The Princeton Review: Cracking the AP calculus AB exam (2019)

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Editorial
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Trade Paperback ISBN 9780525568155
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AP is a trademark registered and owned by the College Board, which is not affiliated with, and
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Acknowledgments
First of all, I would like to thank Arnold Feingold and Peter B. Kahn
for once again doing every problem, reading every word, and
otherwise lending their invaluable assistance. I also want to thank
Aaron Riccio for being a terrific editor on this edition of the book, Gary
King for his first-rate reading, analysis, and contributions, and the
production team of Liz Rutzel, Harmony Quiroz, and Deborah Weber.
Thanks to Frank, without whose advice I never would have taken this
path. Thanks to Jeffrey, Miriam, and Vicki for moral support. Thanks
Mom.
Finally, I would like to thank the people who really made all of this
effort worthwhile—my students. I hope that I haven’t omitted anyone,
but if I have, the fault is entirely mine.
Aaron and Sasha, Aaron, Aaron T., Abby B., Abby F., Abby H.,
Abby L., Abbye, Abigail H., Addie, Aidan, Alan M., Albert G., Alec G,
Alec M., Alec R., Alex and Claire, Alex A., Alex B., Alex and Daniela
B., Alex F., Alex D., Alex G., Alex H., Alex I., Alex S., Alex and Gabe,
Alexa, Alexandra, Natalie and Jason, Alexes, Alexis and Brittany, Ali
and Jon, Ali and Amy Z., Ali H., Alice, Alice C., Alicia, Alisha, Allegra,
Allie and Lauren, Allison and Andrew, Allison H., Allison and Matt S.,
Allison R., Ally, Ally M., Ally T., Ally W., Aly and Lauren, Alyssa C.,
Alyssa TDC, Alyssa and Courtney, Alvin, Amanda, Brittany, and Nick
A., Amanda and Pamela B., Amanda C., Amanda E., Amanda H.,
Amanda M., Amanda R., Amanda S., Amanda Y., Amber and Teal,
Amparo, Andrea T., Andrea V., Andrew A., Andrei, Andrew B., Andrew
C., Andrew D., Andrew E., Andrew H., Andrew M., Andrew S., Andy
and Sarah R., Andy and Allison, Angela, Angela F., Angela L., Anisha,
Ann, Anna C-S., Anna D., Anna and Jon, Anna and Max, Anna L.,
Anna M., Anna S., Anna W., Annabelle M., Annerose, Annie, Annie B.,
Annie W., Antigone, Antonio, Anu, April, Ares, Ariadne, Ariane, Ariel,

