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## SecondEdition| ©2017New Edition Available Michael Sullivan; Kathleen Miranda

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From one of today’s most accomplished and trusted mathematics authors comes a new textbook that offers unmatched support for students facing the AP® calculus exam, and the teachers helping them prepare for it.

Sullivan and Miranda’s *Calculus for the AP®* Course covers every Big Idea,...

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## A proven approach to the AP® Calculus course.

From one of today’s most accomplished and trusted mathematics authors comes a new textbook that offers unmatched support for students facing the AP® calculus exam, and the teachers helping them prepare for it.

Sullivan and Miranda’s *Calculus for the AP®* Course covers every Big Idea, Essential Knowledge statement, Learning Objective, and Math Practice described in the 2016-2017 redesigned College Board™ Curriculum Framework. Its concise, focused narrative and integrated conceptual and problem-solving tools give students just the help they needas they learn calculus and prepare for the redesigned AP® Exam. And its accompanying Teacher’s Edition provides an in depth correlation and abundant tips, examples, projects, and resources to ensure close adherence the new Curriculum Framework.

Features

**FeaturesChapter Projects**Introduced at the opening of each chapter and are revisited at the end, these projects relate calculus to everyday life and provide students with a chance to apply the math they are learning.

**Learning Objectives**Each section begins with a point-by-point listing of the important concepts students need to take away.

**Examples with Worked Solutions**Hundreds of examples with fully detailed solutions give students clear guidance for applying essential concepts to solving problems.

**Sullivan’s Trademark “Now Work” Examples**Students working through the exercise in each example are directed to problems in the section exercise set that require the same type of work to solve, for immediate reinforcement and extra practice. This acclaimed feature, a hallmark of Sullivan textbooks, recognizes that many students begin with the exercises and seek help only as needed, providing a backwards/forwards navigation system that guides students from exercise to example and vice-versa.

**Effective Art Program** •

**Consistent color usage**in the book’s graphics help students to visualize and understand concepts

•

**Dynamic Figures (available online)**illustrate the principles of calculus in an interactive format that allows students to manipulate variables and gauge the results.

**Videos from Experienced AP© Teachers**These videos provide additional “just-in-time” help by walking students step-by-step through key examples, exercises, and concepts.

**Integrated Margin Tips** •AP® EXAM INSIGHTS: Contain tips and suggestions based on the redesigned curriculum framework to help students see how the text prepares for the exam.

•NEED TO REVIEW: offers just in time review of prior topics that students need to know to learn new content

•RECALL: provides a quick refresher of information that is being used theorems, definitions, examples, etc.

•NOTE: feature provides supporting details to content on that page to help further explain topics

•CAUTION: warns about common misconceptions and pitfalls.

•ORIGINS: provides historical and biographical information about key figures and discoveries in the development of Calculus

•IN WORDS helps to translate formulas, theorems, proofs, rules and definitions in clear, plain language to support your students when and where they need it.

**End-of-Section Practice Problems:** •

**Concepts and Vocabulary**—a chance to review comprehension of the main points with these fill in the blank, multiple choice and true/false questions

•

**Skill Building**– provided to help students develop problem-solving and computational skills

•

**Applications and Extensions**–a diverse set of problems that put calculus in real-world contexts

•

**AP® Practice Problems**— multiple choice and open-ended questions for practice all year round

**End-of-Chapter Features** •

**Things to Know**reviews the key concepts

•

**Objectives Table**reviews the chapter’s learning objectives, with directions to Chapter Review and AP Review Problems related to each objective

•

**Chapter Review Problems**review chapter content

•

**AP Review Problems**provide additional multiple-choice and open ended questions to practice for the exam all year round.

**Four (4) Cumulative AP® Practice Exams ** •Cumulative Exam 1: after Chapter 1 that covers: (1) Limits

•Cumulative Exam 2: after Chapter 4 that covers: (1) Limits, (2) Derivatives

•Cumulative Exam 3: after Chapter 7 (full length test modeled after the AB version of the exam) that covers: (1) Limits, (2) Derivatives, and (3) Integrals and Fundamental Theorem of Calculus)

•Cumulative Exam 4: after Chapter 11(full length exam modeled after the BC course) that covers (1) Limits, (2) Derivatives, (3) Integrals &Fundamental Theorem of Calculus, (4) Series

**LaunchPadLaunchpadworks.com**Built to address the biggest classroom issues instructors face, LaunchPad gives students everything they need to prepare for class and exams, while giving instructors everything they need to quickly set up a course, shape the content to their syllabus, craft presentations and lectures, assign and assess homework, and guide the progress of individual students and the class as a whole.

