Hello Future AP Calculus Student!

I’m so excited to work with you for the 2021 - 2022 school year! AP Calculus is a college level mathclass that will challenge your problem solving skills, build upon the prior math you’ve learned, andprepare you for future college classes. In order to cover the content before the AP test in May, we havea summer homework packet. Do not fear! The packet is just review! It is your guide to make sureyou’re comfortable with the material needed to be successful in AP Calculus.

Guidelines for AP Calculus Summer Homework

● This packet is not a required assignment, because you’re doing it before the school year.However, I highly recommend completing it. This assignment will be your way to check tomake sure AP Calc is truly a good fit for you. Believe in yourself!

● Helpful materials: old notes, khan academy, and a graphing calculator (you will need for class).If you do not have access to a graphing calculator, you can reach out to me or go todesmos.com/graphing to use a magical online graphing calculator.

● The best way to work on this packet is by doing a chunk of problems each week duringsummer. I know this homework may look overwhelming to start, but you’re going to be soamazed at how much your brain knows! Just like working out muscles to train for a marathonor sport, this packet is training your brain for all of the amazing math we’re going to do nextyear. Do a few problems each day, and before you know it you’ll be done! You can do it!

● I will post step-by-step solutions on my website (go to Freedom’s website, find the ‘Staff’ page,and then look for Reyburn). This is for you to use to check your work, not copy. I will also beavailable through email (reyburns@luhsd.net), remind (@apcreyburn), and my teacherinstagram(@ms.reyburn).

● I advise you reach out to peers also taking this course, or me if you need questions answered.Form a study group! Despite what you may think or have experienced in the past, the bestmathematicians work with teams, not in isolation. Your classmates will show you ways to thinkthrough problems that you never would have discovered yourself. I’m sure you will do the same forthem.

● As with all work, your answers should be yours. Photomath, your friend’s answers, and Googlemight be able to help you now, but they won’t on the AP test so let’s make good choices.

● Enjoy your summer! It’s no fun to do math when you’re stressed out. If you get stuck on someproblems, check out the examples or wait to review with a friend or in class. It’s okay if you don’tremember all of these topics right away, use this time to remind yourself that you can do this!

I look forward to working with you all next school year!

Ms. Reyburn

Part 1: Calculus Readiness Quiz

You do not need to remember everything from PreCalc in order to be successful in Calculus. However,there are skills and topics you should be comfortable with in order to be successful with this class. Wewill have a ‘readiness’ quiz during the first two weeks of school on this material. The quiz will be agreat indicator of whether or not you will feel prepared to take this course!

The practice problems below are similar to problems you could see on the quiz. The formatting isorganized so if you need extra practice, you can find it on Khan Academy. Each section matches asection from the course called ‘Getting Ready for AP Calculus.

Getting Ready for Limits and Continuity (Unit 1)

Functions

𝑞(𝑥) = − 𝑥2 + 2𝑥

𝑟(𝑥) = 5𝑥

ℎ(𝑥) = 3𝑥 − 5

1. Evaluate each of the following:

a. 𝑞(4)

b. ℎ(4)

c. 𝑔(6)

d. 𝑟(0)

2. Evaluate each of the following:

a. 𝑝(0)

b. 𝑝(4)

c. 𝑝(7)

3. Evaluate each of the following:

a. 𝑟(ℎ(35)) b. 𝑓(𝑔(3))

c. 𝑞(ℎ(𝑥))

45 2 4 16 8 80fgs f undefined

3 Sco

g 3 5

h 35 tog Tof Of 5

735 2537r lo Fo _3x 5t2F

Factoring

4. Factor each completely:

a. 𝑥2 − 11𝑥 + 30

b. 4𝑥2 + 8𝑥 − 32

c. 3𝑥3 + 2𝑥2 − 12𝑥 − 8

d. 16𝑥4 − 81

e. 3𝑥2 + 7𝑥 + 4

Trig

5. Evaluate the following: (you may need to use the trig table or the unit circle on the last page)

a. 𝑠𝑖𝑛( π4 )

b. 𝑐𝑜𝑠( 4π3 )

c. 𝑡𝑎𝑛(π)

d. 𝑠𝑒𝑐( π2 )

e. 𝑐𝑠𝑐( 7π4 )

f. Given and the angle𝑐𝑜𝑠(θ) =− 35 θ

is located in quadrant 3, find .𝑠𝑖𝑛(θ)

