A.P. Calculus Summer Packet Going into AP calculus, there are certain skills that have been taught to you over the previous years that we assume you have. If you do not have these skills, you will find that you will consistently get problems incorrect net year, even though you understand the calculus concepts. It is frustrating for students when they are tripped up by the algebra and not the calculus. This summer packet is intended for you to brush up and possibly relearn these topics. We assume that you have basic skills in algebra. Being able to solve equations, work with algebraic epressions, and basic factoring, for eample should now be a part of you. If not, you would not be going onto AP calculus. So, only the topics I see that students consistently do not have down in their basic skill set are included here. These are skills that are used continually in A.P. Calculus. On the following 15 pages, you have 9 to 1 problems per page. Each problem should be done in the space provided. Rather than give you a tetbook to remind you of the techniques necessary to solve the problem, I have given you several websites that have full instructions on the techniques. If and when you are unsure of how to attempt these problems, eamine these websites. Don t fake your way through these problems. As stated, students are notoriously weak in them, even students who have achieved well prior to AP Calculus. Use the websites. Realize also that certain concepts are interrelated. Domain, for eample, may require you to be epert at working with inequalities. Solving quadratic equations may involve techniques used in solving fractional equations. It is a mistake to decide to do this now. Let it go until mid-summer. I want these techniques to be relatively fresh in your mind in the fall. Also, do not wait to do them at the very last minute. These take time. If you do two concepts a day, the whole packet will take you about a week to complete. If you have questions about any of these problems or techniques used in solving them, contact me at the school website/email address. Have a good summer and see you in the fall. You need to get off to a good start so spend some quality time on this packet this summer. Tear off these first two sheets and return the 17 sheets stapled together. Be sure your name appears on the first sheet. Work needs to be shown when needed. Also do not rely on the calculator. Half of your AP eam net year is taken without the calculator. So paper and pencil techniques only. Grading This packet is due on the first day of school. It will count as a formative grade. There will be a summative assessment over the content of this packet by the end of the first week.
The topics listed are below. You can certainly do Google searches for any of these topics. But I have given you several sites that will cover pretty much all of these topics. Here is a good site for most algebra topics: http://www.purplemath.com/modules/inde.htm Beginning algebra topics Eponents 1. Negative and fractional eponents Intermediate algebra topics. Domain 3. Solving inequalities: absolute value 4. Solving inequalities: quadratic 5. Special Factoring formulas 6. Function transformation 7. Factor theorem (p over q method) 8. Even and odd functions 9. Solving quadratic equations and quadratic formula Advanced algebra topics 10. Asymptotes 11. Comple fractions 1. Composition of functions 13. Inverse, Eponential and Logarithmic Functions 14. Solving Rational (fractional) equations 15. Difference Quotient Trig Information 16. Basic right angle trig 17. Trig equations
Name Topic 1: Fractional & Negative Eponents Simplify using only positive eponents 3 1. 3 3. 5 ( 4 9 ) 1 ( 9) 3. ( ) 1 ( ) 3 4 5. 4. 16 y sin 6. 4 16 4 ( 4) 3 1 3 7. 4 +1 +1 +1 ( ) ( 1) ( ) 8. 1 + 5 ( ) 3 3 9. 1 + 4 1 y + 1 1 y 1
Topic : Domain Find the domain of the following functions: 1. 3 y =. 4 + 1 4 y = 3. + 4 y = 5 6 3 18 4. y = 5. y = 3 + 3 6. y = 9 + 9 7. + 8 + 1 y = 8. y = 5 14 9. 4 + 5 y = 3 6 30 10. = log( 1) y 11. y = tan 1. y = cos
