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Hello Future AP Calculus Student!
Iโm 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 youโve 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 sureyouโre comfortable with the material needed to be successful in AP Calculus.
Guidelines for AP Calculus Summer Homework
โ This packet is not a required assignment, because youโre 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!
โ 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.
โ 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 youโre 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 weโre going to do nextyear. Do a few problems each day, and before you know it youโll be done! You can do it!
โ I will post step-by-step solutions on my website (go to Freedomโs website, find the โStaffโ page,and then look for Reyburn). This is for you to use to check your work, not copy. I will also beavailable through email ([email protected]), remind (@apcreyburn), and my teacherinstagram(@ms.reyburn).
โ 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. Iโm sure you will do the same forthem.
โ As with all work, your answers should be yours. Photomath, your friendโs answers, and Googlemight be able to help you now, but they wonโt on the AP test so letโs make good choices.
โ Enjoy your summer! Itโs no fun to do math when youโre stressed out. If you get stuck on someproblems, check out the examples or wait to review with a friend or in class. Itโs okay if you donโtremember 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 โreadinessโ 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 โGetting Ready for AP Calculus.
Getting Ready for Limits and Continuity (Unit 1)
Functions
๐(๐ฅ) = โ ๐ฅ2 + 2๐ฅ
๐(๐ฅ) = 5๐ฅ
โ(๐ฅ) = 3๐ฅ โ 5
1. Evaluate each of the following:
a. ๐(4)
b. โ(4)
c. ๐(6)
d. ๐(0)
2. Evaluate each of the following:
a. ๐(0)
b. ๐(4)
c. ๐(7)
3. Evaluate each of the following:
a. ๐(โ(35)) b. ๐(๐(3))
c. ๐(โ(๐ฅ))
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. ๐ฅ2 โ 11๐ฅ + 30
b. 4๐ฅ2 + 8๐ฅ โ 32
c. 3๐ฅ3 + 2๐ฅ2 โ 12๐ฅ โ 8
d. 16๐ฅ4 โ 81
e. 3๐ฅ2 + 7๐ฅ + 4
Trig
5. Evaluate the following: (you may need to use the trig table or the unit circle on the last page)
a. ๐ ๐๐( ฯ4 )
b. ๐๐๐ ( 4ฯ3 )
c. ๐ก๐๐(ฯ)
d. ๐ ๐๐( ฯ2 )
e. ๐๐ ๐( 7ฯ4 )
f. Given and the angle๐๐๐ (ฮธ) =โ 35 ฮธ
is located in quadrant 3, find .๐ ๐๐(ฮธ)
6. Convert the following degrees to radians or radians to degrees:
a. 5ฯ3 b. 135๐
7. Evaluate the following: (you will need to use your trig identities!)
a. ๐ ๐๐ 105๐
b. ๐๐๐ 195๐
c. ๐๐๐ 75๐
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. ๐ฅ๐ฅ + 3 + 1
๐ฅ
b. ๐ฅ2 + 3
๐ฅ3 รท ๐ฅ + 1
๐ฅ4
c.1๐ฅ +11๐ฅ โ1
d. 3๐ฅ + 1
9. Simplify the following. Which x values make the original expression undefined?
a. 15๐ฅ2โ25๐ฅ3๐ฅโ5 b. 2๐ฅ2โ14๐ฅโ36
๐ฅ2โ9๐ฅ
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.14๐ฅ3๐ฆโ1๐ง4
2๐ฅโ3๐ฆ๐ง3b.
2๐ฅ3/2
๐ง8๐ฆ6( )12
12. Rewrite each radical in exponential form and then simplify:
a. 64๐ฅ4b.
