AP CALCULUS AB

SUMMER ASSIGNMENT 2019-2020

Attached is your summer assignment for AP Calculus (AB). It will probably take you 2-3 hours to

complete depending on how well you know your material. I would not do the packet at the beginning of the

summer. Give your brain a break and wait until the middle of July to start the packet and see what information

you have been able to retain and what you need to work on over the summer. If there are topics you are

unsure of read the notes section carefully. One of us will be available one day in August to help students. We

will post that date and time on the Calculus AB blog located at (http://halsteadcalculus.blogspot.com/ ). In

addition, the opening week of school for teachers we will be available by appointment. Please e-mail us at

halsteadr@calvertnet.k12.md.us or straubj@calvertnet.k12.md.us if you have any questions or want to meet

with us that opening week of school for teachers and we will respond as soon as possible.

There are six sections to the packet, each section highlights some critical concepts which you need to

retain from pre-calc. At the end of each section there is an assignment that correspond to that sections notes

{You can also use websites, old Pre-Calculus notes, and friends if you get stuck}. The assignments are worth 20

points each for a grand total of 120 process points, this will be your first process grade of the first quarter. The

assignments will be checked for effort on the first day of school with 2 points deducted from the twenty for

every problem not attempted. We will assess these topics with a 100 point take home test the first week of

school.

We cannot emphasize enough the importance of knowing your basic algebra skills: factoring,

common denominating, rational expressions, etc…, and the skills learned in Pre-Calc/Accelerated Algebra 2

specifically the unit circle, trig functions, trig identities and using trig identities to manipulate a trig function,

so please be sure to study these over the summer. The packet does not contain everything from Pre-Calculus

just what is critical for success in AP Calculus. Have a great summer and we look forward to seeing you in the

fall as our students.

If you are struggling and need to watch videos on a certain topic I suggest the following youtuber

https://www.youtube.com/user/patrickJMT . Just go to the search option on his channel and type the topic

you are unsure of.

Sincerely,

Mr. Halstead Mrs. Straub

AP CALC 1 (AB) : SUMMER ASSIGNMENT Name:__________________________________

This assignment is due the first day of class for 120 process points. Pd:____________ Date:___________________

PART 1

Interval Notation:

4,7 This means the Real numbers from four to seven including four and seven.

)1,5− This means the Real numbers from negative one to five including negative one but not five.

( )3,5 This means the Real numbers from three to five not including either endpoint.

( ) ( )3,5 5,9 This means the Real numbers from three to nine not including the endpoints or five.

( ),− This means all Real numbers.

We will not use inequalities a ton in Calc, but they are a good way to practice interval notation.

Ex 1: Solve and graph the solution set: Ex 2: Solve and graph the solution set:

2 5 7

2 12

6

( ,6)

x

x

x

−

−

3 2 5 12

5 5 10

1 2 (remember to flip the inequality symbol)

[-2, 1]

x

x

x

− −

− −

−

Ex 3: Solve the following quadratic inequality: (Remember on these find critical #’s and make a chart)

( )( )

2

2

6

6 0

3 2 0

critical numbers: 3, 2

( 2,3)

x x

x x

x x

x x

+

− −

− +

= = −

−

Piece-wise Function:

A function made up of the restricted domains of multiple other functions:

Ex 1: Graph

7 0

( ) 0 4

2 4

x

f x x x

x x

= − −

Absolute Value Function: (This is really just a piecewise function)

( )f x x= is really x 0

( ) 0

xf x

x x

− =

Ex 1: Solve and sketch the solution set to 3 2 (to solve this you must do both the positive and negative case)x −

Positive Case: Negative Case:

3 2

5

( ,5]

x

x

−

−

3 2

1

[1, )

x

x

− −

answer is what two inequalities have in common

Ex 2: Solve and sketch the solution set to 3 2x +

Positive Case: Negative Case:

3 2

1

(1, )

x

x

+

3 2

5

( , 5)

x

x

− +

−

− −

Assignment #1 :

1. Graph the piecewise functions given.

a.

2 1

( ) 4 1 2

2 3 2

x x

f x x

x x

−

= − +

b.

