AP Calculus AB – 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 next 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 expressions, and basic factoring, for example should now be a part of you in order to be successful in AP Calculus. Only the topics students consistently do not have down in their basic skill set are included here. These are skills that are used continually in AP Calculus. On the following six pages, you have assorted problems related to specific topics. Each problem should be done in the space provided. Rather than give you a textbook to remind you of the techniques necessary to solve the problem, there are a few websites listed that have full instructions on the techniques. If and when you are unsure of how to attempt these problems, examine 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 example, may require you to be an expert at working with inequalities. Solving quadratic equations may involve techniques used in solving fractional equations. This packet is due the first day back in school in the fall. It will be graded. You need to get off to a good start so spend some quality time on this packet this summer. You may either print this packet out or you can work out the problems on separate paper. If using separate paper, please label every problem and circle your answers. Please work the problems in order. Be sure your name appears on the first sheet. Work needs to be shown when needed. No credit will be given on problems requiring work if only the final answer is recorded. Also, do not rely on the calculator. Two-thirds of your AP exam next year is taken without the calculator, so paper and pencil techniques only. It is a mistake to decide to do all of this now. Let it go until mid-summer, or work on it periodically throughout the summer. Focus on MASTERY of the concepts, not just “getting it done”. We want you to be comfortable with these techniques 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 any questions about any of these problems or techniques used in solving them, contact Mr. Brust at the school website/e-mail address below. Have a good summer and see you in August! Mr. Brust patricki.brust@cms.k12.nc.us

Here are some good sites for most algebra topics: http://www.purplemath.com/modules/index.htm http://www.shef.ac.uk/mash/maths http://www.khanacademy.org Here is a good site for solving many equations with a guided step-by-step process: http://www.wolframalpha.com You Tube/Google There are many math concepts taught through short videos on You Tube. Many more concepts are discussed via Google. Beginning Algebra Topics Exponents 1. Negative and fractional exponents Intermediate Algebra Topics 2. 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. Complex fractions 12. Composition of functions 13. Solving rational (fractional) equations Trig Information 14. Basic right angle trig 15. Trig equations (YOU MUST KNOW UNIT CIRCLE LIKE THE BACK OF YOUR HAND!)

Common Algebra Errors to Remember:

Trig Identities Formula Sheet

Reciprocal identities

Pythagorean Identities

Quotient Identities

Even-Odd Identities

Sum-Difference Formulas

Double Angle Formulas

AP Calculus AB Name _______________________________ Summer Packet Problems with an * involve unique techniques to solve. The techniques have been learned, though not necessarily put together in the way required to use.

Section1: Monomial Factors (work must be shown) Factor as indicated

1. (____)43 2234 xxxx =−+ 2. (____)262 2

3

xxx =+

3. (____)2 2 xxxx eexxee −−−− =+− 4. (____)sintansin 2 xxx =+ 5. (____)2

1

42

13 xxx

=+

Section 2: Binomial Factors (work must be shown) Factor as indicated

1. )(____)1()1(3)()1( 2 −=−−− xxxx 2. (____)1

1

11

22

22

+=

+−+

xx

xx

*3. )(____)()12()()12()()12( 2

1

2

3

2

1

2

5

2

1

2

3−−

+=+++ xxxxxx

Section 3: Factoring Quadratic Expressions Factor the polynomials as a product of linear factors. For exponential expressions factor as indicated.

1. 2 5 6x x+ − = 2. 2 264 121x z− = 3. 22 5 3x x+ − = 4. 3sin sinx x− =

Section 4: Cancellation (work must be shown) Reduce each expression to lowest terms:

1. 4

23

)1(

)1(3)2()1(

+

++−+

x

xxx 2.

6

1

3

1

2

1

x

xx − 3.

1

)1(1 2

3

−

−+−

x

xx 4.

x

xx

sin2

)cos(sin1 2+−

Section 5: Quadratic Formula (work must be shown) For each equation, solve for the indicated expression.

