 0 cancels from both f ( ) and g( ) E h( ) + ( ) h ( + )( ) ( + )( ) ( ) h ( + )( ) ( + )( ) Therefore, h( ) has a hole when Note that h( ) ( + ) ( + ) because these two functions do not have the same domain h( ) has a horizontal asymptote at y a when lim h ( ) a or lim h( ) a To determine lim h( ) consider first the largest eponent of f ( ) and g( ) If f ( ) has the larger eponent, then lim h ( ) (DNE) If g( ) has the larger eponent, then lim h( ) 0 If the eponents are the same, consider the leading coefficient E h( ) + Leading coefficients Therefore, lim h( ) and ( ) h has a horizontal asymptote at y
Once the basic characteristics of rational epressions are determined, the functions can be sketched without a calculator: E h( ) + Zero at Vertical Asymptote: Hole when Horizontal Asymptote: y Graph points as needed until you see the shape For each rational function below: Find all zeros b) Write the equations of all vertical asymptotes c) Write the equations of all horizontal asymptotes d) Find the value of any holes e) Sketch the graph (no calculator) showing all characteristics listed (Not all functions will have all characteristics listed above) + f ( ) f ( ) + - - - - - - - - f ( ) f ( ) - - - - - - - - 5 f ( ) + 6 6 f ( ) - - - - - - - -
V Properties of Natural Logarithms Recall that y ln ( ) and y e (eponential function) are inverse to each other Properties of the Natural Log: ( AB) ( A) + ( B) E: ln ( ) + ln ( 5) ln ( 0) ln ln ln ln A ln ln B ( A) ( B) 6 E: ln ( 6) ln ( ) ln ln ( ) ln p ( A ) pln ( A) E: ln ( ) ln ( ) and ln ( ) ln ( ) ln ( 8) ln ( e ), ln ( ) e, ( ) ln 0, 0 e Use the properties of natural logs to solve for E: 5 7 5 7 5 ln ln 7 ( ) ( ) ( ) ( ) ln 5 ln 7 ln ln ( ) ( ) ( ) ( ) ( ) ( ) ln ( ) ln ( ) ln ( 5) ln ( 7) ln 5 ln 7 ln ln ( ln 5 ln 7 ) ln ( ) ln ( ) 0 5 e e 0 5e + 7
VI Calculator Skills No matter what type of calculator you own, you should be able to do the following problems Use the inde of the owner s manual of your calculator to look up instructions or ask a classmate how to do each Evaluations: ) Fractions and more Be careful to use enough parentheses 6 79 + b) + π ) Logarithms and Eponents ln b) log c) ln (e) d) 5 e) e f) 0 f) 8 g) ( ) 8 h) ( )(0) ) Trigonometry Evaluations Make sure you watch the mode degree versus radians sin b) π sin 5 c) π sec d) tan ( ) 7 5 in radians & degrees Graphing Skill #: You should be able to graph a function in a viewing window that shows the important features You should be familiar with the built-in zoom options for setting the window such as zoom-decimal and zoom-standard You should also be able to set the window conditions to values you choose Graph y using the built in zoom-decimal and zoom-standard options Draw each
Find the appropriate viewing window to see the intercepts and the verte defined by y + Use the window editor to enter the and y values Window: Xmin Xma Xscl Ymin Yma Yscl Find the appropriate viewing windows for the following functions: y 0 + 5 y ( ) y 00 06 y + 0 Xmin Xmin Xmin Xmin Xma Xma Xma Xma Xscl Xscl Xscl Xscl Ymin Ymin Ymin Ymin Yma Yma Yma Yma Yscl Yscl Yscl Yscl Graphing Skill #: You should be able to graph a function in a viewing window that shows the -intercepts (also called roots and zeros) You should be able to accurately estimate the -intercepts to decimal places Use the built-in root or zero command Find the -intercepts of y Window [-7, 7] [-, ] (Write intercepts as points) -intercepts: Find the -intercepts of y -intercepts:
Graphing Skill #: You should be able to graph two functions in a viewing window that shows the intersection points Sometimes it is impossible to see all points of intersection in the same viewing window You should be able to accurately estimate the coordinates of the intersection points to decimal places Use the built-in intersection command Find the coordinates of the intersection points for the functions: f ( ) + g ( ) + 7 Intersection points: Find the coordinates of the intersection points of: f ( ) g ( ) Intersection points: Graphing Skill #: You should be able to graph a function and estimate the local maimum or minimum values to decimal places Use the built-in ma/min command Find the maimum and minimum values of the function y (Value means the y-value) Minimum value: Maimum value: Find the maimum and minimum values of the function y +
REVIEW: Do the following problems on your calculator + 8556 + 9 ln e sin 8 ) cos ( 75) in radians and degrees 5) 5π tan 6 6) Find the -intercepts, relative maimum, and relative minimum of y + 7) Find the coordinates of the intersection points for the functions g ( ) + f( ) + 9 and
Multiple Choice Section Directions: Please read questions carefully It is recommended that you do the Short Answer Section prior to doing the Multiple Choice Show all work on this packet If no work is required, eplain how you arrived at your answer Follow calculator instructions as given in each section * A choice of none is short for none of these A choice of DNE means does not eist These problems are due the first class You will be placed in a group to discuss the answers and questions will be answered during class and/or after school Do not epect there to be time in class for ALL questions to be resolved There will be a short quiz on the material during the rd class Functions NO CALCULATOR ) Given f( ) and g ( ) +, find f( g ( )) 8 + + b) + c) 6 d) + of these ) Given f( ) and g ( ) 6, find f( g( )) ) If f( + h) f( ) f( ), find h b) c) + h h d) ) Is the function f( ) + even, odd, or neither? Show why 5) If f is a one-to-one function on its domain, the graph respect to: f ( ) is a reflection of the graph of f( ) with the -ais b) the y-ais c) y d) y -
6) In which graph does y not represent a one-to-one function of? b) c) d) All of these are one-to-one functions of e) None of these are one-to-one function of 7) Given f ( ), find f ( ) b) + c) ( + ) d) NO CALCULATOR Solving equations 8) Solve for + 6 5 b) 5 c) 5 8 d) 9) Solve for + 9 b) c) - d) - and 0) Solve for 7 + 9 + 8 b) 5 c) d) 5 5 8
) Solve for p: g π p r ) Solve for ( ) + 6 8 ± 5 b) 0 ± 6 c) 0 ± 6 d) 8 ± 5 ) Solve for ( ) 5, b) -, c) d) -, ) Solve for + 0 7, b) -7, - c) 0,, 7 d) 0, -, -7 + 5 5) Solve for ( ) -58, 58 b) -6, 6 c) d) -, 6) Solve for 5 7, b) 5 7, c) 5, 5 d) 5 7) Solve by factoring 9 8 + + 9, b), 9 c) 9 d) 9
8) Solve by completing the square 6 + 0 ± 6 b) ± 0 c) ± 7 d) ± 9) Solve for + + b) - c), d), 0) Solve for 6 + 0 ± b) ± c) ± 5 d), ) Solve for + 5 9 5, b) 5, c) 5 9, d) ± 6 ) Solve the inequality algebraically 9 (, ] b) (,] c) [, ) d) [ ), ) Find all the real zeros of the polynomial function f( ) 6 0 b) 0, c) d) 0,, -
NO CALCULATOR Factoring and division ) Which polynomial function has zeros of 0, - and? f( ) ( )( + ) b) f( ) ( + )( ) c) f( ) ( + )( ) d) f( ) ( ) ( ) + 5) Use long division to find the quotient ( 6 + 7 5 + 6) ( ) 7 5 + ( ) b) + 5 5 + ( ) c) + 5 + 5 + ( ) d) 9 / + 7 + ( ) NO CALCULATOR Graphs 6) Find the domain of the relation shown at the right (, ) b) (,] c) (,) d) [, ) of these 7) Find the range of the function shown at the right (, ) b) ( 8, ) c) [, ) d) [, 5] of these