**LaunchPad for this title features: •An interactive e-Book** with integrated book-specific student media.

•

**LearningCurve adaptive quizzing**offering individualized question sets and feedback (with e-book links) for each student based on correct and incorrect responses.

•

**Dynamic Figures (available online)**illustrating calculus principles in an interactive format that allows students to manipulate variables and gauge the results.

•

**Videos from Experienced AP© Teachers**providing additional “just-in-time” help by walking students step-by-step through key examples, exercises, and concepts.

•

**New! Video Assignment Tool**makes it easy to create assessable coursework based on videos from LaunchPad, YouTube, or elsewhere on the Internet.

•An easy-to-use

**Gradebook**to monitor the whole class or individual students.

•

**Optional pre-built units,**curated by the book’s authors or other experienced educators and fully customizable. These units make assigning all of the rich content found in LaunchPad simple and effective!

•

**All instructor resources**(book images, teaching suggestions, and more) in one convenient location.

•

**Easy LMS integration**into any learning management system. LaunchPad also works independently outside of LMS systems.

New to This Edition

** Calculus for the AP® Course**

SecondEdition| ©2017

Michael Sullivan; Kathleen Miranda

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**Calculus for the AP® Course**

Second Edition| 2017

Michael Sullivan; Kathleen Miranda

## Table of Contents

Preface

Applications Index

Chapter P: Precalculus

Chapter 1: Limits and Conitnuity

Cumulative AP® Practice Exam 1

Chapter 2: The Derivative

Chapter 3: More About Derivatives

Chapter 4: Applications of the Derivative

Cumulative AP® Practice Exam 2

Chapter 5: The Integral

Chapter 6: Applications of the Integral

Chapter 7: Techniques of Integration

Cumulative AP® Practice 3:

Chapter 8: Infinite Series

Chapter 9: Parametric Equations, Polar Equations

Chapter 10: Vectors; Lines, Planes, and Quadratic Surfaces in Space

Chapter 11: Vector Functions

Cumulative AP® Practice Exam 4

Appendix A: Precacluus Used in Calculus

Appendix B: Theorems and Proofs

Appendix C: Technology Used in Calculus

**Calculus for the AP® Course**

Second Edition| 2017

Michael Sullivan; Kathleen Miranda

Find Your Rep

### Michael Sullivan

**Michael Sullivan**, Emeritus Professor of Mathematics at Chicago State University, received a Ph.D. in mathematics from the Illinois Institute of Technology. Before retiring, Mike taught at Chicago State for 35 years, where he honed an approach to teaching and writing that forms the foundation for his textbooks. Mike has been writing for more than 35 years and currently has 15 books in print. His books have been awarded both Texty and McGuffey awards from the Textbook and Academic Authors Association (TAA). In 2007, he received the TAA Lifetime Achievement Award. Mike is a member of the American Mathematical Association of Two Year Colleges, the American Mathematical Society, the Mathematical Association of America, and the Textbook and Academic Authors Association.

### Kathleen Miranda

**Kathleen Miranda**, Ed.D from St. John’s University, is an Emeritus Associate Professor of the State University of New York (SUNY), where she taught for 25 years. Kathleen is a recipient of the prestigious New York State Chancellor’s Award for Excellence in Teaching, and she particularly enjoys teaching mathematics to underprepared and fearful students. Kathleen served as director of Curriculum and Assessment Development at Old Westbury. In addition to her extensive classroom experience, Kathleen has worked as accuracy reviewer and solutions author on several mathematics textbooks, including Michael Sullivan’s Brief Calculus and Finite Mathematics. Kathleen’s goal is to help students unlock the complexities of calculus and appreciate its many applications.

**Calculus for the AP® Course**

SecondEdition| 2017

Michael Sullivan; Kathleen Miranda

Find Your Rep

## See a preview of this program’s resources

**1.** Sign in to unlock your preview. | **2.** Select your desired resources| **3.** Submit a request.| **4.** Download.

### Chapter 2 Bell Ringers

### Chapter 2 Student Edition

### Chapter 2 Teacher's Edition

### ExamView Assessment Suite for Calculus for AP®

Michael Sullivan; Kathy Miranda | SecondEdition | ©2017 | ISBN:9781464155383This comprehensive test bank in ExamView Assessment Suite includes thousands of questions in the AP® exam format. Questions are correl...

### ExamView Assessment Suite for Calculus for the AP®, Redesign Edition

Jon Rogawski | SecondEdition | ©2016 | ISBN:9781319046989All questions on our updated ExamView ® test bank will be correlated to the Learning Objectives and Essential Knowledge items in the redesig...