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Arielle and Gabrielle, Arthur and Annie, Arya, Asheley and Freddy,
Ashley A., Ashley K., Ashley and Sarah, Ashley and Lauren, Aubrey,
Avra, Becca A., Becky B., Becky S., Becky H., Ben S., Ben and
Andrew Y., Ben D., Ben S., Benjamin D., Benjamin H., Benji, Beth and
Sarah, Bethany and Lesley, Betsy and Jon, Blythe, Bianca and
Isabella, Biz, Bonnie, Bonnie C., Boris, Braedan, Brendan, Brett, Brett
A., Brette and Josh, Brian, Brian C., Brian N., Brian W., Brian Z.,
Brigid, Brin, Brittany E., Brittany S., Brooke, Devon, and Megan,
Brooke and Lindsay E., Butch, Cailyn H., Caitlin, Caitlin F., Caitlin M.,
Caitlin and Anna S., Camilla and Eloise, Camryn, Candace, Cara,
Cara G., Carinna, Caroline A., Caroline C., Caroline H., Caroline S.,
Caroline and Peter W., Carrie M., Catherine W., Cecilia, Celia, Chad,
Channing, Charlie, Charlotte, Charlotte B., Charlotte M., Chase,
Chelsea, Chloe, Chloe K., Chris B., Chris C., Chris K., Chris P.,
Chrissie, Christa S., Christen C., Christian, Christian C., Christina F.,
Christine, Christine W., CJ, Claire H., Clare, Claudia, Clio, Colette,
Conor G., Corey, Corinne, Coryn, Courtney and Keith, Courtney B.,
Courtney F., Courtney S., Courtney W., Craig, Dan M., Dana J., Dani
and Adam, Daniel, Daniel and Jen, Daniella, Daniella C., Danielle and
Andreas, Danielle and Nikki D., Danielle G., Danielle H., Danny K.,
Dara and Stacey, Dara M., Darcy, Darya, David B., David C., David
R., David S., Deborah and Matthew, Deniz and Destine, Deshaun,
Deval, Devin, Devon and Jenna, Deyshaun, Diana H., Dilly, Disha,
Dong Yi, Dora, Dorrien, Eairinn, Eddy, Elana, Eleanor K., Elexa and
Nicky, Elisa, Eliza, Elizabeth, Elizabeth and Mary C., Elizabeth F.,
Ella, Ellie, Elly B., Elyssa T., Emily A., Emily B., Emily and Allison,
Emily and Catie A., Emily C., Emily G., Emily H., Emily K., Emily L.,
Emily and Pete M., Emily R., Emily R-H., Emily S., Emily T., Emma,
Emma and Sophie, Emma D., Emma S., Eric N., Erica F., Erica H.,
Erica R., Erica S., Eric and Lauren, Erica and Annie, Erika, Erin, Erin
I., Ernesto, Esther, Ethan, Eugenie, Eva, Evan, Eve H., Eve M., Fifi,
Francesca, Frank, Franki, Gabby, Geoffrey, George and Julian.,
George M., Gloria, Grace M., Grace P., Gracie, Graham and Will,
Greg, Greg F., Griffin, Gussie, Gylianne, Haley, Halle R., Hallie,
Hannah C., Hannah J., Hannah R., Hannah and Paige, Harrison,
Annabel and Gillian, Harry, Hayley and Tim D., Hazel, Heather D.,

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Heather E., Heather F., Heather and Gillian, Hernando and Vicki,
Hilary and Lindsay, Hilary F., Holli, Holly G., Holly K., Honor, Ian, Ian
P., India, Ingrid, Ira, Isa, Isabel, Isabella and Simone, Isabelle T.,
Ismini, Isobel, Ivy, Jabari, Jacob and Kara, Jack, Jack B., Jackie and
Vicky, Jackie, Jackie A., Jackie S., Jackson, Jaclyn and Adam, Jacob
D., Jada, James S., Jamie, Jason P., Jason and Andrew, Jay, Jay K.,
Jae and Gideon, Jake T., Jarrod, Jason L., Jayne and Johnny, Jed D.,
Jed F., Jeffrey M., Jenna, Jen, Jen M., Jenn N., Jennifer B., Jennifer
W., Jenny K, Jenny and Missy, Jeremy C., Jeremy and Caleb, Jess,
Jess P., Jesse, Jessica, Jessica and Eric H., Jessica L., Jessica T.,
Jessie, Jessie C., Jessie and Perry N., Jill, Jillian, Daniel and Olivia,
Jillian S., Jimin, Jimmy C., Jimmy P., Joanna, Joanna C., Joanna G.,
Joanna M., Joanna and Julia M., Joanna W., Jocelyn, Jody and Kim,
Joe B., Johanna, John, John and Dan, Jonah and Zoe, Jonathan G.,
Jonathan P., Jonathan W., Jongwoo, Jordan C., Jordan and Blair F.,
Jordan G., Jordan P., Jordana, Jordyn, Jordyn K., Josh and Jesse,
Josh and Noah, Judie and Rob, Julia and Caroline, Julia, Julia F.,
Julia G., Julia H., Julia and Charlotte P., Julia T., Julie, Julie H., Julie
J., Julie and Dana, Julie P., Juliet, Juliette R., Justin L., Kara B., Kara
O., Kasey, Kasia and Amy, Kat C., Kat R., Kate D., Kate F., Kate G.,
Kate L., Kate P., Kate S., Kate W., Katie C., Katie F., Katherine C.,
Kathryn, Leslie, and Travis, Katie, Katie M., Katrina, Keith M., Kelly C.,
Ken D., Kenzie, Keri, Kerri, Kim H., Kimberly, Kimberly H., Kirsten,
Kitty and Alex, Kori, Kripali, Krista, Erika, and Karoline, Kristen,
Kristen, Kristen Y., Magan, and Alexa, Laila and Olivia R., Laura B.,
Laura F., Laura G., Laura R., Laura S., Laura T., Laura Z., Lauren and
Eric, Lauren R., Lauren S., Lauren T., Lauren and Allie, Lea K., Leah,
Lee R., Leigh and Ruthie, Leigh, Lexi S., Lila, Lila M., Lilaj, Lili C.,
Lilla, Lily, Lily M-R., LilyHayes, Lisbeth and Charlotte, Lina, Lindsay F.,
Lindsay K., Lindsay L., Lindsay N., Lindsay R, Lindsay and Jessie S.,
Lindsey and Kari, Lisa, Liz D., Liz H., Liz M., Lizzi B., Lizzie A., Lizzie
M., Lizzie W., Lizzy, Lizzy C., Lizzy R., Lizzy T., Louis, Lucas, Luke
C., Lucinda, Lucy, Lucy D., Mackenzie, Maddie and JD, Maddie H.,
Maddie P., Maddy N., Maddy T., Maddy W., Madeline, Magnolia,
Maggie L., Maggie T., Manuela, Mara and Steffie, Marcia, Margaret
S., Maria B., Mariel C., Mariel L., Mariel S., Marielle K., Marielle S.,

