Name__________________________________

AP Calculus AB Summer Assignment

Mrs. Fava(TCFava@fcps.edu)/Mr. May(DMMay@fcps.edu)

Dear Prospective AP Calculus AB Student,

Congratulations on committing to a rigorous course of study. The study of calculus will draw on so much prior experience from Algebra 1, Geometry, Algebra 2 and Trigonometry. This course will be taught at a college level and, as such, this background knowledge is expected and will not be retaught.

The following packet was designed to help you remember the very basics of what you are supposed to know. This represents the very minimum of what you should complete over the summer. If are having difficulty remembering some of these topics, the onus is on you to research the topic and to find more practice. Here are some websites to get you started:

(1) Hippo Campus: http://www.hippocampus.org/ (2) Khan Academy: https://www.khanacademy.org/ (3) Wolfram Alpha: http://www.wolframalpha.com/

Of course, a google or youtube search will also yield some great results. Answers to this review assignment will be posted in Blackboard the week before you return to school. It is the student’s responsibility to check their answer and arrive to class with any questions. Time will be spent in class during the first week of school assisting students on these review topics.

During the second week of school, students will be quizzed on this material. Reviewing the concepts in the packet is the only way to ensure success.

We look forward to a productive and rewarding school year. If there are any questions, you can contact either Mr. May or Mrs. Fava via e-mail.

Have a great summer,

Mrs. Fava and Mr. May

The following Trigonometric Identities MUST be memorized Reciprocal Identities Quotient Identities Pythagorean Identities

1 1 sin x sin 2 x + cos 2 x =1sin x = csc x = tan x = csc x sin x cos x

1 1 tan 2 x + =1 sec 2 xcos x = sec x = cos xsec x cos x cot x =

1 1tan x = cot x = cot x tan x

π π sin θ cos θ cos − =θ − = sin θ 2 2

π π csc θ sec θ sec − =θ − = csc θ 2 2

π π tan θ cot cot θ tan − = θ − = θ

2 2

sin x 1+ cot 2 x = csc 2 x

Co-Function Identities Odd/Even Identities

Odd Even

sin ( )− = −sin cos ( )− = θθ θ θ cos

− = θ − = θcsc ( )θ −csc sec ( )θ sec tan ( ) − = − tan θ θ

cot ( ) − = −cot θ θ

Double Angle Identities

sin 2 x = 2sin x cos x

cos 2 x = cos 2 x − sin 2 x

cos 2 x = 2cos 2 x −1

cos 2 x =1− 2sin 2 x

The Radian Measures and Coordinates MUST be memorized

y y y − coordinate sinθ = = y − coordinate cosθ = x = x − coordinate tanθ = =Remember: , , and

r r x x − coordinate

AP

Calculus Name: ______________________________ Summer Assignment

You should be able to answer all of these questions without a calculator!

I. Algebraic Manipulation: Simplify the expression

1. ( )( )( 1 )232 +−+ xxxx 2. x + 3

x

3. ( )( )

( )( 8)27

4133 19

239 3

2

2

2

+

++•

−

−+

x xx

x xx 4.