### Review Problem Videos

### Teacher's Edition Front Matter: Pacing Guides & Walkthrough

### Teacher's Edition for Calculus for AP*

Michael Sullivan; Kathy Miranda | SecondEdition | ©2017 | ISBN:9781464155437The Teacher’s Edition provides answers at the point-of-use, abundant tips, examples, and resources to ensure close adherence to the new Curriculum Fra...

### Teacher's Resource Flash Drive for Calculus for AP®

Michael Sullivan; Kathy Miranda | SecondEdition | ©2017 | ISBN:9781464155444The convenient TRFD includes additional resources that work in tandem with Teacher's Edition including: Professional Development Videos, Worked Exampl...

### Teacher's Solutions Manual for Calculus for AP® (Online Only)

Michael Sullivan; Kathy Miranda | SecondEdition | ©2017 | ISBN:9781464155413**Calculus for the AP® Course**

Second Edition| 2017

Michael Sullivan; Kathleen Miranda

## FAQs

### Is AP calc the hardest? ›

Finally, **AP Calculus BC is the toughest AP math exam**, if not one of the hardest AP exams period. AP Calculus AB is also challenging, but covers less material and moves more slowly.

**Is it hard to get a 5 on AP calculus? ›**

Is it Easy to Get a 5 on AP Calc BC? At first glance, it may appear that **it is easier to get a 5 on the AP Calculus BC exam than on the AP Calculus AB exam**. In 2021, for example, about 38% of Calculus BC test-takers received a 5 on the exam, whereas only 18% of Calculus AB test-takers received a 5 that same year.

**Does AP Calculus get a formula sheet? ›**

**There's no formula sheet given on the AP exam**, so you'll have to memorize the formulas you'll need.

**Which AP is the easiest? ›**

**The Three Easiest AP Classes**

- AP Psychology. With a reputation as one of the easiest AP classes, it comes as no surprise that AP Psychology is also one of the most popular—292,501 students took the exam in 2021. ...
- AP Comparative Government and Politics. ...
- AP Environmental Science.

**What math is higher than Calc? ›**

After completing Calculus I and II, you may continue to Calculus III, **Linear Algebra**, and Differential Equations. These three may be taken in any order that fits your schedule, but the listed order is most common.

**How many people fail calculus? ›**

The Mathematics Association of America (MAA) has reported the national average of unsuccessful Calculus 1 students to be **25%**.

**Is a 70 A 5 on an AP exam? ›**

**Usually, a 70 to 75 percent out of 100 translates to a 5**. However, there are some exams that are exceptions to this rule of thumb. The AP Grades that are reported to students, high schools, colleges, and universities in July are on AP's five-point scale: 5: Extremely well qualified.

**What is the pass rate for AP Calculus? ›**

The Pass Rate

In 2021, about 71% of test-takers passed AP exams with a score of 3 or higher. AP Calculus AB students had a pass rate around **61%**.

**What grade is a 5 on AP Calc? ›**

Receiving a 3, 4, or 5 is generally accepted as scoring well on an AP® exam. According to the College Board a 3 is 'qualified,' a 4 'well qualified,' and a 5 '**extremely well qualified**.

**Does AP Calc look good for college? ›**

**Calculus, and particularly AP calculus, is considered a must-have for students aiming for competitive STEM colleges or majors**. Strong performance in high school calculus demonstrates readiness for college mathematics.

### Is calculus AB or BC harder? ›

**AP Calculus BC is the more advanced of the two courses** and is designed to build off the skills students learned in AP Calculus AB. Because this is a very advanced course, students are expected to have a substantial math background. Learners may find it helpful to have taken AP Calculus AB before taking AP Calculus BC.

**What percent of students get a 5 on the AP calc exam? ›**

**How do you get a 5 on AP Calc? ›**

**How Can I Get a 5 in AP Calculus AB/BC?**

- Know the test 📚 Knowing how you'll be tested and what you'll be tested over is key to getting a 5. ...
- Memorize derivative and integral rules ✅ ...
- Understand application problems 🚗 ...
- Practice, practice, practice!

**What do you memorize for AP calculus? ›**

**AP® Calculus Content Tips:**

- Always remember to include +C in your answer for indefinite integrals. ...
- Review past trigonometric concepts. ...
- In doing derivatives, remember the difference between y = x n y = x^n y=xn, y = a x y = a^x y=ax, y = a n y = a^n y=an, and y = x x y = x^x y=xx.