6. Convert the following degrees to radians or radians to degrees:

a. 5π3 b. 135𝑜

7. Evaluate the following: (you will need to use your trig identities!)

a. 𝑠𝑖𝑛 105𝑜

b. 𝑐𝑜𝑠 195𝑜

c. 𝑐𝑜𝑠 75𝑜

4 44670 43 21 413 2

41 2 2 8 45 9 4 9 x 3 2x 3 4x

4 x 2 h

3xt4

rs

sinoto Undefined

c 1,8 3000 o

o3

5 7 5353OJh sin 135 cos30 Sin 3 05435

E3 Loft It cos Erationalize7546 542 FYI 1hedenom

qcos 15 0445 sincisosings 2

Iz E I

E ETE EET

rare TFF

Rational Expressions

8. Simplify the following. Make sure there are no radicals in the denominator.

a. 𝑥𝑥 + 3 + 1

𝑥

b. 𝑥2 + 3

𝑥3 ÷ 𝑥 + 1

𝑥4

c.1𝑥 +11𝑥 −1

d. 3𝑥 + 1

9. Simplify the following. Which x values make the original expression undefined?

a. 15𝑥2−25𝑥3𝑥−5 b. 2𝑥2−14𝑥−36

𝑥2−9𝑥

Getting Ready for Differentiation: Definition and Basic Rules (Unit 2)

Equations of Lines

10. Create a linear equation for the given information/representation:

a. Through the point (2, 1) with slope 4. b. Through points (4, 10) and (6, 10).

1 Change the15 to2 Divide

I IIIT

K537rTx I

X3t3X

5 3 5 262 7 185X TE

undefined 0 denom_O undefinedwhen

3 5 0 Xcx 9 O 2 xt

3415 X

y 1 41 2m w6 E O

OR Y 1 4 8

y 4 7 y 10 06 6 OR y lo

c. Through points (2, 7) and (4, 5)

d. Through the point (5, 9) with an

undefined slope.

e. For the graph below:

Exponents and Logs

11. Simplify Completely:

a.14𝑥3𝑦−1𝑧4

2𝑥−3𝑦𝑧3b.

2𝑥3/2

𝑧8𝑦6( )12

12. Rewrite each radical in exponential form and then simplify:

a. 64𝑥4b.

324𝑥9

13. Evaluate each logarithm without a calculator:

a. 𝑙𝑜𝑔981

b. 𝑙𝑜𝑔27

3

c. 𝑙𝑜𝑔2

132

Getting Ready for Differentiation: Composite, Implicit, and Inverse Functions (Unit 3)

Literal Equations

14. Solve for y:

a. 𝑥 = 2𝑦𝑦 + 3 b. 3𝑥 + 4𝑦2 = 19

7 5m I

y 5 1 x 4

OR y xt9 y x 4

undefined

I X 5 b

k

71,975 x yx EYy z4y3762y2

249s 3h24

64 4200

2 2 2.232353

23353X 9 81 X 2 312

X Z

X 5X

27 3 X s

YG x

II y use.a fsF

3x y2 19342

Inverse Functions

15. Find inverse of each of the following:

a. 𝑔(𝑥) = 2𝑥 + 1 b. ℎ(𝑥) = 𝑥3

3

16. Given the following representations, find the indicated inverses:

x -3 4 7 10 13

g(x) 7 2 -3 0 9

a. 𝑔−1(− 3)b. 𝑓−1(− 2)

Get ready for contextual applications of differentiation (Unit 4)

Word Problems and Modeling

17. Avis car rental charges a flat rental fee, plus an additional 10 cents per mile driven. When I rented

a car with them, I drove 150 miles and paid $330. Write a linear equation to represent the

relationship between the amount of money you are charged (y) and how many miles you drive (x).

18. A kiddie pool has sprung a leak. Yikes. It starts with 7.5 gal of water. Five minutes after the leak

began, there was 4.45 gallons left in the kiddie pool. Write a linear equation to represent the

gallons left in the pool (y) and how much time has passed since the leak started (x).

X 2y I 3x 3 J

h G T3xX l 2gg x 3x y3

O0

where does wheredoesfCx 2

gcx 3 6gC3 7

y 330 10 x 150

OR Y 1 1 315

m 61 y 4.45 061 x 5

OR

y o61 7.5

19. Lennox owns a big apple orchard. She ships her apples to various markets using a fleet of trucks.

Every week, each truck goes on 3 trips, and for each trip Lennox gets 300 dollars. On a single trip,

a truck delivers 50 packs, and each pack contains 12 kilograms of apples. Overall, Lennox sells

4500 dollars worth of apples in a week. How many trucks does she have?