xy

y x

1

1

+

+

II. Solving Equations

5. 5 431 =−

x

6. 1 23 12

13 1

= − −

++

−

+

x x

x x

7. x 4 −13x 2 + 37 = 7

III. Exponents and Logarithms

9. Solve for x: log3 (x −1) = 2

1 111. Evaluate: 2 log 4 + log 5 − log 202 2 22 2

8. 2x3 − 3x2 − 8x +12 = 0

x x+13 3 10. Solve for x: • = 9

12. Graph: y = log2 (x −1)+ 3

5

4

3

2

1

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

-2

-3

-4

-5

Domain: ______________

Range: _______________

Asymptote: ____________

IV. Functions and Graphs

x 1 x13. If f (x) = , find f 14. If f (x) = , find f (1− x)1− x x 1− x

15. Sketch the graph of y = x 2 − 2x and find the domain and range

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

-7 -6 -5 -4 -3 -2 -1

1 2 3 4 5 6 7

Domain: ________________________

Range: _________________________

3x16. Sketch the graph of y = and find the domain and range x −1

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

-7 -6 -5 -4 -3 -2 -1

1 2 3 4 5 6 7

Domain: ________________________

Range: _________________________

Asymptote (s): __________________

17. Sketch the graph of ( ) = ln( x −1) and find the domain, range and identify the equation for the asymptote f x

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

-7 -6 -5 -4 -3 -2 -1

1 2 3 4 5 6 7

Domain: ________________________

Range: _________________________

Asymptote: _________________

( ) x+318. Sketch the graph of f x = −e − 2 and find the domain and range

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

-7 -6 -5 -4 -3 -2 -1

1 2 3 4 5 6 7

Domain: ________________________

Range: _________________________

Asymptote: _____________________

19. Solve each function by completing the square.

a. x2 − + = 8x 5 0 b. x2 + − = 5x 3 0

Answer: ______________________ Answer: ______________________

4 − x x ≤ 0

20. Sketch the graph of f (x) = 2 − 3x 0 < x ≤ 2 2x x > 2

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

-7 -6 -5 -4 -3 -2 -1

1 2 3 4 5 6 7

lim ( ) = ______ lim f x = ______ =−

f x → −

( ) f (2) ______ x→1 x −5

lim ( ) = ______ lim f x = ______ f ( 5)− = ______f x ( ) x→1+ x→−5+

lim ( ) = ______ f (1) = ______f x x→2

V. Trigonometry

5π21. Find tan − = ____________

3

23. Solve: tan x = 2sin x

7π 22. Find sin = ____________ 6

24. Solve: 2sin 2 θ − cos θ = 1

VI. Limits: Evaluate each limit.

4 − 18 − x25. lim x→2 x − 2

x + 227. lim x→− 2 x2 + 5x + 6

cos 2x26. lim cos xx→π

( x − 3)28. lim

x→2 ( x − 4)

VII. Piecewise Functions.

Part I.the graph at any specified domain value.

2 + 1 x ≤ 1 1) f x( ) =

x

− + 4 >x x 1

Function? Yes or No

f 4( ) − =

f ( ) 1 =

f ( ) 2 =

Carefully graph each of the following equations. Identify whether or not the graph is a function. Then, evaluate

1 3 x + x ≤ −1

2) f x( ) = 2 2 −x + 3 x > −1

Function? Yes or No

− =

( ) − =

f ( ) 1

f 5

f ( ) 6 =

Part II. Write equations for the piecewise functions whose graphs are shown below. Every tick mark represents one unit.

3) 4)

5)

6) Sammy’s Silkscreening Shop has an initial charge of $20 to create the silk screen. Then, for orders of 50 or fewer shirts, they charge $17 per shirt. If the order is over 50 shirts, they charge $15.80 a shirt. Write a piecewise function that gives the cost, C, for an order of x shirts.

7) The city of Alexandria is about to open a new parking garage. On the weekends, they plan to charge $3 per half hour. There will be a maximum charge of $8 though (up to 12 hours). Write a piecewise function for the weekend parking charges.

Graph Matching (Meet the Parents)

1. Y = rx 11.

2. y = sin x 12.

3. y = lxl 13.

4. y = x 14.

5. y = .Jx2 - a2 (Stuck? Squ.u-e both s;des.) __ 15.

6. y = x2 16.

7. y = cos x 17.

8. y = x• 18.

9. y = tan x 19.

10. y = x• 20.

y = secx

y = ln x

Y = [xll or y = lxl or y = [x]

1 y =­x

y = cscx

y = .Ja2 - x2

y = cotx

VII. Graphs of Functions and Properties.

A B

E

I J

M

Q R

c

'

0

K

0

s

' '

D

H

L

p

T