AP Calculus Prep for Summer 2025
Teacher: Caleb Alons
Email: calebscottalonsmathematics@gmail.com
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 2025 AP Calculus Prep syllabus).
TUITION: $320
WHY THIS COURSE?
Every Calculus course that I have tutored or lectured for has shared a common obstacle. Students struggle with Calculus for one or both of these reasons: (a) insufficient prerequisite ability in algebraic properties and mechanics and (b) the pacing of the course is too rapid to fully grasp each new concept. My heart in designing this course was to address these two problems head-on, so I wrote my own Calculus prep workbook that walks students through in a conversational writing style how they can leverage their existing understanding to understand Calculus concepts. By taking this course, the issue of insufficient prerequisite skill can be addressed preemptively before the student enters a full-year AP Calculus course. Additionally, the issue of pacing is addressed by nature of early exposure; early exposure to all fundamental concepts of AP Calculus over the summer will guarantee higher student comfortability in their full-year AP course.
Syllabus for
AP Calculus Prep
Summer 2025
I. COURSE DESCRIPTION
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. Built upon 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. 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.
Prerequisite(s): Students should have a firm grasp of Algebra 1, Algebra 2, Geometry, and Precalculus. Any Calculus teacher will testify that the majority of student struggle in Calculus comes not from the Calculus concepts themselves, but rather from a deficit in algebraic abilities or mastery of prerequisite concepts.
II. STUDENT LEARNING OUTCOMES FOR THIS COURSES
A. 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
B. 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
III. TEXTBOOKS AND OTHER LEARNING RESOURCES
A. REQUIRED MATERIALS
1. Textbooks:
a. Alons, Caleb S. Calculus: Prep for the AP Class. Unpublished manuscript, August 2022, typescript.
Access to the textbook is provided with the course; a PDF copy will be provided via the course website.
2. Other:
a. Access to the Internet to log onto the course website
IV. POLICIES AND PROCEDURES
A. 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:
a. Weekly Homework Assignments:
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
Self-grade work with the provided solution keys: 2 points
___________________________________________
Expected Section Total: 10 points
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). For Unit Reviews and Practice Days, there is no written lesson, so students should simply reward themselves automatically for the "read the textbook" points. 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.
b. Student Engagement:
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.
c. Final Examination:
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.
d. 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.
e. Grading Scale:
A: 486–540 points
B: 432–485 points
C: 378–431 points
D: 324–377 points
F: 0–323 points
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.
V. 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.A.2.d of the syllabus above for information concerning due dates for the final examination.
To see the full week-by-week course calendar, please send an email to calebscottalonsmathematics@gmail.com so that I can email you a PDF copy of the entire syllabus that includes all calendar information.
IMPORTANT DATES AT A GLANCE:
First Day of Class: Jun 9, 2025
Final Examination: Aug 8 (9-week course)
VI. INSTRUCTOR QUALIFICATIONS
Caleb Scott Alons will receive his B.S. in Mathematics in May 2025. Caleb is a returning AP Homeschoolers instructor, having previously taught two years of supplementary and preparatory summer courses in both mathematics and science. After developing a love for teaching mathematics, Caleb started professionally tutoring mathematics in 2020 and has been an assistant lecturer at Oral Roberts University for the past four years, teaching several hundred students in Calculus I/II/III, Vector Calculus, Differential Equations, Discrete Mathematics, Linear & Matrix Algebra, Abstract Algebra, Real Analysis, General (Point-Set) Topology, and Algebraic Topology. 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