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Marietta, Marisa B., Marisa C., Marissa, Marlee, Marnie and Sam,
Mary M., Matt, Matthew B., Matthew K., Matt F., Matt G., Matt and
Caroline, Matt V., Matthew G., Mavis H., Max B., Max and Chloe K.,
Max M., Maxx, Ben, and Sam, Maya and Rohit, Maya N., Mckyla,
Meesh, Megan G., Meliha, Melissa, Melissa and Ashley I., Meredith,
Meredith and Gordie B., Meredith and Katie D., Meredith R., Meredith
S-K., Michael and Dan, Michael H., Michael R., Michaela, Michal,
Michele S., Mike C., Mike R., Miles, Milton, Miranda, Moira, Molly L.,
Molly R., Molly and Annie, Morgan, Morgan C., Morgan, Zoe, and Lila,
Morgan K., Nadia and Alexis, NaEun, Nancy, Nanette, Naomi, Natalie
and Andrew B., Natasha and Mikaela B., Natasha L., Nathania, Nico,
Nick, Nick A., Nicky S., Nicole B., Nicole J., Nicole N., Nicole P.,
Nicole S., Nicolette, Nidhi, Nikki G., Nina R., Nina V., Noah, Nora,
Nushien, Odean, Ogechi, Oli and Arni, Oliver, Oliver W., Olivia and
Daphne, Olivia P., Omar, Oren, Pam, Paige, Pasan, Paul A., Peter,
Philip and Peter, Phoebe, Pierce, Pooja, Priscilla, Quanquan, Quinn
and Chris, Rachel A, Rachel B., Rachel and Adam, Rachel H., Rachel
I., Rachel and Jake, Rachel K., Rachel L., Rachel and Mitchell M.,
Rachel S., Rachel and Eli, Rachel and Steven, Rachel T., Rachel W.,
Rae, Ramit, Randi and Samantha, Rasheeda, Ray B., Rayna,
Rebecca A., Rebecca B., Rebecca G., Rebecca and Gaby K.,
Rebecca R., Rebecca W., Renee F., Resala, Richard, Tina, and Alice,
Richie D., Ricky, Riley C., Rob L., Robin P., Romanah, Rose, Ryan,
Ryan B., Ryan S., Saahil, Sabrina and Arianna, Sabrina O., Sally,
Sam B., Sam C., Sam G., Sam L., Sam R., Sam S., Sam W., Sam Z.,
Samantha and Savannah, Samantha M., Samuel C., Samar, Sara G.,
Sara H., Sara L., Sara R., Sara S., Sara W., Sarah, Patty, and Kat,
Sarah and Beth, Sarah A., Sarah C., Sarah D., Sarah F., Sarah G.,
Sarah and Michelle K., Sarah L., Sarah M-D., Sarah and Michael,
Sarah P., Sarah S., Sascha, Samara, Saya, Sean, Selena G., Serena,
Seung Woo, Shadae, Shannon, Shayne, Shayne, Shorty, Siegfried,
Simon W., Simone, Sinead, Siobhan, Skye F., Skye L., Sofia G., Sofia
M., Sonja, Sonja and Talya, Sophia A., Sophia B., Sophia J., Sophia
S., Sophie and Tess, Sophie D., Sophie S., Sophie W., Stacey, Stacy,
Steffi, Steph, Stephanie, Stephanie K., Stephanie L., Stephen C.,
Sumair, Sunaina, Suzie, Syd, Sydney, Sydney S., Tammy and Hayley,

