AP Calculus AB Summer Packet DUE August 8, Welcome!
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1 AP Calculus AB Summer Packet 06--DUE August 8, 06 Welcome! This packet includes a sampling of problems that students entering AP Calculus AB should be able to answer without hesitation. The questions are organized by topic: A Fundamental Algebra Skills T Trigonometry F Higher-Order Factoring L Operations with Logarithms and Eponential Functions R Rational Epressions and Equations G Graphing Students entering AP Calculus AB must have a strong foundation in algebra as well as trigonometry. Most questions in this packet concern skills and concepts that will be used etensively in AP Calculus. 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 the packet has been provided at the end of this file. Although this packet will not be collected, 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 that may take place within the first week of school. A low or failing score on the pre-requisite assessment may indicate that a student either a) lacks fundamental prerequisite skills necessary for success in AP Calculus AB, or b) lacks the work ethic necessary for success in AP Calculus. The course requires (and your teacher epects) that problems be represented in multiple ways (numerically, algebraically, graphically), solved using an appropriate analysis, and justified using a logical mathematical reasoning. This is not a course where every problem given on a test or quiz is identical to problems done repeatedly during class. The AP eam requires students to take skills and concepts and apply these to seemingly different situations or to connect multiple mathematical ideas. Take a deep breath and begin working pacing yourself rather than completing the packet in one sitting at the last minute. If you have the basics down and you put in the necessary work, you will see how amazing Calculus is and how it is the capstone to all mathematics you have learned thus far! AP Calculus AB is challenging, demanding, and rewarding. Mrs. Ramsey or Dr. Wynne may be contacted at RamseyB@FultonSchools.org or WynneB@FultonSchools.org If you have need a refresher on a particular topic, you are encouraged to consult YouTube or Khan Academy
2 A: Fundamental 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. ( + ) = 6( + ) T F f. 5 T F g. If ( + )( 0) =, then + = or 0 =. T F A. More basic algebra. a. If 6 is a zero of f, then is a solution of f () = 0. Lucy has the equation ( + 6) 8 = 6. She multiplies both sides by ½. If she does this correctly, what is the resulting equation? Simplify 0 Rationalize the denominator of e. If f () = + + 5, then f ( + h) f () = (Give your answer in simplest form.) f. A cone s volume is given by V r h. If r = h, write V in terms of h. g. Write an epression for the area of an equilateral triangle with side length s. h. Suppose an isosceles right triangle has hypotenuse h. Write an epression for its perimeter in terms of h.
3 T: Trigonometry You should be able to answer these quickly, without referring to (or drawing) a unit circle. T. Find the value of each epression, in eact form. a. sin cos 6 tan cos tan e. tan f. tan T. Find the value(s) of in [0, ) which solve each equation. a. sin cos tan sec T. Find the value(s) of in [0, ) which solve each equation. a. cos sin 0 sin = cos T. Solve by factoring. Give all real solutions, if any. a. sin + sin + =0 cos cos = 0 sin cos sin = 0 tan + tan = + T5. 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 T6. Find an equivalent algebraic epression. sin tan a. cos( arctan ) You will need to know your trig identities as well as Double-Angle Formulas
4 F. Solve by factoring. a = 0 F: Higher-Level Factoring + 6 = 0 ( 6) + ( 6) 0 = = + 8 F. 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. ( + ) ( + 6) + ( + )( + 6) = 0 ( ) ( 9) + ( ) 5 ( 9) = 0 ( + ) 5 ( + 5) ( ) + ( + ) ( + 5) ( ) = ( ) ( ) F. Solve. Each question can be solved by factoring, but there are other methods, too. a. a ( a ) (a ) ( )
5 L: Logarithms and Eponential Functions L. Epand as much as possible. a. ln y ln y ln ln y L. Condense into the logarithm of a single epression. a. ln + 5lny ln a 5ln 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 L. Solve. Give your answer in eact form and rounded to three decimal places. a. e + 5 = = 8 00e ln = 50 = (need rounded answer only on d) L5. Round final answers to decimal places. a. At t = 0 there were 0 million bacteria cells in a petri dish. After 6 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 5568 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 R. a. f ( ) Function 8 R: Rational Epressions and Equations Domain Hole(s): (, y) if any Horiz. Asym., if any ( ) 8 f ( ) 6 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 For what real value(s) of, if any, is the output of the function a. f ( ) e 6 equal to zero? undefined? For what real value(s) of, if any, is the output of g ( ) equal to zero? undefined? cos sin R. Simplify completely. a. (Don t worry about rationalizing) (Your final answer should have just one numerator and one denominator) 5 ( ) ( ) 5 (Don t worry about rationalizing)
7 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(-)? 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 () +
8
9 Answer Key A. a. true false; 7/ false; false; + e. true f. false; g. false; 0, 0, 0 A. a. = ( + 6) = 8 0 ( ) or 0 e. 6h + h + h f. V h g. s 6 0 h. h ( ) or anything equivalent T. a. - e. f. T. a.,,, T. a. T. 7,, 6 6 a. n n 5,,, 6 6 n n n n n = - T6. a. + + T5. a. e. f. F. a. -5, -, -, -,,, -,, F. a. -6, -, - -, 0,, 5, -9, -5, -,,,, 5 5 F. a., 7 -,,
10 L. a. ln + ln y ln ( + ) ln ln y ln ln ln + ln + ln y L. a. ln y 5 ln a ln log (change of base) e L. a. = e.89 = (- is etraneous) = 7 L. a. = 5 = =.70 L5. a million cells.70 hours.7% years R. a., 5 7 5, y = = 6 0 (0, ) none none (, ) (, ) skip skip = R. a. Answers vary. One possibility: ( )( ) ( )( ) Answers vary. One possibility: ( )( )( ) ( )( ) R. a. = 0: never undefined: at = 0 = 0: at = n undefined: never R. a. 6 ( ( ) ) 6 ( ) 5 ( ) ( ) 5 G. a. see graph 5 D: [-5, 5] R: [-, 6] see graph e. D: [-5, 5] R: [-6, 6] f. -5 G. a. 5.5 that notation means the same thing as g (f ()) =.5 e. = f. (, 6) g. (-, ) h. [0, ) i. [, 5]
11 G. a.
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