A.P. Calculus – Summer Work 2014

Instructions for AP Calculus AB summer work.

Before leaving school this year (due Friday, May 30, 2014):

1. Log onto your school email account and send me an email message (laurie.brandvold@thompsonschools.org) with the following information:

Your name (as you would like me to address you),

your career plans (if you have an idea),

which class(es) you are taking from me next year

the best way to contact you over the summer, AND

the best way to contact you during the school year. We will be using this email account to correspond and to log into Google-related applications. Your login should be fmlastXX@thompsonschools.org where f is your first initial, m is your middle initial, last is your entire last name, and XX is your 20XX graduation year. Your password should be your student number. However, it is possible that you used BHS or 00 followed by your student number for a password. This is why I want you to login before you leave school so the IMC folks can help you if you need to get something changed before everyone goes home for the summer.

Due on the first day of class next year:

2. Purchase the textbook for class. You may order the textbook, new or used, through various websites or students who had the class this year. Please be sure to allow for shipping time, which can sometimes be several weeks! If you search on the ISBN number, you should get several hits.

Here is the information you will need for ordering the text:

Title: Calculus: Graphical, Numerical, Algebraic, 3rd Edition / AP* Edition By Finney, Demana, Waits and Kennedy Published by Pearson / Prentice Hall ISBN 0-13-201408-4

3. Secure a graphing calculator to use on a daily basis. TI – 83/84’s and TI-Nspires are the most popular and widely

supported models. In any case, be sure to familiarize yourself with the use of your calculator before classes begin.

4. There are certain skills that have been taught to you over the previous years that we assume you have when you begin AP Calculus. 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. This is frustrating for students and teachers. 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. If not, you should do a much more significant review of algebra than is given here. Only the skills that I see students consistently do not have are included here.

On the following pages, you have 9 to 12 problems per page. Each problem should be done in the space provided. I have given you a couple of websites that have full instructions on the techniques, if you need a reminder. When you are unsure of how to attempt these problems, examine these websites. Do not 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 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. 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 exam next year is taken without the calculator, so use paper and pencil techniques first.

5. Other supplies that you should always bring to class include pencils, lined paper , graph paper, a 3-ring binder and dividers.

6. Have a restful summer! I look forward to spending the school year with you. If you have any questions, e-mail

me at laurie.brandvold@thompsonschools.org or call me at home (690-7013). Please no calls between 10pm and 5:30am.

The topics for review are listed below. You can certainly do Google searches for any of these topics. But two great sites that I recommend for most of them are: http://www.purplemath.com/modules/index.htm https://www.khanacademy.org/ (If you find other helpful websites, please email me the link and I will share it with the class.)

Beginning algebra topics

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

Pre-Calculus

16. Limits and Difference Quotients

Name_______________________________________

Topic 1: Fractional & Negative Exponents Simplify using only positive exponents

1.

!

"3x"3 2. ( ) ( )994

2

35 2

1

!!"#

$%&

'!

!

x 3.

!

22

2 " x

#

$ %

&

' (

"2

2 " x( )2

)

*

+ +

,

-

.

.

4.

!

16x2y( )

3

4 5.

!

"x

"1

2

2sin x 6.

!

4x "16

x " 4( )3

4

7.

!

"42x "1

2x +1

#

$ %

&

' (

"3

2 2x +1( ) " 2 2x "1( )2x +1( )

2

)

*

+ +

,

-

.

. 8.

!

1

22x + 5( )

"3

2

3

2

9.

!

1

x"2

+4

x"1

y"1

+1

y"2

#

$ %

&

' (

"1

2

Topic 2: Domain Find the domain of the following functions:

1. 14

23

+

!=x

xy 2.

42

42

+

!=x

xy 3.

183

65

2

2

!!

!!=

xx

xxy

4. x

y

x!

=22 5. 33 +!!= xxy 6.

92

92

+

!=

x

xy

7. 4

2

5

128

+

++=

x

xxy 8. 145

2!!= xxy 9.

30

6

2

3

!!

!=

xx

xy

10. ( )122log != xy 11. xy tan= 12. x

xy

cos=

Topic 3: Solving inequalities (absolute value) Write the following absolute value expressions as piecewise expressions 1.

!

y = 2x " 4 2.

!

y = 6 + 2x +1 3.

!

y = 4x +1 + 2x " 3 Solve the following absolute value inequalities 4.

!

x " 3 >12 5.

!

x " 3 # 4 6.

!

10x + 8 > 2 7.

!

3x " 4 > "2 8.

!

x " 6 > "8 9.

!

x +1 " x # 3

Topic 4: Solving inequalities (quadratic) Write the following absolute value expressions as piecewise expressions 1.

!

x2"1 2.

!

x2

+ x "12 3.

!

x2

+ 4x + 4 Solve the following by factoring and making appropriate sign charts. 4.

!

x2"16 > 0 5.

!

x2

+ 6x "16 > 0 6.

!

x2" 3x #10

7.

!

2x2

+ 4x " 3 8.

!

x3

+ 4x2" x # 4 9.

!

2sin2

x " sin x 0 # x < 2$

Topic 5: Special factorization Factor completely 1.

!

x3

+ 8 2.

!

x3" 8 3.

!

27x3"125y

3 4.

!

x4

+11x2" 80 5.

!

ac + cd " ab " bd 6.

!

2x2

+ 50y2" 20xy

7.

!

x2

+12x + 36 " 9 y2 8.

!

x3" xy

2+ x

2y " y

3 9.

!

x " 3( )2

2x +1( )3

+ x " 3( )3

2x +1( )2

Topic 6: Function transformation

!

If f x( ) = x2"1, describe in words what the following would do to the graph of f x( ) :

1.