1. xforxx ,0142 =−− 2. xforxx ,032 2 =−+

3. 2cos 3cos 2 0 , cos ,x x for x then for x+ + = 4. xforyxyx ,0)1( 22 =+−−

Section 6: Simplifying (work must be shown) Rewrite each of the following in simplest form:

1. 1

)1()3)(1( 2

+

+−+−

x

xxx 2.

1

1

11

2

2

2

+

+−+

x

xx

3. 1

2

1

1

1

12 −

−−

−+ xxx

4. 2

2

2 1

11

12

)2(

xx

x

xx

−+−+

−

−

Section 7: Rationalizing (work must be shown) Remove the sum or difference from the denominator by multiplying the numerator and denominator by the conjugate of the denominator.

1. xcos1

1

− 2.

11 2 +− x

x

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

Section 8: Function Transformation If 𝑓(𝑥) = 𝑥2, describe in words what the following would do to the graph of ( )f x .

1. ( ) 4f x − 2. ( 4)f x − 3. ( 2)f x− +

4. 5 ( ) 3f x + 5. (2 )f x 6. ( )f x

Given the graph of ( )y f x= . Sketch the following:

7. 2 ( )y f x= 8. ( )y f x= − 9. ( 1)y f x= −

10. ( 2)y f x= + 11. ( )y f x= 12. y f x=

Section 9: Trigonometry Review Solve the following problems. If point P is on the terminal side of , find all 6 trig functions of . Draw a picture.

1. ( 2, 4)P − 2. ( )5, 2P −

3. If 5

cos13

−

= , is quadrant II, 4. If cot 3 = , in quadrant III,

find all 6 trig functions of . find all 6 trig functions of .

For problems 5-10, solve each equation on the interval without using a calculator (show work)

)0, 2 .

5. 1

sin2

x = 6. 2cos cosx x= 7. 2cos 3 0x+ =

8. 22sin sin 1x x+ = 9. 2cos 2cos 3x x+ = 10. 2sin cos sin 0x x x+ =

Section 10: Solving Inequalities (show work) Solve the following inequalities and graph the solution on the real number line.

1. 129 − x 2. 0, − bbax

Section 11: Writing Equations (show work)

Write an equation for each of the problems below. Not all will be linear. 1. The line that passes through (-3,6) and (4,10). 2. The line parallel to 3x-2y=8 that passes through (-1,-7). 3. The line perpendicular to 𝑥 = 7 that passes through (6, -4). 4. If the slope of a line can be calculated by 𝑚 = 3𝑥, what is the equation of the line passing through (4, 9) ? 5. If the length of a rectangle is 3 more than twice its width, write an equation that would calculate the area of the rectangle. 6. A piece of cardboard measuring 10 in by 13 in is going to be used to make a box with an open top by cutting out equal squares from each of the corners. Write an equation that would calculate the volume of the box. 7. In a research experiment, a population of fruit flies is increasing according to the law of exponential growth. After 2 days there are 100 flies, and after 4 days there are 300 flies. Write an equation to calculate the number of flies after t days.

Section 12: Using the calculator

Solve the following using a graphing calculator. Round all answers to 3 decimal places.

1. 3 2 5x x= − 2. 3xe x= 3. 16 4

9log log

4x x+ =

Section 13: Exponential/Logarithmic Properties (show work) Find the exact value without a calculator:

1. 20 15

4log 4 ln e− 2. 3

8

log 81

log 9

3

8 3. 8 4log 2 log 2+ 4. 3 3log 5 log 4

3−

Use the rules of logarithms to rewrite the expression as a single logarithm:

5. 2 2

1log log

3x x+ 6. 3log 4log 2logu v w+ −

7. 2 2

4 4 4log ( 1) log ( 1) log ( 2 1)x x x x− + + − + + 8. 21

ln 2ln ln( 1)1

x xx

x x

+ + − −

−

Section 14: Solving Exponential/Logarithmic Equations (show work)

1. 3 2x xe e −= 2. 2 6 125

125

x+ = 3. 2 7 23 27x x− =

4. 2 24 2 16x x = 5. 35(2 ) 8x =

6. 5 5log (2 3) log 7x+ = 7. log log( 21) 2x x+ − =