20.Jesse is filling spherical balloons. When full, one of these balloons has a diameter of 24cm. Jesse

can fill a balloon at a rate of 820 cm3 per breath. How many breaths will it take for Jesse to fill a

balloon?

21. Let represent the gallons of water in a pool t minutes after it began to drain. Interpret the𝐺(𝑡)

following:

a. G(3) = 100

b. G(0) = 120

c. G(20) > G(25)

Get ready for applying derivatives to analyze functions (Unit 5)

Interpreting Features of a Graph

22. identify the following: maxima/minima, intervals of

increasing/decreasing, zeroes (you can estimate to the

nearest half)

riot needed Each week

450050packs 12kg I truck 3tripsX 300 900

I truck 900 5 trucks

7,234.56Vol of sphereTtr's 8.8

43,2443820

7,234.56cm's 9 breaths

at 3minutes There's more

100gal in the pool water at 20

min than at

25 minutesat the start120 gal in the pool

Max Ost a o

Min 1 l C 2 7 5

Zeroes C2 5,0 f 5,0 65,0 G5,0Inc C2,0 U i a

Dec co 2 Ufo iintervalnotation

Polynomials

23.Find zeros of each function:

a. 𝑓(𝑥) = 𝑥3 + 11𝑥2 − 𝑥 − 11 b. ℎ(𝑥) = 𝑥4 − 6𝑥2

24.Based on the zeroes you found above, determine on which intervals the function is positive and

which intervals the function is negative. You may write your answer as inequalities or in interval

notation.

a. 𝑓(𝑥) = 𝑥3 + 11𝑥2 − 𝑥 − 11 b. ℎ(𝑥) = 𝑥4 − 6𝑥2

Get ready for integration and accumulation of change (Unit 6)

Polynomial Division

25. Solve

a. (𝑣3 + 8𝑣2 + 2𝑣 − 31) ÷ (𝑣 + 7) b. (9𝑥3 − 2𝑥2 + 6) ÷ (9𝑥 − 2)

Basic Graphing

26.Sketch a graph of each function.

a. 𝑔(𝑥) = 𝑥 b. ℎ(𝑥) = − |𝑥| + 3

1J I il l 11 So X2 2_6X I

I 10 11 LO so X 0,12 56 X 562 10 11 x 1 x111

So find out if it is or BETWEENtest pts

the Zeroes

12 z 0 I 2 3 TG e O p TG 3Ii a

ffiz fC2 Fco f 2 hfz h.fih i hfs

143 27 H 39 27 S S 27t t 1

neg Con aUft 1 posCn DUG.co neg fro o U0,56 posits FduFcs

27J I 8 2 31 X

7 7 35 9 2 19372 70 7619 3 2 7I l 5 44 TEG

V S 7 2 692

i

r thI l l l l l l l l l l l l l l l

c. 𝑓(𝑥) = (𝑥 − 2)2

d. 𝑝(𝑥) = 𝑠𝑖𝑛 𝑥

e. 𝑞(𝑥) = 𝑐𝑜𝑠 𝑥

f. 𝑟(𝑥) = 𝑒𝑥

Get ready for applications of integration (Units 7 and 8)

Solving Equations

27. Solve.

a. − 7(3𝑛 + 8) + 1 = 36 − 8𝑛

b. 𝑝2 + 8𝑝 + 20 = 5

c. 4𝑥3 − 12𝑥 = 0

28. Solve. Be sure to check for extraneous solutions:

a. − 6 − 𝑛 = − 16 − 2𝑛

b. 34 − 3𝑝 = 𝑝 − 8

c. (𝑛 − 2)3/4 − 7 = 20

d. −2𝑥 + 3 = 1

𝑥 + 1

inrII i iWl l l l l i i 2T T F za

ro to

A hI l l l l l l l

2in 561 1 36 8h 4 2 3 021h 55 36 8h

n 7 o 21 53 X Fs91 13n

P218ps 115 0

pts pt3 O

P 5 p 3

z z

7 7G n 16 2n 31443 2743n 10

h 2 4527A 83

Zy y n 2 81

34 3p p 16,0 64 CHECK26 17 16 3P2 Bp 130 0 54303 3 8

525 5 2x 2 X 13p lo p 3 0 no negative 5 3 xf 10 10 3 10 10 X 513

29.Solve. (For part b you can leave your answer in logarithmic form.)

