AP Calculus Prep

Summer 2026

Teacher: Caleb Alons

Email: calebscottalonsmathematics@gmail.com

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COURSE OVERVIEW:

The syllabus for this 0.5 credit course is provided below (email calebscottalonsmathematics@gmail.com to request a PDF copy of the Summer 2026 AP Calculus Prep syllabus).

TUITION: $350

WHY THIS COURSE?

AP Calculus tells a compelling story that comes in three major movements. I have spent the past six years researching methods for teaching calculus and have discovered that the vast majority of mainline textbooks are often organized and written in such a way that is only helpful to those who already understand calculus. This "retrospectively helpful" organization for textbooks (and consequently courses) motivated me to write my own text—one that would guide newcomers to calculus through its intuitive and well-motivated plot in an accessible way. Students registered for AP Calculus Prep will receive access to a one-of-a-kind textbook written in conversational language and well-paced lessons paired with example exercises that function to motivate and progress student curiosity and understanding. Because I believe that excellent style in presenting solutions has a major impact on speed and accuracy in the problem-solving process, I include full-length solutions to every single exercise in the text to model good form, style, and solution presentation (a mandatory skill for AP Calculus FRQs). As of now, this is a completely unique feature of my curriculum that cannot be found in any existing AP Calculus curricula online. Ultimately, I adore AP Calculus and have fallen in love with the story it tells; it is my sincere hope to share my joy for this story with all my students and for them to also find excitement in the great Epic of Calculus.