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Tara and Max, Taryn, Tasnia, Tayler, Taylor, Tenley and Galen,
Terrence, Tess, Theodora, Thomas B., Timothy H., Tina, Tita, Tom,
Tori, Torri, Tracy, Tracy K., Tripp H., Tripp W., Tyler, Tyrik, Uma,
Vana, Vanessa and John, Veda, Veronica M., Vicky B., Victor Z.,
Victoria, Victoria H., Victoria J., Victoria M., Vinny and Eric, Vivek,
Vladimir, Waleed, Will A., Will, Teddy, and Ellie, William M., William Z.,
Wyatt and Ryan, Wyna, Xianyuan, Yakir, Yesha, Yujin, Yuou, Zach,
Zach B., Zach S., Zach Y., Zachary N., Zaria, Zoë, Zoe L., and Zoey
and Rachel.

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Contents
Cover
Title Page
Copyright
Acknowledgments
Get More (Free) Content
Part I: Using This Book to Improve Your AP Score
Preview: Your Knowledge, Your Expectations
Your Guide to Using This Book
How to Begin
Part II: Practice Test 1
Practice Test 1
Practice Test 1: Answers and Explanations
Part III: About the AP Calculus AB Exam
AB Calculus vs. BC Calculus
The Structure of the Calculus Exam
How the AP Calculus AB Exam is Scored
Overview of Content Topics
General Overview of This Book
How AP Exams Are Used
Other Resources
Designing Your Study Plan
Part IV: Test-Taking Strategies for the AP Calculus AB
Exam

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1 How to Approach Multiple-Choice Questions
2 How to Approach Free-Response Questions
Part V: Content Review for the AP Calculus AB Exam
Unit 1: Differential Calculus Essentials
3 Limits
What is a Limit?
Limits of Trigonometric Functions
Practice Problem Set 1
End of Chapter 3 Drill
4 Continuity
The Definition of Continuity
Types of Discontinuities
Practice Problem Set 2
End of Chapter 4 Drill
5 The Definition of the Derivative
Deriving the Formula
Differentiability
Practice Problem Set 3
End of Chapter 5 Drill
6 Basic Differentiation
Notation
The Power Rule
The Addition Rule
Higher Order Derivatives
Practice Problem Set 4
The Product Rule
The Quotient Rule
The Chain Rule

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Practice Problem Set 5
Derivatives of Trig Functions
Graphical Derivatives
Practice Problem Set 6
End of Chapter 6 Drill
7 Implicit Differentiation
How to Do It
Second Derivatives
Practice Problem Set 7
End of Chapter 7 Drill
Unit 2: Differential Calculus Applications
8 Basic Applications of the Derivative
Equations of Tangent Lines and Normal Lines
Practice Problem Set 8
The Mean Value Theorem for Derivatives
Rolle’s Theorem
Practice Problem Set 9
End of Chapter 8 Drill
9 Maxima, Minima, and Curve Stretching
Applied Maxima and Minima Problems
Practice Problem Set 10
Curve Sketching
Practice Problem Set 11
End of Chapter 9 Drill
10 Motion
Related Rates
Practice Problem Set 12
Position, Velocity, and Acceleration
Practice Problem Set 13

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End of Chapter 10 Drill
11 Derivatives of Exponential and Logarithmic Functions
The Derivative of ln x
The Derivative of ex
The Derivative of logax
The Derivative of ax
Practice Problem Set 14
End of Chapter 11 Drill
12 Other Topics in Differential Calculus
The Derivative of an Inverse Function
Practice Problem Set 15
Linearization
Practice Problem Set 16
L’Hôpital’s Rule
Practice Problem Set 17
End of Chapter 12 Drill
Unit 3: Integral Calculus Essentials
13 The Integral
The Antiderivative
Integrals of Trig Functions
Addition and Subtraction
Practice Problem Set 18
u-Substitution
Practice Problem Set 19
End of Chapter 13 Drill
14 Definite Integrals
Area Under a Curve
Tabular Riemann Sums

