* A graphing calculator is highly recommended for this class!!!

Transcription

1 AP Calculus AB Summer Packet Ms. Febus - García marlene_febus@gwinnett.k.ga.us This packet includes a sampling of problems that students entering AP Calculus AB should be able to answer. The questions are organized by topic: A Basic Algebra Skills S Solving F Functions G Graphing T Trigonometry L Logarithms and Eponential Functions R Rational Epressions and Equations Students entering AP Calculus AB MUST have a strong foundation in algebra and trigonometry(pre-calculus). Most questions in this packet were included because they are the skills and concepts that will be used etensively in AP Calculus AB. Others have been included not so much because they are skills that are used frequently but because being able to answer them indicates a strong grasp of important mathematical concepts and more importantly the ability to problem-solve. An answer key to this packet has been provided at the end of this file. This packet will not be collected but you should complete it anyway. It is etremely important for all students to review the concepts contained in this packet and to be prepared for an assessment of prerequisite skills to take place within the first week of school. * A graphing calculator is highly recommended for this class!!!

2 A: Basic Algebra Skills A. True or false. If false change what is underlined to make the statement true. a. T F T F ( + ) = + 9 T F T F e. ( + ) = ( + ) T F f. 5 T F g. If ( + )( 0) = then + = or 0 =. T F ================================================== T: Trigonometry You should be able to answer these quickly without using calculator and without referring to (or drawing) a unit circle. T. Evaluate Trig Functions without a calculator:. cos.sin.sec 0. tan90 5. csc (-50). csc cos0 tan sin 8. sin 9. Cos. arcsin0. Tan. Sin 5. arctan0 T. Find the value of each epression in eact form. a. sin cos tan sec 5 7 e. csc f. cot 5

3 Note: You will need to know your trig identities Sum & Difference & Double Angle Formulas: Memorize the following Trig Identities: sin cos sec tan csc cot sin( ) sin cos( ) cos tan( ) tan cos cos sin sin cos cos sin cos cos sin T Find the value(s) of in [0 ) which solve each equation. a. sin cos tan sec e. csc is undefined f. cot T. Solve the equation. Give all real solutions if any. a. sin cos( ) tan 0 sec 9 e. csc( ) 0 f. cot 0 T5. Solve by factoring. Give all real solutions if any. a. sin + sin + = 0 cos cos = 0 sin cos sin = 0 tan + tan = + T. Graph each function identifying - and y-intercepts if any and asymptotes if any. a. y = -sin () y = + cos y = tan y = sec + e. y = csc () f. y = cot

4 S. Solve by factoring. a = 0 S: Solving + = 0 ( ) + ( ) 0 = = + 8 S. Solve by factoring. You should be able to solve each of these without multiplying the whole thing out. (In fact for goodness sake please don t multiply it all out!) a. ( + ) ( + ) + ( + )( + ) = 0 ( ) ( 9) + ( ) 5 ( 9) = 0 ( + ) 5 ( + 5) ( ) + ( + ) ( + 5) ( ) = 0 5 ( ) ( ) S. Solve. (Hint: Each question can be solved by factoring but there are other methods too) a. a( a ) (a ) 0 0 ( ) S. Solving Inequalities: Solve and graph the solution a e. 0 f. 0

5 L: Logarithms and Eponential Functions L. Evaluate Logarithms and Eponentials without a calculator a. log log 9 log0 ln e e. ln f. ln e g. log 7 h. log sin L. Epand as much as possible. a. ln y ln y ln lny L. Condense into the logarithm of a single epression. a. ln + 5ln y lna 5 ln ln ln ln ln (contrast with part c) L. Solve. Give your answer in eact form and rounded to three decimal places. a. ln ( + ) = ln + ln = ln + ln ( + ) = ln ln ( + ) ln ( ) = ln L5. Solve. Give your answer in eact form and rounded to three decimal places. a. e + 5 = = 8 00e ln = 50 = (need rounded answer only in d) L. Round final answers to decimal places. a. At t = 0 there were 0 million bacteria cells in a petri dish. After hours there were 0 million cells. If the population grew eponentially for t 0 how many cells were in the dish hours after the eperiment began? after how many hours will there be billion cells? The half-life of a substance is the time it takes for half of the substance to decay. The half-life of Carbon- is 558 years. If the decay is eponential what percentage of a Carbon- specimen decays in 00 years? how many years does it take for 90% of a Carbon- specimen to decay?

