Calculus

Calculus, at its core, considers and answers two questions:

• “How is it changing right now?”
• “How much has accumulated?”

Prior to calculus students are asked questions like “Given a car’s position at two different times, what was the car’s average speed between those times?” Derivative calculus asks questions like “What is the speed of the car at exactly 5 seconds from the moment it first began to accelerate given a position function in miles (d) as a function of time in seconds (t)?”

Prior to calculus students are asked questions like “During a sale, a business makes an average of profit per hour of $720. How much profit will it make in 5 hours?”

Integral calculus asks questions like, “What is the total profit (P) during the first 5 hours of a sale given a function H(t) that represents thousands of dollars of profit per hour as a function of time in hours (t) since the start of the sale?”

Calculus enables people to solve many real-world problems. Developing proficiency in calculus enables people to serve God, help people, and become employed in many rewarding fields of work.

This course prepares students to do well on an AP Calculus AB Exam and/or be deeply prepared to succeed in a college Calc 1 course.

Experience Calculus is built on a foundation of algebra, geometry, and trigonometry. Those skills will be relied on, built upon, and honed. The most frequent method is a Socratic-style discussion that the entire class participates in. A discussion takes place that reveals the relevance of this truth in our mathematical studies or our daily lives (real-life application). Lessons are presented through questions, problem-solving, and proof. The students have a puzzle or idea that they want to explore. As we delve deeper into answering the question at hand, we begin to discover the new mathematical concept. We discuss questions and problems in order to understand them and creatively and systematically problem-solve. We develop proofs in calculus as a way to construct understanding so that our application of theorems is not built on memorization alone but on deep conceptual understanding.

Methodology Students continue to study the new truth and deepen their understanding of it via homework, classwork, projects, and assessments. Since our classes do not meet 5 days per week, students must spend time studying the concepts outside of class. This also helps students grow in their ability to think and work math concepts independently which is required for future math classes. For this reason, we ask that students not use resources that answer problems for them. Doing so undercuts learning and will not lead to proficiency or the competency necessary to pass an AP Calculus AB exam. If a student receives an A on an assignment, it should be a true indication of their independent ability as the teacher interprets this A as independent mastery. 

Assignments or classroom review activities are utilized to assist students with opportunities to retain or improve their mastery. Assessments are directly or indirectly cumulative and serve as an opportunity to deepen their learning. Assessments are not meant to be a regurgitation of previous homework assignments.

Preparedness Students who are adequately prepared to take this course will have mastered (can work independently without prompting):

• Completed a comparable Pre-Calculus Course

Additionally, students who are adequately prepared to take this course will have the following commitments:

• Commitment to working diligently and persistently to learn
• Commitment to ask probing questions and communicate at the risk of needing correction

Quarter 1

1. The two central questions of calculus
2. Limits
3. Derivative Functions
4. The limit definition of derivative
5. Tangent lines
6. Continuity and other features of functions
7. Statement-reason proofs
8. Calculus proofs

• The order in which topics are presented may vary according to instructor and course section.

Quarter 2

1. Derivative rules
2. Theorems: squeeze theorem, mean value theorem, intermediate value theorem
3. Derivatives of inverse functions
4. Implicit differentiation
5. Second derivatives
6. Derivatives as they relate to position, velocity, acceleration, and jerk
7. Contextual problems requiring derivative calculus

• The order in which topics are presented may vary according to instructor and course section.

Quarter 3

1. More contextual problems requiring derivative calculus
2. The central question of integral calculus
3. Meanings and properties of integrals
4. Approximating integrals using Riemann sums
5. Integral as the limit of a Riemann sum
6. Fundamental theorem of calculus: how derivatives and integrals are related
7. Reverse power rule
8. More proofs

• The order in which topics are presented may vary according to instructor and course section.

Quarter 4

1. Common integrals (with proofs)
2. Methods of integration: u-substitution, using long division, completing the square, integration by parts
3. Differential equations and slope fields
4. Contextual problems requiring integral calculus: motion problems, the area between curves, volumes of solids including cross-sectional solids and solids formed by rotation

• The order in which topics are presented may vary according to instructor and course section.

• Computer: You will need a stable, reliable computer, running with a processor with a speed of 1 GHz or better on one of the following operating systems: Mac OS X with Mac OS 10.7 or later; Windows 8, 7, Vista (with SP1 or later), or XP (with SP3 or later). We do not recommend using an iPad or other tablet for joining classes. An inexpensive laptop or netbook would be much better solutions, as they enable you to plug an Ethernet cable directly into your computer. Please note that Chromebooks are allowed but not preferred, as they do not support certain features of the Zoom video conference software such as breakout sessions and annotation, which may be used by our teachers for class activities.
• High-Speed Internet Connection: You will also need access to high-speed Internet, preferably accessible via Ethernet cable right into your computer. Using Wi-Fi may work, but will not guarantee you the optimal use of your bandwidth. The faster your Internet, the better. We recommend using a connection with a download/upload speed of 5/1 Mbps or better. You can test your Internet connection here.
• Webcam: You may use an external webcam or one that is built in to the computer. Webcam Recommendations: Good (PC only) | Best (Mac and PC)
• Headset: We recommend using a headset rather than a built-in microphone and speakers. Using a headset reduces the level of background noise heard by the entire class. Headset Recommendations: USB | 3.5mm
• Zoom: We use a web conferencing software called Zoom for our classes, which enables students and teachers to gather from around the globe face to face in real time. Zoom is free to download and easy to use. To download Zoom:
  1. Visit zoom.us/download.
  2. Click to download the first option listed, Zoom Client for Meetings.
  3. Open and run the installer on your computer.
  4. In August, students will be provided with instructions and a link for joining their particular class.
• Scanner: In this class, students frequently submit homework assignments by scanning pages from their workbooks. Students and/or their parents should have easy access to a scanner and the ability to use it.