8) Find the domain of the function f( ) 5 (,5] b) (,5) c) [ 5, ) d) ( 5, ) e)none 9) Describe the transformation of the graph of f( ) which yields the graph of g ( ) 0 vertical shift 0 units up b) vertical shift 0 units down c) horizontal shift 0 units right d) horizontal shift 0 units left 0) Graph g ( ) ( ) using a transformation of the graph of f( ) b) c) d) ) Which sequence of transformations will yield the graph of g f( )? ( ) ( ) 0 horizontal shift 0 units right b) horizontal shift unit left vertical shift unit up vertical shift 0 units up c) horizontal shift unit right d) horizontal shift 0 units left vertical shirt 0 units up vertical shift unit up + + from the graph of ) Find the -intercept(s) of + y + y 0 ± b) (, 0) ( 6, 0) ± c) (, 0) d) (6, 0)
) Find the intercepts of the graph of + 7y -int: (0, 7) b) -int: (0, ) c) -int: (, 0) y-int: (, 0) y-int: (7, 0) y-int: (0, 7) d) -int: (7, 0) y-int: (0, ) ) Find the and y-intercepts: y 5 + (0, -), (0, ), (, 0) b) (0, ), (, 0), (, 0) c) (0, -), (-, 0), (-, 0) d) (0, ), (-, 0), (-, 0) of these 5) Determine the left and right behaviors of the graph of 5 f( ) + up to the left, down to the right b) down to the left, up to the right c) up to the left, up to the right d) down to the left, down to the right of these 6) Determine the left and right behaviors of the graph of f( ) 5 + + up to the left, down to the right b) down to the left, up to the right c) up to the left, up to the right d) down to the left, down to the right of these 7) Which function is graphed? c) f( ) + 6 b) f( ) 6 + d) f( ) + 6 f( ) + 6 of these
8) Graph the following: +, 0 f( ) +, > 0 9) Find the domain of the function f( ) + (, ), (, ), (, ) b) (, ), (, ), (, ) c) (, ) d) (, ), (, ) of these 0) Find the domain of f( ) + + all real numbers ecept -,, and b) all real numbers ecept - c) all real numbers ecept and d) all real numbers ) Find the domain of f( ) + 9 all real numbers b) all real numbers ecept ± c) all real numbers ecept d) all real numbers ecept, ± ) Find the vertical asymptote(s) of the graph of f( ) + ( )( + 5) y, y -5, y - b), -5, -, c) d), -5 ) Find the horizontal asymptote(s) of the graph of f( ) + y 0 b) - c) d) y
) Find the horizontal asymptote(s) of the graph of f( ) + 7 6 ± 7 b) y c) y ± 7 d) y 0 5) Find all intercepts of the graph of f( ) + 7 (0, -), (, 0) b) (-, 0), (, 0) c) (, 0), (0, ) 7 d) (, 0), (0, ) 6) Match the rational function with the correct graph f( ) + b) c) d) 7) Match the graph with the correct function + f( ) c) f( ) + b) f( ) + d) f( ) + e) None of these 8) What is the domain of f( ) e? (, ) b) [ 0, ) c) (, ) d) (,)
9) Without using a graphing utility, sketch the graph of f( ) Trigonometry NO CALCULATOR 50) Give the eact value of cos π b) c) d) 5) Find all solutions to cos 0 in the interval [ 0, π ] π π, 6 6 b) 5 π, 7 π 6 6 c) π 5π, d) π, π 5) Give the eact value of csc π b) undefined c) - d) of these 5) Find all solutions to sec sec 0, π + in the interval [ ] π π π π,,, b) π 5π, π, c) π π, d) π π, π, 6 6 5) Find the eact value of 5 tan 6 π b) c) - d) 55) Evaluate sec π
b) c) d) 56) Find all solutions of sin cos + cos 0 in the interval [ 0, π ) π π 5π π,,, 6 6 b) π 7π π π,,, 6 6 c) 5 π, π 6 6 d) 0, π of these NO CALCULATOR Logarithms and natural logarithms 57) Solve for 7 8 b) c) d) 58) Evaluate ln e e b) e c) d) ln ( ) 59) Simplify ln e 5 e + ln b) e + ln c) + ln d) + ln 5 5 5 5 5 5 5 5 60) Simplify ln e ln b) ln c) d) 6) Solve for ln e + 9
+ ln 9 b) 9 c) d) ln e 6) Simplify 5 7 + ln e 5 + ln 7 b) 7 + 5 c) ln 7 5 d) 5 6) Solve for ln ln 6 b) ln c) ln d) ln + ln 6) Solve for ln (7 ) + ln ( + 5) ln ( ) 6 b) 7 c) 7, 5 d) 6,5 65) Find the domain of the function f( ) ln( ) (, ) b) ( 0, ) c) (, ) d) (,) Limits NO CALCULATOR 66) Use the graph to estimate lim f( ) DNE b) 0 c) - d) 67) Use the graph to find lim f( ), if it eists
b) - c) DNE d) - e) - 68) Find + lim ( ) 7 b) 9 c) -7 d) ± 69) Find lim 5 + 70) Find lim f( ) if +, f( ), 7) If lim f ( ) and c lim g ( ) c, find lim [ f( ) g ( )] c 7) Find lim 5 6 + 0 b) -7 c) d) + 7) Find lim + 8 0 b) 0 c) d) e) DNE 7) Find the limit ( + ) ( + ) ( ) lim 0 b) - c) d) DNE