!

f x( ) " 4 2.

!

f x " 4( ) 3.

!

" f x + 2( ) 4.

!

5 f x( ) + 3 5.

!

f 2x( ) 6.

!

f x( )

Here is a graph of

!

y = f x( ). Sketch the following graphs

7.

!

y = 2 f x( ) 8.

!

y = " f x( ) 9.

!

y = f x "1( )

10.

!

y = f x + 2( ) 11.

!

y = f x( ) 12.

!

y = f x

Topic 7: Factor theorem (p over q method/synthetic division)

Use the p over q method and synthetic division to factor the polynomial ( )xP . Then solve ( ) .0=xP 1. ( ) 64

23!++= xxxxP 2. ( ) 2425

23!!+= xxxxP

3. ( ) 1036

23!+!= xxxxP 4. ( ) 20192

23!!+= xxxxP

5. ( ) 8465

234!!++= xxxxxP 6. ( ) 30614111

234 ++++= xxxxxP

Topic 8: Even and odd functions

Show work to determine if the relation is even, odd, or neither 1. ( ) 72

2!= xxf 2. ( ) xxxf 24

3!!= 3. ( ) 444

2 +!= xxxf

4. ( )x

xxf1

!= 5. 12+!= xxxf 6. 165

2=! yx

7.

!

y = ex"

1

ex

8.

!

3y3

= 4x3

+1 9.

!

3x = y

Topic 9: Solving quadratic equations and quadratic formula Solve each equation 1. 037

2=! xx 2. ( ) ( ) 21524 =!!! xxxx 3. 046

2=++ xx

4. 03322

=+! xx 5. ( )( ) 123222 =!+! xxx 6.

6

131=+

xx

7.

!

x4" 9x

2+ 8 = 0 8.

!

x "10 x + 9 = 0 9.

!

1

x2"

1

x= 6

Topic 10: Asymptotes For each function, find the equations of both the vertical asymptote(s) and horizontal asymptotes (if they exist)

1.

!

y =x

x " 3 2.

!

y =x + 4

x2"1

3.

!

y =x + 4

x2

+1

4.

!

y =x

2" 2x +1

x2" 3x " 4

5.

!

y =x

2" 9

x3

+ 3x2"18x

6.

!

y =2x

2+ 6x

x3" 3x

2" 4x

7.

!

y =x

2" x " 6

x3" x

2+ x " 6

8.

!

y =2x

3

x3"1

9.

!

y =x

2x2"10

Topic 11: Complex fractions

Simplify the following

1.

!

x

x "1

2

2.

!

1

x+ 4

1

x" 2

3.

!

x "1

x

x +1

x

4.

!

3

x"

4

y

4

x"

3

y

5.

!

1"2

3x

x "4

9x

6.

!

x2" y

2

xy

x + y

y

7.

!

x"3" x

x"2"1

8.

!

x

1" x+

1+ x

x

1" x

x+

x

1+ x

9.

!

4

x " 5+

2

x + 2

2x

x2" 3x "10

+ 3

Topic 12: Composition of functions If

!

f x( ) = x2 , g x( ) = 2x "1, and h x( ) = 2x , find the following 1.

!

f g 2( )( ) 2..

!

f g 2( )( ) 3.

!

f h "1( )( )

4.

!

h f "1( )( ) 5.

!

g f h1

2

"

# $ %

& '

"

# $

%

& '

"

# $ $

%

& ' ' 6.

!

f g x( )( )

7.

!

g f x( )( ) 8.

!

g g x( )( ) 9.

!

f h x( )( )

Topic 13: Solving Rational (fractional) equations

Solve each equation for x

1.

!

2

3"

5

6=

1

x 2.

!

x +6

x= 5 3.

!

x +1

3"

x "1

2=1

4.

!

x " 5

x +1=

3

5 5.

!

60

x"

60

x " 5=

2

x 6.

!

2

x + 5+

1

x " 5=

16

x2" 25

7.

!

x

x " 2+

2x

4 " x2

=5

x + 2 8.

!

x

2x " 6"

3

x2" 6x + 9

=x " 2

3x " 9 9.

!

2x + 3

x "1=

10

x2"1

+2x " 3

x +1

Topic 14: Solving Rational (fractional) equations Solve the following problems.

If point P is on the terminal side of

!

" , find all 6 trig functions of

!

" . Draw a picture. 1.

!

P "2,4( ) 2.

!

P 5,"2( )

3. If

!

cos" =5

13," in quadrant II, 4. If

!

cot" = 3," in quadrant III,

find

!

sin" and tan" find

!

sin" and cos" Find the exact value of the following without calculators: 5.

!

sin2225°" cos

2300° 6.

!

6sec180°" 4cot 90°( )2

7.

!

4cos30°" 6sin120°( )"2

Solve the following triangles (3 decimal place accuracy)

8.

!

A = a = 21.7

B =16° b =

C = 90° c =

9.

!

A = a = 6 feet

B = b =

C = 90° c = 95 inches

Topic 15: Solving Trigonometric equations

Solve each equation on the interval

!

0,2"[ )

1.

!

sin x =1

2 2.

!

cos2

x = cos x 3.

!

2cos x + 3 = 0

4.

!

4sin2

x =1 5.

!

2sin2

x + sin x =1 6.

!

cos2

x + 2cos x = 3 7.

!

2sin x cos x + sin x = 0 8.

!

8cos2

x " 2cos x =1 9.

!

sin2

x " cos2

x = 0

Topic 16: Limits and Difference Quotients

1. (

)

2.

3.

4.

5.

6.

7. Simplify the forward difference quotient,

, for .

8. Simplify the forward difference quotient,

, for .