Syllabus for

AP Calculus Prep

Summer 2026

  1. COURSE DESCRIPTION
    1. AP Calculus Prep is designed to give students preparing to take AP Calculus AB/BC an advanced standing that ultimately will propel them to a mastery of calculus and excellent understanding of several core concepts on the AP exam. Motivated by the principle of layered learning, AP Calculus Prep identifies the most useful prerequisite skills from algebra and precalculus and introduces students to limits, continuity, discontinuity, derivative theory, applications of derivatives, integrals, and the Fundamental Theorem of Calculus. The curriculum attentively previews the overarching "story" that AP Calculus tells, and students are invited to understand calculus as a compelling, intuitive, and unified narrative.
    2. Most students should anticipate dedicating 4–5 hours each week to keep up with the pace and to get the most out of this course.
    3. Prerequisite(s): Students should have a firm grasp of Algebra 1, Algebra 2, Geometry, and Precalculus. The purpose of AP Calculus Prep is advancement to calculus concepts and NOT remedial for calculus prerequisites.
    4. Most calculus teachers will attest that the source of most student struggle is NOT the calculus material itself; rather, the inability to perform and remember prerequisite skills in algebra, geometry, and precalculus dramatically stunts student performance in AP Calculus. Although the first week of the course reviews many of these prerequisites, students should NOT expect this course to be remedial or a review of algebra and precalculus.
  2. STUDENT LEARNAING OUTCOMES FOR THIS COURSE
    1. COURSE OUTCOMES
      • After completing this course successfully, students will have studied and practiced the following concepts:
        1. Algebraic Properties and Laws
        2. Trigonometric Properties and Identities
        3. Fundamental Function Properties and Mechanics
        4. Advanced Algebraic Manipulation
        5. Interpreting Limits Graphically
        6. Defining Continuity
        7. Continuous Functions and Properties of Limits
        8. Jump Discontinuity and Limits of Piecewise Functions
        9. Removable Discontinuity and Factoring Limits
        10. Removable Discontinuity and Rationalizing Limits
        11. Infinite Discontinuity and Asymptotes
        12. Oscillating Discontinuity and the Squeeze Theorem
        13. Average Rate of Change over an Interval and the Secant Line
        14. Instantaneous Rate of Change at a Point and the Tangent Line
        15. Differentiability and Derivative Notation
        16. Properties of Derivatives
        17. Graphically Relating Functions and Their Derivatives
        18. Derivatives of Constants and Polynomials
        19. Derivatives of Exponentials and Logarithms
        20. Derivatives of Fundamental Trigonometric Functions
        21. Product Rule
        22. Quotient Rule
        23. Chain Rule
        24. Implicit Differentiation
        25. First Derivative Test
        26. Optimization
        27. Related Rates
        28. Fundamental Theorem of Calculus (Antiderivative Part)
        29. Properties of Indefinite Integrals
        30. Introduction to u-Substitution
        31. Fundamental Theorem of Calculus (Evaluation Part)
        32. Area Under a Curve
        33. Average Value of a Function
        34. Net Area vs. Total Area
    2. GENERAL OUTCOMES
      • Students will develop the following skills that do not necessarily have restricted application to this course specifically:
        1. Advanced creative problem solving
        2. Study discipline and mental toughness
        3. Seeking help and offering help to peers
        4. Collaborative learning
        5. Increased proficiency in standardized math testing
        6. Mathematical argumentation and presentation
  3. TEXTBOOKS AND OTHER LEARNING RESOURCES
    1. REQUIRED MATERIALS
      1. Textbooks:
        1. Alons, Caleb S. Calculus: Prep for the AP Class. Unpublished manuscript, August 2022, typescript.
        2. Access to the textbook is provided with the course; a PDF copy will be provided via the course website.
      2. Other:
        1. Access to the Internet to log onto the course website
  4. POLICIES AND PROCEDURES
    1. COURSE POLICIES AND PROCEDURES
      1. Attendance: AP Calculus Prep does not have a live component. Instead, students will engage on the class website throughout the week with the instructor and their fellow peers as they progress through the course material. Students desiring additional help from the instructor need only reach out to the instructor via email.
      2. Evaluation Procedures:
        1. Weekly Homework Assignments:
        2. Sections from the textbook are assigned each week. For each section, the student must do the following: read the textbook section and take notes (4 points), complete the exercises at the end of the section (4 points), and self-grade work with the provided solution keys (2 points). The expected section total is 10 points.
        3. Each item listed above is based on completion. Students will report their own completion scores per section by the Sunday of each week before 11:59 PM (EST).
        4. For Unit Reviews and Practice Days, there is no written lesson, so students should simply reward themselves automatically for the "read the textbook" points. There are a handful of sections that have only a lesson portion and no exercises; students should simply reward themselves automatically for the "complete the exercises" and "self-grade" points whenever the assigned textbook section has only a lesson portion. Otherwise, all lessons have all three components that must be completed to earn the completion points. Since there are 43 total sections, the expected weekly homework total is 430 points.
        5. Student Engagement:
        6. Student engagement and peer collaboration is STRONGLY encouraged in this course. Positive, collaborative, and productive engagement on the course website strongly and positively influences a student’s overall course grade.
        7. Final Examination:
        8. Students will take a final examination at the end of the course. The final examination contains 11 questions each worth 10 points for a total of 110 points. The final examination is comprehensive. The final examination must be proctored by a parent/guardian and taken in one uninterrupted sitting; aside from this, students may take as much time as needed to finish. The use of class notes, formula sheets, calculators, digital aids, the Internet, etc. are strictly prohibited. Access to the final examination will be given on the Friday of the final week of the course from 12 PM EST–11:59 PM (EST). If a student has special circumstances and needs to take the final examination at a different time/date, the student should email the instructor directly in advance for accommodation, which will be given for all legitimate situations.
        9. Grading Scale:
          1. 486–540 points
          2. 432–485 points
          3. 378–431 points
          4. 324–377 points
          5. 0–323 points
        10. The numeric grading scale is not completely fixed, as the instructor will consider other holistic factors into the overall final grade. The final grade is designed to reflect the student’s holistic growth and progress made throughout the course. Students who score poorly earlier in the course, but who demonstrate an eagerness and initiative to truly learn and overcome challenges, can earn higher grades than their total point sum is at the end of the course. A student’s progress over time is also factored into potentially raising a student’s final letter grade: steady improvements from poor to excellent scores will be awarded. Other factors for raising a student’s final letter grade include the student’s level of engagement, participation, activity on the course website (asking questions, answering questions, cultivating community with peers, creating a welcoming learning environment, etc.), and communication with the instructor throughout the course. The end goal of this course is for the student to acquire genuine comprehension and mastery of the material, so the final course grade is designed to reflect the student’s ability to learn, willingness to struggle, and desire to overcome deficits and setbacks. Since mathematics in the real world is messy and characterized by "being stuck," the grading system for this course hopes to incentivize students to think independently and creatively and to reward students for embracing challenges and learning how to handle struggle.
  5. COURSE CALENDAR
    • Note on Due Dates:
      • Completion scores for assigned sections will ALWAYS be due by the Sunday of their assigned week at 11:59 PM (EST).
      • Review section IV of the syllabus above for all information concerning the final examination.

