Welcome to AP Calculus AB!

Transcription

1 Welcome to AP Calculus AB! This packet includes a sampling of problems that students entering AP Calculus should be able to answer. The questions are organized by topic: A Super-Basic Algebra Skills T Trigonometry F Higher-Level Factoring L Logarithms and Eponential Functions R Rational Epressions and Equations G Graphing In Calculus, it s rarely the calculus that ll get you; it s the algebra. Students entering AP Calculus absolutely must have a strong foundation in algebra. Most questions in this packet were included because they 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. 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 -5 days of school. Students whose scores show they were not prepared for the assessment probably either a) don t have the mathematical prerequisite skills necessary for success in AP Calculus, or b) don t have the work ethic necessary for success in AP Calculus. If you are struggling on a topic, please contact me over the summer and I will help you find resources to review the content. The curriculum (and your teacher) will epect you to approach problems with the mathematical toolkit needed to do the calculations and the mathematical understanding needed to make sense of unusual problems. This is not a class where every problem you see on tests and quizzes is identical to problems you ve done dozens of times in class. This is because the AP test itself (and, truly, all real mathematics) requires you to take what you know and apply it, rather than to simply regurgitate a rote process. Now that I ve said all that, I encourage you to take a deep breath and start working. If you have the basics down and you put in the work needed, you ll see how amazing Calculus is! AP Calculus is challenging, demanding, rewarding, and to put it simply totally awesome. I can t wait to see you in the fall! Mrs. McDonald bibowman@k1.wv.us Remind Tet Alerts (please sign up now) PS: It is STRONGLY recommended that you purchase a TI-84 Plus CE calculator before the fall. Calculators are used regularly during class and at home and I do not have enough to send home with you daily. Usually before school begins they go on sale at Wal-Mart/Target (or other similar stores) for around $100.

2 A: Super-Basic Algebra Skills A1. True or false. If false, change what is underlined to make the statement true. 4 a. 1 T F 1 b. T F c. ( + ) = + 9 T F 1 d. 1 T F e. (4 + 1) = 16( + ) T F f. 5 T F g. If ( + )( 10) =, then + = or 10 =. T F A. More basic algebra. a. If 6 is a zero of f, then is a solution of f () = 0. b. Lucy has the equation (4 + 6) 8 = 16. She multiplies both sides by ½. If she does this correctly, what is the resulting equation? c. Simplify 4 10 d. Rationalize the denominator of 1 1 e. If f () = + + 5, then f ( + h) f () = (Give your answer in simplest form.) f. A cone s volume is given by V 1 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. T1. Find the value of each epression, in eact form. a. sin b. cos 11 6 c. 4 tan d. sec 5 csc f. cot e. 4 T. Find the value(s) of in [0, ) which solve each equation. a. sin b. cos 1 c. tan d. sec e. csc is undefined f. cot 1 T. Solve the equation. Give all real solutions, if any. a. sin 1 b. cos( ) c. tan 0 d. 4 sec 1 9 e. csc( 4 ) 0 f. cot 6 0 T4. Solve by factoring. Give all real solutions, if any. a. 4sin + 4 sin + 1 = 0 b. cos cos = 0 c. sin cos sin = 0 d. tan + tan = + T5. Graph each function, identifying - and y-intercepts, if any, and asymptotes, if any. a. y = -sin () b. y = 4 + cos c. y = tan 1 d. y = sec + 1 e. y = csc () f. y = cot

4 F1. Solve by factoring. a = 0 F: Higher-Level Factoring b = 40 c. ( 6) + ( 6) 10 = 0 d = + 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) 4 = 0 b. ( ) ( 9) + ( ) 5 ( 9) = 0 c. ( + 11) 5 ( + 5) ( 1) + ( + 11) 4 ( + 5) 4 ( 1) = 0 d ( 1) ( 4) F. Solve. Each question can be solved by factoring, but there are other methods, too. 1 a. a ( a ) (a ) 0 b. 6 0 c. d. 6 1 ( 1) 1 1

5 L: Logarithms and Eponential Functions L1. Epand as much as possible. a. ln y b. ln 4y c. ln d. ln 4y L. Condense into the logarithm of a single epression. a. 4ln + 5ln y b. lna 5ln c. ln ln d. ln ln (contrast with part c) L. Solve. Give your answer in eact form and rounded to three decimal places. a. ln ( + ) = b. ln + ln 4 = 1 c. ln + ln ( + ) = ln d. ln ( + 1) ln ( ) = ln L4. Solve. Give your answer in eact form and rounded to three decimal places. a. e = 1 b. = c. 100e ln 4 = 50 d. = 1 (need rounded answer only on d) L5. Round final answers to decimal places. a. At t = 0 there were 140 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 11 hours after the eperiment began? after how many hours will there be 1 billion cells? b. The half-life of a substance is the time it takes for half of the substance to decay. The half-life of Carbon-14 is 5568 years. If the decay is eponential what percentage of a Carbon-14 specimen decays in 100 years? how many years does it take for 90% of a Carbon-14 specimen to decay?

6 R1. a. b. c. f ( ) Function R: Rational Epressions and Equations Domain Hole(s): (, y) if any Horiz. Asym., if any (4 ) 48 f ( ) 6 4 f ( ) skip skip 10 8 Vert. Asym.(s), if any R. Write the equation of a function that has a. asymptotes y = 4 and = 1, and a hole at (, 5) b. holes at (-, 1) and (, -1), an asymptote = 0, and no horizontal asymptote R. Find the -coordinates where the function s output is zero and where it is undefined. a. For what real value(s) of, if any, is the output of the function f ( ) e equal to zero? undefined? b. For what real value(s) of, if any, is the output of g ( ) equal to zero? undefined? cos sin R4. Simplify completely. a. 4 4 (Don t worry about rationalizing) b. 1 4 (Your final answer should have just one numerator and one denominator) c. 5 1 d. ( ) 1 ( ) 5 (Don t worry about rationalizing)

7 G: Graphing G1. PART of the graph of f is given. Each gridline represents 1 unit. a. Complete the graph to make f an EVEN function. b. What are the domain and range of feven? c. What is feven(-)? d. 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 1 unit. a. f -1 (5) = b. f (g(5)) = c. (g f )() = d. 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 () < 4 i. Solve for : f () g() G. Given the graph of y = f () (dashed graph), sketch each transformed graph. a. b. c. d. y = f ( + ) y = f () y = f () y = f () + 1

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