Topic 3: Solving inequalities (absolute value) Write the following absolute value epressions as piecewise epressions 1. y = 4. y = 6 + +1 3. y = 4 +1 + 3 Solve the following absolute value inequalities 4. 3 >1 5. 3 4 6. 10 + 8 > 7. 3 4 > 8. 6 > 8 9. +1 3
Topic 4: Solving inequalities (quadratic) Write the following absolute value epressions as piecewise epressions 1. 1. + 1 3. + 4 + 4 Solve the following by factoring and making appropriate sign charts. 4. 16 > 0 5. + 6 16 > 0 6. 3 10 7. + 4 3 8. 3 + 4 4 9. sin sin 0 <
Topic 5: Special factorization Factor completely 1. 3 + 8. 3 8 3. 73 15y 3 4. 4 +11 80 5. ac + cd ab bd 6. + 50 y 0y 7. +1 + 36 9y 8. 3 y + y y 3 9. ( 3) ( +1) 3 + ( 3) 3 ( +1)
Topic 6: Function transformation If f ( ) = 1, describe in words what the following would do to the graph of f ( ) : 1. f ( ) 4. f ( 4) 3. f ( + ) 4. 5 f ( ) + 3 5. f ( ) 6. f ( ) Here is a graph of y = f ( ). Sketch the following graphs 7. y = f ( ) 8. y = f ( ) 9. y = f ( 1) 10. y = f ( + ) 11. y = f ( ) 1. y = f
Topic 7: Factor theorem (p over q method/synthetic division) Use the p over q method and synthetic division to factor the polynomial P ( ). Then solve P( ) = 0. 1. P( ) = 3 + 4 + 6. P( ) = 3 + 5 4 3. P( ) = 3 6 + 3 10 4. P( ) = 3 + 19 0 4 3 4 3 5. P( ) = + 5 + 6 4 8 6. P( ) = + 11 + 41 + 61 + 30
Topic 8: Even and odd functions Show work to determine if the relation is even, odd, or neither 1. f ( ) = 7. f ( ) = 4 3 3. f ( ) = 4 4 + 4 4. f ( ) 1 = 5. f = + 1 6. 5 6y = 1 7. y = e 1 e 8. 3y3 = 4 3 +1 9. 3 = y
Topic 9: Solving quadratic equations and quadratic formula Solve each equation. 4 ( ) 5 ( 1) = 1. 7 3 = 0 3. + 6 + 4 = 0 4. 3 + 3 = 0 5. ( + )( 3) = 1 6. 1 13 + = 6 7. 4 9 + 8 = 0 8. 10 + 9 = 0 9. 1 1 = 6
Topic 10: Asymptotes For each function, find the equations of both the vertical asymptote(s) and horizontal asymptotes (if they eist) 1. y = 3. y = + 4 1 3. y = + 4 +1 4. y = +1 3 4 5. y = 9 3 + 3 18 6. y = + 6 3 3 4 7. y = 6 3 + 6 8. y = 3 3 1 9. y = 10
Topic 11: Comple fractions Simplify the following 1. 1. 1 + 4 1 3. 1 + 1 4. 3 4 y 4 3 y 5. 1 3 4 9 6. y y + y y 7. 3 1 8. 1 + 1+ 1 + 1+ 9. 4 5 + + 3 10 + 3
Topic 1: Composition of functions If f ( ) =, g( ) = 1, and h( ) =, find the following ( ).. f ( g( ) ) 3. f ( h( 1) ) 1. f g( ) ( ) 5. g f h 1 4. h f ( 1) 6. f ( g( ) ) ( ) 8. g( g( ) ) 9. f ( h( ) ) 7. g f ( )
Topic 13: Inverse, Logs, and Eponential Functions For Each problem, Find the inverse and identify the domain and range of both f and f 1. 3 1. f() = + 10. f() = 1 3. f() = +3 + 4. f() = 3 + 4 5. f() = + 4, 0 6. f() = ln( + 4) Solve each Equation: 7. e 3 = e e 8. 5( 3 ) = 9 9. 3 1 = 4 10. e 4 e = e 1 11. log ( + 1) = 3 1. ln( + 3) = ln 3
Topic 14: Solving Rational (fractional) equations Solve each equation for 1. 3 5 6 = 1. + 6 = 5 3. +1 3 1 =1 4. 5 +1 = 3 5 5. 60 60 5 = 6. + 5 + 1 5 = 16 5 7. + 4 = 5 + 8. 6 3 6 + 9 = 3 9 9. + 3 1 = 10 1 + 3 +1
Topic 15: Difference Quotient For Questions 1-6, Find the difference Quotient of f 1. f() = 3 + 1. f() = 1 +3 3. f() = + 7 4. f() = +7 5. f() = + 6. f() = ( + 3)
Solve the following problems. Topic 16: Solving Rational (fractional) equations If point P is on the terminal side of, find all 6 trig functions of. Draw a picture. 1. P(,4). P( 5,) 3. If cos = 5, in quadrant II, 13 4. If cot = 3, in quadrant III, find sin and tan find sin and cos Find the eact value of the following without calculators: 5. sin 5 cos 300 6. ( 6sec180 4cot 90 ) 7. ( 4cos30 6sin10 ) Solve the following triangles (3 decimal place accuracy) 8. A = a = 1.7 B =16 b = C = 90 c = 9. A = a = 6 feet B = b = C = 90 c = 95 inches
Topic 17: Solving Trigonometric equations Solve each equation on the interval [0,) 1. sin = 1. cos = cos 3. cos + 3 = 0 4. 4sin =1 5. sin + sin =1 6. cos + cos = 3 7. sin cos + sin = 0 8. 8cos cos =1 9. sin cos = 0