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Practice Problem Set 20
The Mean Value Theorem for Integrals
Practice Problem Set 21
End of Chapter 14 Drill
15 Integrals of Exponential, Logarithmic, and Trig Functions
Inverse Trig Functions
Practice Problem Set 22
More Integrals of Trig Functions
Integrating ex and ax
Practice Problem Set 23
End of Chapter 15 Drill
Unit 4: Integral Calculus Applications
16 The Area Between Two Curves
Vertical Slices
Horizontal Slices
Practice Problem Set 24
End of Chapter 16 Drill
17 The Volume of a Solid of Revolution
Washers and Disks
Cylindrical Shells
Volumes of Solids with Known Cross-Sections
Practice Problem Set 25
End of Chapter 17 Drill
18 Differential Equations
Separation of Variables
Slope Fields
Practice Problem Set 26
End of Chapter 18 Drill

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full title of the book
ISBN
page number
Need to report a technical issue?
Contact TPRStudentTech@review.com and provide:
your full name
email address used to register the book
full book title and ISBN
Operating system (Mac/PC) and browser (Firefox, Safari, etc.)
Once you’ve registered, you can…
Get our take on any recent or pending updates to the AP Calculus
AB Exam
Access and print Practice Test 6 as well as the corresponding
Answers and Explanations
Take a full-length practice SAT and ACT
Get valuable advice about the college application process,
including tips for writing a great essay and where to apply for
financial aid
If you’re still choosing between colleges, use our searchable
rankings of The Best 385 Colleges to find out more information
about your dream school.

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PREVIEW: YOUR KNOWLEDGE, YOUR
EXPECTATIONS
Your route to a high score on the AP Calculus AB Exam depends a lot
on how you plan to use this book. Start thinking about your plan by
responding to the following questions.
1. Rate your level of confidence about your knowledge of the
content tested by the AP Calculus AB Exam:
A. Very confident—I know it all
B. I’m pretty confident, but there are topics for which I could use
help
C. Not confident—I need quite a bit of support
D. I’m not sure
2. If you have a goal score in mind, choose your goal score for the
AP Calculus AB Exam:
5 4 3 2 1
I’m not sure yet
3. What do you expect to learn from this book? Choose all that
apply to you.
A. A general overview of the test and what to expect
B. Strategies for how to approach the test
C. The content tested by this exam
D. I’m not sure yet

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Ch-ch-ch-changes
The organizational structure of the AP Calculus AB
Exam is changing for the 2019–2020 course. The
material in this book covers all currently announced
content for the course and exam, but please check
your online student tools for any late-breaking news.
YOUR GUIDE TO USING THIS BOOK
This book is organized to provide as much—or as little—support as you
need, so you can use this book in whatever way will be most helpful for
improving your score on the AP Calculus AB Exam.
The remainder of Part I will provide guidance on how to use this
book and help you determine your strengths and weaknesses.
Part II of this book contains Practice Test 1 and its answers and
explanations. (Bubble sheets can be printed out from your free
online student tools for easy use.) We strongly recommend that
you take this test before going any further, in order to realistically
determine:
your starting point right now
which question types you’re ready for and which you might
need to practice
which content topics you are familiar with and which you will
want to carefully review
Once you have nailed down your strengths and weaknesses with
regard to this exam, you can focus your test preparation, build a
study plan, and be efficient with your time.