a. 𝑙𝑜𝑔7(𝑥 − 1) − 𝑙𝑜𝑔

74 = 1 b. 4𝑛−2 + 6 = 36

30.Solve each system.

a. b.𝑥 + 𝑦 = 13 𝑥2 + 𝑦2 = 252𝑥 − 4𝑦 = 14 𝑦 = 𝑥 + 5

31. Solve: (write your answer in radians, your answers should be on the interval )0 ≤ θ ≤ 2π

a. − 6𝑠𝑖𝑛θ = 3

b. 2𝑐𝑜𝑠θ + 3 = 0

c. 𝑐𝑜𝑠2θ = 𝑐𝑜𝑠θ

G 6log I

7 IT zg

4h 2 30

µIn4h7 In 3028 X l

n 2 In4 1h30

GsI4 4y 522x 4y 14 I ti l l l I l 5,0Gx 66

11,2 X H

1Ity 13y 2

Sino Lz

andI

COS cos 0 0

6 Cos0 cos D 0

cos 0 0 cos Q2cos 0 53Cos

and and O

2

If and 6

Part 2: Memorization (No memorization on the first quiz, but you need to start memorizingthese in order to be successful in AP Calculus)

Trig Identities𝑐𝑠𝑐 𝑥 = 1

𝑠𝑖𝑛 𝑥 𝑠𝑒𝑐 𝑥 = 1𝑐𝑜𝑠 𝑥 𝑐𝑜𝑡 𝑥 = 1

𝑡𝑎𝑛 𝑥 𝑡𝑎𝑛 𝑥 = 𝑠𝑖𝑛 𝑥𝑐𝑜𝑠 𝑥

𝑠𝑖𝑛2𝑥 + 𝑐𝑜𝑠2𝑥 = 1 𝑡𝑎𝑛2𝑥 + 1 = 𝑠𝑒𝑐2𝑥 1 + 𝑐𝑜𝑡2𝑥 = 𝑐𝑠𝑐2𝑥

Double angle identities, half angle identities, sum and difference identities

While you do not need to memorize the entire trig table/unit circle, it will be valuableto know:~ How to use the unit circle and special righttriangles to find any trig ratio.

~ On the unit circle, points = (sin 𝞱, cos𝞱)~ Converting between degrees and radians:

𝑟𝑎𝑑𝑖𝑎𝑛𝑠 = 𝑑𝑒𝑔𝑟𝑒𝑒 × π180

𝑑𝑒𝑔𝑟𝑒𝑒 = 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 × 180π

~ Key trig values:

Sin, cos, and tan at 0, , , andπ2 π 3π

2

Exponents and Logarithms

is equal to𝑦 = 𝑙𝑜𝑔𝑏𝑥 𝑥 = 𝑏𝑦

𝑙𝑜𝑔𝑏𝑚𝑛 = 𝑙𝑜𝑔

𝑏𝑚 + 𝑙𝑜𝑔

𝑏𝑛

𝑙𝑜𝑔𝑏

𝑚𝑛 = 𝑙𝑜𝑔

𝑏𝑚 − 𝑙𝑜𝑔

𝑏𝑛 𝑙𝑜𝑔

𝑏𝑚𝑛 = 𝑛 𝑙𝑜𝑔

𝑏𝑚

𝑛𝑥𝑚 = 𝑥

𝑚𝑛 𝑥−1 = 1

𝑥

Other things to Consider~Geometry formulas for area, surface area, and volume~ Parent functions and properties (linear, exponential, quadratic, cubic, circles, conics, logs, e)~ Slope/Rate of change and how to identify it in different representations

The Trig TableFill it out, know the key values or how to find them.

Radian Degree sin x cos x tan x cot x sec x csc x

0

π6

π4

π3

π2

2π3

3π4

5π6

π

7π6

5π4

4π3

3π2

5π3

7π4

11π6

2π

Oo O l O f l

300 Yz 53 2 53 3 53 2533 2

450 52 2 52 2 1 1 52 52

600 53 2 1 2 53 53 3 2 2533

900 I O I1200 Fk Iz Tz 53

3 z 2531350 542 52 2 l l FL FL1500 Yz 53

2 53 3 Fg 2533 2

180 O l O f l

2100 Yz 53253

3 Fg 2533 z

2250 542 52 2 I l Tz Tz

2400 532 1 2 53 53 3 Z 2532700 I 0 O I

3000 53 2 1 2 53 B Z23

3150 52 2 52 2 I I 52 I

330012 5312 53 3 53 3 253

3 z

360 0 I 0 I