SUMMER 2026

Unit 1: Prerequisite Skills Review

  • Week 1: Jun 8–14
    • Lesson: Overlooked Algebra
    • Lesson: Trigonometric Trickery
    • Lesson: Fundamental Function Mechanics
    • Lesson: Reducing Expressions
    • Unit 1 Review
    • Submit completion scores

Unit 2: Limits, Continuity, and Discontinuity

  • Week 2: Jun 15–21
    • Lesson: Graphical Interpretation of Limits
    • Practice Day: Graphical Interpretation of Limits
    • Lesson: Contintuity
    • Lesson: Continuous Functions and Properties of Limits
    • Lesson: Jump Discontinuity and Limits of Piecewise Functions
    • Submit completion scores
  • Week 3: Jun 22–28
    • Lesson: Removable Discontinuity and Factoring Limits
    • Lesson: Removable Discontinuity and Rationalizing Limits
    • Lesson: Infinite Discontinuity and Asymptotes
    • Lesson: Oscillating Discontinuity and the Squeeze Theorem
    • Unit 2 Review
    • Submit completion scores

Unit 3: Derivative Theory and Applications

  • Week 4: Jun 29–Jul 5
    • Lesson: Average Rate of Change over an Interval and the Secant Line
    • Lesson: Instantaneous Rate of Change at a Point and the Tangent Line
    • Lesson: Differentiability and Derivative Notation
    • Lesson: Properties of Derivatives
    • Submit completion scores
  • Week 5: Jul 6–12
    • Lesson: Graphically Relating Functions and Their Derivatives
    • Lesson: Derivatives of Constants and Polynomials
    • Lesson: Derivatives of Exponentials and Logarithms
    • Lesson: Derivatives of Fundamental Trigonometric Functions
    • Practice Day: Derivatives Review
    • Submit completion scores
  • Week 6: Jul 13–19
    • Lesson: The Product Rule
    • Lesson: The Quotient Rule
    • Lesson: The Chain Rule
    • Practice Day: Combining Derivative Rules
    • Practice Day: Full Derivative Rules Review
    • Submit completion scores
  • Week 7: Jul 20–26
    • Lesson: Implicit Differentiation
    • Lesson: The First Derivative Test
    • Lesson: Optimization
    • Lesson: Related Rates
    • Unit 3 Review
    • Submit completion scores

Unit 4: Integrals and the Fundamental Theorem of Calculus

  • Week 8: Jul 27–Aug 2
    • Lesson: Fundamental Theorem of Calculus (Antiderivative Part)
    • Practice Day: Antiderivative Practice
    • Lesson: Properties of Indefinite Integrals
    • Lesson: Introduction to u-Substitution
    • Practice Day: u-Substitution
    • Submit completion scores
  • Week 9: Aug 3–9
    • Lesson: Fundamental Theorem of Calculus (Evaluation Part)
    • Practice Day: Area Under a Curve
    • Lesson: Average Value of a Function
    • Lesson: Net Area vs. Total Area
    • Submit completion scores
    • FINAL EXAMINATION

INSTRUCTOR QUALIFICATIONS

Caleb Scott Alons holds a B.S. in Mathematics, a B.A. in Biblical Literature, and is currently pursuing a Ph.D. in Mathematics. Caleb launched the AP Homeschoolers summer mathematics program four years ago and has been teaching with AP Homeschoolers ever since. After developing a love for teaching mathematics as a senior in high school, Caleb started professionally tutoring mathematics in 2020 and has since served hundreds of students in primary mathematics, Algebra I–II, Geometry, Precalculus/Trigonometry, AP Calculus AB/BC, undergraduate Calculus I–III, Differential Equations, Linear & Matrix Algebra, Discrete Mathematics, Abstract Algebra, Real Analysis, General Topology, and Algebraic Topology. Caleb also lectured undergraduate mathematics at Oral Roberts University for four years. In his undergrad, Caleb led as president the KME Honors Mathematical Society and Association of Computing Machinery in addition to being an active participant in the Mathematical Association of America, presenting award-winning research in MAA sectionals and the MAKO UG Research Conference. Caleb also scored three consecutive years in the William Lowell Putnam Mathematical Competition, which is esteemed as the hardest UG mathematical exam in the world.

As someone who formerly struggled with mathematics in high school himself, Caleb is deeply invested in seeing every student succeed no matter the circumstances and has personally tutored many students with learning disabilities to help them achieve 5's on AP Calculus AB/BC, score above the 90th percentile on the PSAT/SAT/ACT math tests, and rank in the top 5% of their university math courses. Ultimately, teaching mathematics is a source of great joy for Caleb, and he delights in serving each student under his instruction.

The teacher is the servant of his students. – CSA

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