For #8-9 Write the following as piecewise functions.

8. ( ) 2 4f x x= − 9. 2( ) 1f x x= −

Part 2: Slope

Slope of a line: 1 2

1 2

y ym

x x

−=

−

Ex 1: Find the slope of a line segment connecting ( 3,6) and (2, 8)− −

( )6 8 14

3 2 5m

− −= = −

− −

Completing the square will also be used occasionally in Calculus, so here is a quick example of how to do it.

( )

2

2 2 2

2

4 6

4 2 2 6

2 10

x x

x x

x

+ =

+ + − =

+ =

(You could then proceed to solve for x, but not necessary at this time)

Now for a little more complex completing the square problem.

Find the center and radius of a circle with the equation, then give a rough sketch:

( )

( )

( )

( )

2 2

2 2

2 2 2 2

22

22

22

4 4 20 16 25 0

5 54 5 4 4 2 2 25

2 2

54 25 4 2 16 25

2

54 4 2 25 25 16

2

52 4 (Divided everything by 4)

2

x y x y

x x y y

x y

x y

x y

+ + − + =

+ + − + − + − = −

+ − + − − = −

+ + − = − + +

+ + − =

Center : 5

, 22

−

Radius: 2

Parallel lines: Two lines in a plane whose slopes are equal (they don’t intersect as a result)

Perpendicular Lines: First, the AP will sometimes call these Normal Lines, two lines who intersect and whose slopes

are opposite reciprocals of each other. Hence, they meet at 90 degree angles.

Point slope form of a line: You use this when you have the slope and a point and it is going to be the dominant way

we write the equation of a line in this class. ( )1 1y y m x x− = −

Ex 1: Find the equation of a line that has a slope of 3 and passes through the point (1, 2)−

Plug into point-slope:

2 3( 1)

no need to put in slope-intercept form

y x+ = −

Ex 2: Find the equations of a line parallel and perpendicular(normal) to the line in example 1 though the point (4,6)

Parallel: 4 3( 6)y x− = −

Perpendicular(normal): 1

4 ( 6)3

y x− = − − slopes are opposite reciprocals of each other

Summary of equation of a line types:

Vertical lines always in the form: x = a where a is a number

Horizontal lines always in the form: y= a where a is a number

Point-slope form of a line: ( )1 1y y m x x− = −

Slope intercept form of a line: y=mx +b

General form of a line: 0Ax By C+ + =

Ex 3: Find the equation of the lines that pass through (2, 1)− and are:

A) Parallel to 2 3 5x y− = in slope intercept form

B) Normal to 2 3 5x y− = in general form.

A) 2 5

3 3y x= − , slope is two-thirds.

So now get it to go through (2, 1)−

A)

( )2

1 23

2 41

3 3

2 7

3 3

y x

y x

y x

+ = −

= − −

= −

B) Still has a slope of two-thirds but

remember it is normal or perpendicular.

So you need the opposite reciprocal slope

B)

( )3

1 22

33 1

2

32 General form: 3 2 4 0

2

y x

y x

y x x y

+ = − −

= − + −

= − + + − =

Assignment #2:

9. Write an equation of line, ( )f x , given (2) 5f = − and ( 3) 1f − = . Find ( 1)f − .

Part 3: Intercepts and Points of intersection.

x-intercept: The place a graph crosses the x-axis, found by setting y = 0 and solving for x.

y-intercept: The place a graph crosses the y-axis, found by setting x =0 and solving for y.

Ex 1: Find the x and y intercepts of the function 3 4y x x= −

x-intercepts y-intercepts 3

2

0 4

0 ( 4)

0 ( 2)( 2)

(0,0), (2,0), ( 2,0)

x x

x x

x x x

= −

= −

= − +

−

( )30 4 0

0

(0,0)

y

y

= −

=

How do you find points of intersection? You set the equations equal to each other or solve one equation for a

variable and then substitute it into the other equation.