6 F: FUNCTIONS Graph each of the following Parent Functions and be familiar with these graphs. f ( ). f ( ). f ( ). f ( ) 5. f ( ). f ( ) 7. f( ) 8. f ( ) e 9. f ( ) ln 0. f ( ) sin. f ( ) cos. f ( ) tan -. f ( ) tan. f ( ) 5. f( ). f ( ) [ ] 7. f ( ) - 8. f ( ) Analyzing Functions F. Increasing/Decreasing Determine the interval(s) over which f() is: a. Increasing Decreasing Constant Linear e. Concave Up f. What are the zeros of f? g. For what values of is f() discontinuous? F. Compositions 9. Let f ( ) and g ( ) find the following: a. f ( g( )) g( f ( )) f ( ) Domain Range and Zeros of f() e. Domain Range and Zeros of g() Find f and verify that f f f f.. f. f

7 F. Piecewise Functions: Graph and then evaluate the function at the indicated points.. > f( ) a. f() f() f(5). - <- - f( ) << > a. f(-) f(-) f() f(5) e) f() F. Even/Odd Functions Show work to determine if the relation is even odd or neither. a. f ( ) 7 f ( ) f ( ) f ( ) e. f f. f ( ) sin ( ) F5. Domains of Functions: Find the Domain of each. a. y y y 5 8 y e. y f. y 9 9 F. Asymptotes Find the equation of both Horizontal and Vertical Asymptotes for the following functions. Find the coordinates of any holes. a. y y y y 9 8

8 R. a. f ( ) Function 8 R: Rational Epressions and Equations Domain Hole(s): ( y) if any Horiz. Asym. if any ( ) 8 f ( ) f ( ) skip skip 0 8 Vert. Asym.(s) if any R. Write the equation of a function that has a. asymptotes y = and = and a hole at ( 5) holes at (- ) and ( -) an asymptote = 0 and no horizontal asymptote R. Find the -coordinates where the function s output is zero and where it is undefine a. For what real value(s) of if any is the output of the function f ( ) e equal to zero? undefined? For what real value(s) of if any is the output of g ( ) equal to zero? undefined? cos sin ================================================== G: Graphing G. PART of the graph of f is given. Each gridline represents unit. a. Complete the graph to make f an EVEN function. What are the domain and range of feven? What is feven(-)? Complete the graph to make f an ODD function. e. What are the domain and range of fodd? f. What is fodd(-)?

9 G. The graphs of f and g are given. Answer each question if possible. If impossible eplain why. Each gridline represents unit. a. f - (5) = f (g(5)) = (g f )() = Solve for : f (g ()) = 5 e. Solve for : f () = g () For parts f i respond in interval notation. f. For what values of is f () increasing? g. For what values of is g () positive? h. Solve for : f () < i. Solve for : f () g() G. Given the graph of y = f () (dashed graph) sketch each transformed graph. a. y = f ( + ) y = f () y = f () y = f () + ========================================================================

10 A. a. true false; 7/ false; false; + e. true f. false; g. false; T. Answer Key Undefined T. a. T. a. - e. f. e. 0 f. T. a. n n T5. n n n 9 e. no solution f. n a. n 7 n n n n n n = - n T. a. e. f. S. a S. a. < -9 or > 5 7 <- or > 5 - < < S. a S. e f. or 5 S. a. 7 -

11 L. a. - e. 0 f. g.7 L. a. ln + ln y ln ( + ) ln ln y ln ln ln + ln + ln y h. sin L5. a. = = 5 =.70 L. a. ln y 5 ln ln a log (change of base) L. a. = e.89 = (- is etraneous) = e L. a million cells.70 hours.7% 89.9 years F. a. 0 (80) 0 8 F.. a. e. ( 0) f. = 0 g. = - = 9 9 Range: [0 ) Zeros: = 0 Domain: e. Domain: [- -] [- ) f Range: ) (. f. f Zeros: = 9 F.. a. 7. a. 8 7 e. Undefined F. a. even odd neither odd e. even f. odd F5. a. 0 0 e. [ ) 9 f. F. a. VA: = HA: y = Holes: None VA: =- = HA: y = 0 Holes: None VA: =- = HA: y = Holes: None VA: =0 = HA: y = 0 Holes:

12 R. a (0 ) none none ( ) ( ) skip skip = y = = R. a. Answers vary. One possibility: ( )( ) ( )( ) Answers vary. One possibility: ( )( )( ) ( )( ) R. a. = 0: never undefined: at = 0 = 0: at = n undefined: never R. a. ( ( ) ) ( ) 5 ( ) ( 5 ) G. a. see graph D: [-5 5] R: [- ] 5 see graph e. D: [-5 5] R: [- ] f. -5 G. a. 5.5 that notation means the same thing as g (f ()) =.5 e. = f. ( ) g. (- ) h. [0 ) i. [ 5] G. a.