**What is the least passed AP exam? ›**

The most failed AP exams are **Physics 1** (failed by 48.4% of all students), Environmental Science (failed by 46.6% of all students), and Chemistry (failed by 43.9% of all students). For a full chart of the hardest AP exams (those with the lowest passing rate), check out this site.

**What is the least popular AP class? ›**

Many of the least popular AP exams are **world language tests**, as these exams generally target a more niche group of students. What's more, some languages are less commonly taught at high schools than others (e.g., Japanese and Italian courses are rarer than Spanish courses).

**Is it better to get an A in regular or B in AP? ›**

The answer that most colleges will give you is that **it's better to get an A in the Honors/AP class**. And most highly-selective schools will expect that you do. But many colleges would rather see a B in an Honors or AP course than a higher grade in a regular college prep course.

**What is the toughest math? ›**

**The Riemann Hypothesis**, famously called the holy grail of mathematics, is considered to be one of the toughest problems in all of mathematics.

**What's harder than calculus? ›**

At an advanced level, **statistics** is considered harder than calculus, but beginner-level statistics is much easier than beginner calculus. Frankly, it mostly depends upon the student's interest as some students find it hard to comprehend statistics while others find it hard to understand calculus.

**What is the lowest math class in college? ›**

Entry-level math in college is considered the stepping stone to more advanced math. **Algebra 1**, trigonometry, geometry, and calculus 1 are the basic math classes. Once you have successfully navigated through these courses, you can trail blazed through more advanced courses.

### Is it OK to not take calculus in high school? ›

In 2016, the University of California stated that **high school students do not need to take calculus to be eligible for admission**. And in 2020, it added more math courses, such as data science, to the list of courses it accepts for students' third and fourth year of high school math.

**What percent of high schoolers take calculus? ›**

Only 19 percent of high school graduates took calculus in high school, although the numbers are much higher for those going to a competitive college. The numbers vary widely by race.

**How much IQ do you need for calculus? ›**

**115-120** is probably required for a solid understanding of the full calculus sequence.

**What GPA is a 90 in an AP class? ›**

Letter Grade | Numerical Grade | AP/IB GPA |
---|---|---|

A/A+ | 93-100 | 5.0 |

A- | 90-92 | 4.7 |

B+ | 87-89 | 4.3 |

B | 83-86 | 4.0 |

**Can you score a 0 on the AP exam? ›**

...

Step 3: Use the Chart to Estimate Your Scaled Score.

**What GPA is a 95 in an AP class? ›**

**What is the easiest AP class to pass? ›**

What is the easiest AP class? AP Computer Science Principles is rated as the easiest AP class by real AP class alumnae, with an average difficulty rating of just 2.8 / 10 (1 = easiest), and the 2022 pass rate of 69% is about average.

**Is a 2 A failing AP score? ›**

**Students who earn AP scores of 2 are well prepared to succeed in introductory college coursework**. Compared to academically similar college peers who did not take the AP course, AP students who earn scores of 2 perform as well or better when they take those introductory college courses.

**Is a 3 failing on the AP exam? ›**

Students can score a 1, 2, 3, 4, or 5 on the AP Exam, and **a 3 is considered passing**. Post-secondary institutions that offer college credit for AP Exam scores typically begin offering credit to students with a score of 3 or higher.

**What GPA is a C+ in an AP class? ›**

C+ GPA. A C+ letter grade is equivalent to a **2.3 GPA**, or Grade Point Average, on a 4.0 GPA scale, and a percentage grade of 77–79.

### What GPA is an A in an AP class? ›

Many schools treat AP classes differently when calculating a student's grade point average (GPA). Rather than the traditional 4.0 scale, AP classes are weighted on a 5.0 scale — in an AP class, **an A is equivalent to 5.0 instead of 4.0**, and a B is equal to 4.0 instead of 3.0.

**What happens if you get a 1 on an AP exam? ›**

In fact, on the AP® rubric, a 1 is described as “no recommendation”. Because of that, **no college in the United States of overseas will accept an AP® score of 1 and give you college credit**.

**Do AP classes hurt your GPA? ›**

At many high schools, honors and AP classes both offer more heavily weighted training compared to regular classes. While honors courses usually add 0.5 points to your GPA, **AP classes often add 1 point**. In other words, a 3.5 GPA would be boosted to a 4.0 in an honors class and a 4.5 in an AP class.

**Is a B in an AP class good? ›**

On a weighted scale, a B in an AP class would equate to a 4.0 weighted GPA, so it will not hurt your overall GPA too much. Colleges normally set a benchmark GPA that, **if you meet it, then you are good**.