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Part III of this book will:
provide information about the structure, scoring, and content
of the AP Calculus AB Exam
help you to make a study plan
point you towards additional resources
Part IV of this book will explore the following strategies:
how to attack multiple-choice questions
how to write a high scoring free-response answer
how to manage your time to maximize the number of points
available to you
Part V of this book covers the content you need for your exam.
Part VI of this book contains two additional Practice Tests and
their answers and explanations. If you skipped Practice Test 1, we
recommend that you do all three (with at least a day or two
between them) so that you can compare your progress between
them. Additionally, this will help to identify any external issues: if
you get a certain type of question wrong both times, you probably
need to review it. If you only got it wrong once, you may have run
out of time or been distracted by something. In either case, this
will allow you to focus on the factors that caused the discrepancy
and to be as prepared as possible on the day of the test.
You may choose to use some parts of this book over others, or you may
work through the entire book. This will depend on your needs and how
much time you have. Let’s now look at how to make this
determination.

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Once you register your book online, you can
download and print out the bubble sheets for all
three practice tests. You can also find a handy
Appendix that contains all the formulas you should
know.
HOW TO BEGIN
1. Take a Test
Before you can decide how to use this book, you need to take a
practice test. Doing so will give you insight into your strengths
and weaknesses, and the test will also help you make an effective
study plan. If you’re feeling test-phobic, remind yourself that a
practice test is a tool for diagnosing yourself—it’s not how well
you do that matters but how you use information gleaned from
your performance to guide your preparation.
So, before you read further, take the AP Calculus AB Practice Test
1 starting at this page of this book. Be sure to do so in one sitting,
following the instructions that appear before the test.
2. Check Your Answers
Using the answer key on this page, count how many multiple-
choice questions you got right and how many you missed. Don’t
worry about the explanations for now, and don’t worry about why
you missed questions. We’ll get to that soon.
3. Reflect on the Test
After you take your first test, respond to the following questions:

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How much time did you spend on the multiple-choice
questions?
How much time did you spend on each free-response
question?
How many multiple-choice questions did you miss?
Do you feel you had the knowledge to address the subject
matter of the free-response questions?
Do you feel your free responses were well organized and
thoughtful?
Circle the content areas that were most challenging for you
and draw a line through the ones in which you felt
confident/did well.
Functions, Graphs, and Limits
Differential Calculus
Integral Calculus
Applications of Derivatives
Applications of Integrals
4. Read Part III and Complete the Self-Evaluation
Part III will provide information on how the test is structured
and scored. It will also set out areas of content that are tested.
As you read Part III, reevaluate your answers to the questions
above. At the end of Part III, you will revisit the questions above
and refine your answers to them. You will then be able to make a
study plan, based on your needs and time available, that will
allow you to use this book most effectively.

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5. Engage with Parts IV and V as Needed
Notice the word engage. You’ll get more out of this book if you
use it intentionally than if you read it passively, hoping for an
improved score through osmosis.
Strategy chapters will help you think about your approach to the
question types on this exam. Part IV will open with a reminder to
think about how you approach questions now and then close with
a reflection section asking you to think about how/whether you
will change your approach in the future.
Content chapters in Part V are designed to provide a review of the
content tested on the AP Calculus AB Exam, including the level of
detail you need to know and how the content is tested. You will
have the opportunity to assess your mastery of the content of
each chapter through test-appropriate questions and a reflection
section.
6. Take Another Test and Assess Your Performance
Once you feel you have developed the strategies you need and
gained the knowledge you lacked, you should take Test 2. You
should do so in one sitting, following the instructions at the
beginning of the test.
When you are done, check your answers to the multiple-choice
sections. See if a teacher will read your long-form calculus
responses and provide feedback.
Once you have taken the test, reflect on what areas you still need
to work on, and revisit the chapters in this book that address
those topics. Through this type of reflection and engagement, you
will continue to improve.
7. Keep Working

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After you have revisited certain chapters in this book, continue
the process of testing, reflecting, and engaging with Practice Test
3 on this page. You want to be increasing your readiness by
carefully considering the types of questions you are getting wrong
and how you can change your strategic approach to different
parts of the test.
As we will discuss in Part III, there are other resources available
to you, including a wealth of information on the College Board’s
AP Students website:
https://apstudent.collegeboard.org/apcourse/ap-calculus-ab.
You can continue to explore areas that can stand to improve and
engage in those areas right up to the day of the test.
Another Course? Of Course!
If you can’t get enough AP Calculus AB and want to
review this material with an expert, we also offer an
online Cram Course that you can sign up for here:
https://www.princetonreview.com/college/ap-test-
prep.

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