Calculus Summer Work

Transcription

1 Calculus Summer Work About This Packet ( and this class) Welcome! 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. 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. (If you re the sort of student who doesn t do homework unless forced to, Calculus might not be the best place for you ) 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 - 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. 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.

2 A: Super-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. + = T F g. If ( + )( 0) =, then + = or 0 =. T F A. More basic algebra. a. If is a zero of f, then is a solution of f () = 0. Lucy has the equation ( + ) 8 =. 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 () = + +, 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 tan sec 7 e. csc f. cot 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 T. 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 F. Solve by factoring. a. + = 0 F: Higher-Level Factoring + = 0 ( ) + ( ) 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. ( + ) ( + ) + ( + )( + ) = 0 ( ) ( 9) + ( ) ( 9) = 0 ( + ) ( + ) ( ) + ( + ) ( + ) ( ) = 0 = ( + ) ( ) F. Solve. Each question can be solved by factoring, but there are other methods, too. a. a ( a + ) + (a + ) = = 0 + = + + = + ( + ) + +

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 + ln y lna + 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 L. Solve. Give your answer in eact form and rounded to three decimal places. a. e + = = 8 00e ln = 0 = (need rounded answer only on 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 8 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 ( ) = + 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 (, ) 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 + R. Simplify completely. a. + + (Don t worry about rationalizing) ( ) + (Your final answer should have just one numerator and one denominator) ( + ) ( + ) (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 - () = f (g()) = (g f )() = Solve for : f (g ()) = 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 Answer Key A. a. true false; 7/ false; false; + e. true f. false; g. false; 0, 0, 0 A. a. = ( + ) = 8 ± 0 ( ) or 0 e. h + h + h f. V = h g. s 0 h. h ( + ) or anything equivalent T. a. - e. f. T. a.,,, e. 0, f., T. a. = n = + n, T. + = + n = n = n, = e. no solution f. = n a. = n = n = + n = n = n = + n = n = - n T. a. e. f. F. a. -, -, -, -,,, -,, F. a. -, -, - -, 0,,, -9, -, -,,,, ± F. a., 7 -,,

9 L. a. ln + ln y ln ( + ) ln ln y ln + ln ln + ln + ln y L. a. ln y lna ln log (change of base) L. a. = e = (- is etraneous) = 7 L. a. = = =.70 L. a million cells.70 hours.7% 89.9 years R. a., ( 7 ), y = = 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 = 0. + n undefined: never R. a. ( ( + ) + ) + ( + ) + + ( + ) + + ( + ) G. a. see graph D: [-, ] R: [-, ] see graph e. D: [-, ] R: [-, ] f. - G. a.. that notation means the same thing as g (f ()) =. e. = f. (, ) g. (-, ) h. [0, ) i. [, ] G. a.