**Is it worth taking AP Calc in high school? ›**

**Close to 80 percent agreed that students who have taken calculus in high school are more likely to succeed in college** — despite research showing that high school calculus does not necessarily correlate with advanced placement in college math.

**Can you skip AP Calculus AB? ›**

Unlike courses like Algebra I and Algebra II, Calculus AB (Advanced Placement Calculus) and Calculus BC do not have to be taken in a specific sequence. Meaning, **you could take Calculus BC without having ever taken Calculus AB**. Many students do.

**Can you skip Calculus BC? ›**

**No, you do not need Calc BC to apply as a CS major at any school**. While you may face some stiff competition at schools like CalTech or MIT, it's not required. I would also look into taking an online class either over the summer or during the year.

**Is AP Calculus worth it? ›**

All in all, **AP Calculus is definitely worth taking**. It's extremely useful for most STEM fields, like physics, chemistry, and biology. Even if you aren't going into STEM, calculus is a new type of math that you just might like if you try. If you start to find it difficult, don't get discouraged!

**What is the hardest AP to get a 5 on? ›**

AP Class/Exam | Pass Rate (3+) | Perfect Score (5) |
---|---|---|

2. Environmental Science | 53.4% | 11.9% |

3. Chemistry | 56.1% | 10.6% |

4. U.S. Government and Politics | 57.5% | 15.5% |

5. U.S. History | 58.7% | 13.0% |

**What is the easiest AP exam to get a 5? ›**

AP Class/Exam* | Pass Rate (3+) | Perfect Score (5) |
---|---|---|

1. Physics C: Mechanics | 84.3% | 41.6% |

2. Calculus BC | 81.6% | 44.6% |

3. Spanish Literature | 75.1% | 17.6% |

4. Physics C: Electricity and Magnetism | 74.4% | 40.4% |

### How rare is it to get a 5 on an AP exam? ›

The odds of passing with a 5—the highest score—are quite low on any exam: **between 10% and 20%** for most tests. They are even lower for popular tests, such as AP English Lit and AP Environmental Science, which have 5 rates below 10%.

**What happens if you get a 5 on the AP? ›**

A 4 or a 5 is the AP score that will **most likely earn you college AP credit**. Of course, no matter how you do on the AP test, you still get a grade for that AP class from your high school.

**How many people get a 5 on AP stats? ›**

Getting a 5 takes careful content knowledge, targeted practice and dedicated studying. Only around 14% earn the top score for the AP® Statistics exam. To get a 5, start studying ASAP® and focus on applying concepts to specific situations.

**What are the 3 main topics in calculus? ›**

**The Three Calculus Concepts You Need to Know**

- 1) Limits. Limits are a fundamental part of calculus and are among the first things that students learn about in a calculus class. ...
- 2) Derivatives. Derivatives are similar to the algebraic concept of slope. ...
- 3) Integrals.

**Is it possible to self study AP Calculus? ›**

If you plan to self-study for the AP Calculus BC exam without taking an AP class, you may have a few more obstacles and challenges ahead. However, **it is definitely doable**. The biggest challenge will be not having a teacher to introduce concepts and help you improve.

**What is the hardest AP subject? ›**

The Hardest AP Classes

Traditionally, courses like **English Literature, Physics 1, and Chemistry** are difficult to self-study for or complete at home because of the need for conversation and 1:1 instruction — like the lab element necessary to understand the science courses' material.

**Is AP Calculus worth taking? ›**

All in all, **AP Calculus is definitely worth taking**. It's extremely useful for most STEM fields, like physics, chemistry, and biology. Even if you aren't going into STEM, calculus is a new type of math that you just might like if you try. If you start to find it difficult, don't get discouraged!

**What percentage of people get a 5 on AP Calc? ›**

**Is AP Calculus harder than AB? ›**

Is AP Calculus AB Harder or AP Calculus BC? **AP Calculus BC is more difficult than AP Calculus AB**. Not only does it include additional topics, which requires an accelerated pace, but the additional units, especially Unit 10, tend to be more difficult than the Calc AB units.

**What is the most failed AP exam? ›**

The most failed AP exams are **Physics 1 (failed by 48.4% of all students), Environmental Science (failed by 46.6% of all students), and Chemistry (failed by 43.9% of all students)**. For a full chart of the hardest AP exams (those with the lowest passing rate), check out this site.

### What is the most passed AP exam? ›

AP Class/Exam | Pass Rate (3+) | Perfect Score (5) |
---|---|---|

1. Physics 1 | 51.6% | 8.8% |

2. Environmental Science | 53.4% | 11.9% |

3. Chemistry | 56.1% | 10.6% |

4. U.S. Government and Politics | 57.5% | 15.5% |