324๐ฅ9
13. Evaluate each logarithm without a calculator:
a. ๐๐๐981
b. ๐๐๐27
3
c. ๐๐๐2
132
Getting Ready for Differentiation: Composite, Implicit, and Inverse Functions (Unit 3)
Literal Equations
14. Solve for y:
a. ๐ฅ = 2๐ฆ๐ฆ + 3 b. 3๐ฅ + 4๐ฆ2 = 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. ๐(๐ฅ) = 2๐ฅ + 1 b. โ(๐ฅ) = ๐ฅ3
3
16. Given the following representations, find the indicated inverses:
x -3 4 7 10 13
g(x) 7 2 -3 0 9
a. ๐โ1(โ 3)b. ๐โ1(โ 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 the๐บ(๐ก)
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. ๐(๐ฅ) = ๐ฅ3 + 11๐ฅ2 โ ๐ฅ โ 11 b. โ(๐ฅ) = ๐ฅ4 โ 6๐ฅ2
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. ๐(๐ฅ) = ๐ฅ3 + 11๐ฅ2 โ ๐ฅ โ 11 b. โ(๐ฅ) = ๐ฅ4 โ 6๐ฅ2
Get ready for integration and accumulation of change (Unit 6)
Polynomial Division
25. Solve
a. (๐ฃ3 + 8๐ฃ2 + 2๐ฃ โ 31) รท (๐ฃ + 7) b. (9๐ฅ3 โ 2๐ฅ2 + 6) รท (9๐ฅ โ 2)
Basic Graphing
26.Sketch a graph of each function.
a. ๐(๐ฅ) = ๐ฅ b. โ(๐ฅ) = โ |๐ฅ| + 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. ๐(๐ฅ) = (๐ฅ โ 2)2
d. ๐(๐ฅ) = ๐ ๐๐ ๐ฅ
e. ๐(๐ฅ) = ๐๐๐ ๐ฅ
f. ๐(๐ฅ) = ๐๐ฅ
Get ready for applications of integration (Units 7 and 8)
Solving Equations
27. Solve.
a. โ 7(3๐ + 8) + 1 = 36 โ 8๐
b. ๐2 + 8๐ + 20 = 5
c. 4๐ฅ3 โ 12๐ฅ = 0
28. Solve. Be sure to check for extraneous solutions:
a. โ 6 โ ๐ = โ 16 โ 2๐
b. 34 โ 3๐ = ๐ โ 8
c. (๐ โ 2)3/4 โ 7 = 20
d. โ2๐ฅ + 3 = 1
๐ฅ + 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. ๐๐๐7(๐ฅ โ 1) โ ๐๐๐
74 = 1 b. 4๐โ2 + 6 = 36
30.Solve each system.
a. b.๐ฅ + ๐ฆ = 13 ๐ฅ2 + ๐ฆ2 = 252๐ฅ โ 4๐ฆ = 14 ๐ฆ = ๐ฅ + 5
31. Solve: (write your answer in radians, your answers should be on the interval )0 โค ฮธ โค 2ฯ
a. โ 6๐ ๐๐ฮธ = 3
b. 2๐๐๐ ฮธ + 3 = 0
c. ๐๐๐ 2ฮธ = ๐๐๐ ฮธ
G 6log I
7 IT zg
4h 2 30
ยตIn4h7 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 Identities๐๐ ๐ ๐ฅ = 1
๐ ๐๐ ๐ฅ ๐ ๐๐ ๐ฅ = 1๐๐๐ ๐ฅ ๐๐๐ก ๐ฅ = 1
๐ก๐๐ ๐ฅ ๐ก๐๐ ๐ฅ = ๐ ๐๐ ๐ฅ๐๐๐ ๐ฅ
๐ ๐๐2๐ฅ + ๐๐๐ 2๐ฅ = 1 ๐ก๐๐2๐ฅ + 1 = ๐ ๐๐2๐ฅ 1 + ๐๐๐ก2๐ฅ = ๐๐ ๐2๐ฅ
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 ๐ฑ, cos๐ฑ)~ Converting between degrees and radians:
๐๐๐๐๐๐๐ = ๐๐๐๐๐๐ ร ฯ180
๐๐๐๐๐๐ = ๐๐๐๐๐๐๐ ร 180ฯ
~ Key trig values:
Sin, cos, and tan at 0, , , andฯ2 ฯ 3ฯ
2
Exponents and Logarithms
is equal to๐ฆ = ๐๐๐๐๐ฅ ๐ฅ = ๐๐ฆ
๐๐๐๐๐๐ = ๐๐๐
๐๐ + ๐๐๐
๐๐
๐๐๐๐
๐๐ = ๐๐๐
๐๐ โ ๐๐๐
๐๐ ๐๐๐
๐๐๐ = ๐ ๐๐๐
๐๐
๐๐ฅ๐ = ๐ฅ
๐๐ ๐ฅโ1 = 1
๐ฅ
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
ฯ6
ฯ4
ฯ3
ฯ2
2ฯ3
3ฯ4
5ฯ6
ฯ
7ฯ6
5ฯ4
4ฯ3
3ฯ2
5ฯ3
7ฯ4
11ฯ6
2ฯ
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