Ex 1: Find the intersection point of 2 4 and 6 2 16y x x y= + + =

,6 2 16

2 6 16

3 8

Now set equal

2 4 3 8

5 4

4

5

first x y

y x

y x

x x

x

x

+ =

= − +

= − +

+ = − +

=

=

to get y-coordinate plug back into one of original equations.

42 4

5

84

5

28

5

4 28Intersection point is: ,

5 5

y

y

y

= +

= +

=

Ex 2 (Harder one): Find the intersection point(s) of 2 4 and 1y x x y= − − =

2

2

2

1

(1 ) 4

1 2 4

0 2 3

0 ( 3)( 1)

3 or 1

x y

y y

y y y

y y

y y

y y

= +

= + −

= + + −

= + −

= + −

= − =

( ) ( )

1 3 and 1 1

2, 2

Intersection points: 2, 3 and 2,1

x x

x x

= + − = +

= − =

− −

Assignment 3:

4. Create an equation that has x-intercepts at x = -2, x = 4, and x = 6.

5. Find the points of intersections of the graphs of the equations and check your results.

Part 4: Functions and Relations

Function: Every input of x is assigned to exactly one y-output

Relation: An input of x has multiple outputs for y.

Ex 1: Does the set of inputs and outputs given for a mystery equation represent a function or a relation?

A) ( ) ( ) ( ) ( ) ( ) ( ) 1,2 , 2,4 , 3, 5 , 6,4 , 4,7 , 5, 3− − Yes this is a function, each input has only one output. Remember it is

okay for two different inputs like 2 and 6 to have the same output but one input cannot have two different outputs.

B) ( ) ( ) ( ) ( ) ( ) 4,6 , 3, 5 , 2,8 , 4,2 , 6,11− No this is not a function, the input of 4 has two different outputs.

Domain: Acceptable inputs that do not cause a function to be undefined.

Range: The y-values or outputs created by a function.

Ex 2: For the functions given below what are their domains and ranges.

Evaluating a function:

Ex 3: For the function 2( ) 7f x x= + , find A) f(2) B) f(3a) C) 3( )f x D) ( )f x x+ (Relax young Jedi)

A) 22 7 11+ = C) ( )

23

6

7

7

x

x

+

+

B) ( )

2

2

3 7

9 7

a

a

+

+

D) ( )

2

22

( ) 7

2 7

x x

x x x x

+ +

+ + +

Ex 4: (Little bit harder) If 2

2 4

( ) 2 4 0

3 2 0

x x

f x x x

x x

−

= + − − −

, Find A) f(0) B) ( 5)f − C) 2( 4)f t +

A) 20 2 2+ = B) ( )2 5 10− = − C) since t is squared the input must be greater than zero

( )2 2 23 4 2 3 12 2 3 14t t t− + − = − − − = − −

There are six algebraic functions I expect you to know inside and out, backwards, sleeping, eating, standing on your

head drinking a cup of water you get the idea.

1) y = x 2) 2y x=

3) 3y x= 4) y x=

5) y x= 6) 1

yx

=

I also expect you to be able to transform these functions (shift left/right, up/down, flip, take abs value of

Function types:

Polynomial function: 1( ) .......n nf x ax bx c−= + + + , Ex: 4 3 2( ) 9 2 6 7 10f x x x x x= + − + −

Rational Function: ( )

( ) , where ( ) 0( )

p xf x q x

q x= Ex:

23 2 6( )

2 8

x xf x

x

+ −=

−

Radical Function: ( ) ( )nf x p x= Ex: ( )2

5( ) 2 3f x x= +

Transcendental functions: Functions that are not made out of algebraic powers. (Trig functions, logarithmic functions,

exponential functions, inverse trigonometric functions)

Ex: 4( ) , ( ) ln(2 1), ( ) sin , ( ) cos(2 ), ( ) tan , ( ) arcsinxf x e f x x f x x f x x f x x f x x= = + = = = =

I will expect you to know the graphs of sinx, cosx, and tanx without a calculator.

Remember we can add, subtract, multiply, or divide functions

Ex 1: If 2( ) 2, and ( ) 4f x x g x x= − = −

Find

A) ( ) ( ) or ( )( )f x g x f g x+ +

B) ( ) ( ) or ( )( )f x g x f g x− −

C) ( ) ( ) or ( )( )f x g x f g x

D) ( )

or ( / )( )( )

f xf g x

g x

A) 2 22 4 6x x x x− + − = + −

B) ( ) ( )2 2 22 4 2 4 2x x x x x x− − − = − − + = − + +

C) ( ) ( )2 3 2 3 22 4 4 2 8 2 4 8x x x x x x x x− − = − − + = − − +

D) ( )

( )( )

( )( )2

2 2 1 , 2

2 2 24

x xx

x x xx

− −= = −

− + +−

Composite functions: When one function is substituted into another function or substituted into itself

Notations: f(g(x)) or ( )( )f g x

Ex 1: Using the same functions as out last example find: A) f(g(x)) B) g(f(x)) C) f(f(x)) D) g(f(3))

A) ( )2 2 2( 4) 4 2 6f x x x− = − − = − B) ( )2 2 2( 2) 2 4 4 4 4 4g x x x x x x− = − − = − + − = −

C) ( )( 2) 2 2 4f x x x− = − − = − D) 2( (3)) (1) 1 4 3g f g= = − = −

Assignment 4:

8. Use a piece of graph paper and sketch the graphs below without a calculator

A) ( ) 1 1f x x= − +

B) ( ) 4f x x= − +

C) ( )2

( ) 3 1f x x= + −

D) ( ) 4 2f x x= −

E) ( )

1( ) 3

2f x

x= −

+

F) 21

( ) 44

f x x= − −

G) 2( ) 4f x x= −

H) ( )3

( ) 2 1f x x= − − −

Part 5: Trig stuff (You need to have the unit circle memorized. You will not be making a unit circle for each quiz or test

we take.)

Ex: 2

sin3

=3

2 Ex:

51

4Tan

= −

Ex:

25 3

6 6 2Cos Cos

= =

Ex:

1

6 2Sin

− = −

Know these identities: 2 2sin cos 1 + = 2 2tan 1 sec + = 2 2cot 1 csc + =

( )sin 2 2sin cos =

sintan

cos

=

coscot

sin

=

( ) 2 2cos 2 cos sin = −

( ) 2cos 2 1 2sin = −

( ) 2cos 2 2cos 1 = −

1sec

cos

=

1csc

sin

=

1cos

sec

=

1sin

csc

=

Ex 1: Solve on )0,2 Ex 2: Solve on )0,2 Ex 3:

2sin 1 0 + = ( )sin 2 cos 0 + =

( )cos 2 2 3sin = −

1sin

2

7 11,

6 6

= −

=

( )

2sin cos cos 0

cos 2sin 1 0

1cos 0 or sin

2

3 7 11, , ,

2 2 6 6

+ =

+ =

= = −

( )( )

2

2

1 2sin 2 3sin

0 2sin 3sin 1

0 2sin 1 sin 1

1sin or sin 1

2

5, ,

6 6 2

− = −

= − +

= − −

= =

Assignment #5:

Part 6: Miscellaneous

The assignment below has some other topics you need to review. If you are unsure how to do a

problem, please look online, ask friend for help, or look at your notes from PreCalc.

A. Factor completely.

1.

2. 215 16 4x x− +

3.

4.

B. Find all vertical and horizontal asymptotes (if they exist).

1. 3

xy

x=

−

2. 2

4

1

xy

x

+=

−

3. 2

4

1

xy

x

+=

+

4. 2

3 2

9

3 18

xy

x x x

−=

+ −

5. 3

3

2

1

xy

x=

−

C. Simplify each of the following.

1.

1

1

xx

xx

−

+

2.

2 2x y

xy

x y

y

−

+

3. 2

1 3

5 6 3

x x

x x x

++

− + −

4. 2

3 4x − +

− − − −

−

−

−

x

y

D. Function Transformations

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

1. y = 2f(x)

2. y = -f(x)

3. y = f(x-1)

4. y = f(x+2)

5. y =|f(x)|

6. y = f|x|

E. If , find .

F. Rewrite as a single logarithmic expression.