Single Variable Calculus
This course follows (with some exceptions) the curriculum of AP Calculus AB and BC, or Calculus 1 and Calculus 2 on the college level.
I say "with some exceptions" because, as I noted in the discussion of how this class is different from traditional teaching, I de-emphasize the "skills" portion of traditional calculus courses. (I don't eliminate all skills learning because some skills are necessary, but I really don't want students to get bogged down in the nasty algebra that comes with many traditional calculus skills.)
If you are looking to break this into two courses (the way that AP does or the way colleges do) I will endeavor to add, at some point, a "map" that shows you which investigations and projects are in the AB course and which are only in the BC course.
ONE LAST THING.... The documents on this website (and, I hope, the website itself), is 100% AI free, for better or worse. I wrote each of these investigations and projects by myself. I am notorious for making typos, spelling errors, omitting words, or just saying things poorly. I apologize in advance. If you find a mistake or something that could be phrased differently, please email me. I will appreciate that. I am always working on improving this material.
Course Syllabus
Module 0 - Three Key Concepts of Calculus
In this introductory module, we learn about three important, yet simple, concepts of calculus: local linearity, the derivative, and the integral.
Module 1 - Limits and Continuity
In this module we are introduced to limits (what they are and how to analyze them graphically, numerically, and algebraically). We then see how limits give us a tool to analyze continuity (a boring, yet necessary, aspect of calculus). Along the way, however, we are introduced to the important difference quotient and some cool functions and limits.
Module 2 - Limits and Vertical and Horizontal Asymptotes and Holes Revisited
In this module we use limits to define precisely vertical asymptotes, horizontal asymptotes, and holes. We then see some really cool things related to these, such as a way to examine the number pi and the number e.
In this module we use limits and the definition of the derivative to come up with a limit-based formula for the derivative of a function. We then use that limit-based formula to come up with various "derivative rules" for taking the derivatives of functions in general (because using the limit-based formula can be a real pain the ass sometimes). At the end of the module we start examining tangent lines.
Module 4 - The Derivative, A Clean Up Module
I thought it unwise to put every topic about derivatives in Module 3, so I broke off some investigations and put them here. In this module, we deal with higher order derivatives, the relationship between continuity and differentiability, the Mean Value Theorem of Differential Calculus, L'Hopital's Rule for dealing with difficult limits, and a fun investigation into Calculus and the movie "Mean Girls".
Module 5 - Derivatives of Implicit Curves and Other Oddities
In this module we start looking at derivatives of curves that are not necessarily functions. We look at derivatives of implicitly defined curves, derivatives of polar and parametric curves We also look at derivatives of numerically defined object. Along the way, we start seeing derivatives of arctrigonometric functions, the natural log function, and numerically (or data) defined functions.
Module 6 - Using Derivatives (Part 1) - Graphical Analysis
We now know what a derivative is and how to find (basic) derivatives. In the next few modules, we learn how to USE derivatives. In this module we use derivatives to do graphical/functional analysis, including determining when a function is increasing/decreasing, concave up/down, finding relative and absolute extrema. We also analyze implicit curves and paramentric curves.
Module 7 - Using Derivatives (Part 2) - Optimization
In math, optimization is the process of the "best" (smallest, biggest, shortest, longest, etc.) of some "thing" (i.e., a can, a distance, an area). We model mathematically the "thing" we in which we are interested, then apply the techniques of Module 6 to find the absolute extrema for which we are looking. The only additional twist is that because we are working with "real life" things, there are, usually, restrictions on the domain and outcomes we must reject because they don't fit the real world of the problem.
Module 8 - Using Derivatives (Part 3) - Rates of Change, Motion, and Related Rates
In this module we connect derivatives to rates of change and use that analyze motion in one and two dimensions. We also examine the topic of related rates.
Module 9 - Using Derivatives (Part 4) - Tangent Lines and Tangent Polynomials
In this module we explore local linearity in some depth with tangent lines and tangent line approximation. We then examine higher and higher degree Tangent Polynomials (also known as Taylor Polynomials). We then extend Taylor Polynomials to Taylor Series and prove the World's Most Beautiful Equation.
Module 10 - Integration Fundamentals
We move into the "second half" of calculus by learning what a definite integral is, how to evaluate simple definite integrals, and how to approximate more complex integrals. We end this module with the lovely, but practically unusable mathematical definition of the definite integral...which we will never use again, except to prove the upcoming "First Fundamental Theorem of Calculus."
Module 11 - Basic Integration Techniques
We "prove" the first fundamental Theorem of Calculus and use that to solve definite integrals and get a definition for indefinite integrals.
Module 12 - The Integral as an Accumulator
We have seen the integral as representing an area. Now we see the integral as an accumulation of amounts. We also learn about the Mean Value Theorem of Integral Calculus and how to find the average value of a function on a given interval.
Module 13 - Advanced Integration Techniques
Up till now we have relied on basic integration, u-substitution, and technology to evaluate integrals. There are other integration techniques that permit one to evaluate integrals by hand, if you so choose. (One reason for so choosing would be to understand what's going on behind the integral, as we'll see in this module.) This modules covers some of those advanced techniques.
We start using integrals by defining and then analying integral-defined functions.
Module 15 - Using Integrals (Part 2) - Motion Revisited
Two short investigations and one project in this module. First, we derive the equations for velocity and position for bodies in free-fall in one or two dimensions. Second we derive the integral expression for total distance of a body in free-fall in two dimensions.
Module 16 – Using Integrals (Part 3) = Areas, Volumes, and Arc Length
This module is a bit lengthy, but it is (in my view) full of interesting stuff. I just had to put all of that in here!
Module 17 - An Introduction to Differential Equations
A short module with only four investigations and no projects. This gives the briefest of introductions to the world of differential equatiosn.
Module 18 - Some Applications of Differential Equations
We see how differential equations explains Newton's Law of Cooling and Carbon-14 dating, and we apply exponential growth and decay functions to Covid-19 and populations.
Module 19 - Infinite (Numerical) Series
Our penultimate module. We take a step away from "traditional calculus" to examine infinite series of numbers. The goal is to learn how to determine whether an infinite series converges (to a number), or whether the infinite series diverges. While this stuff is fascinating in its own right, we'll use this in Module 20, when we determine the domains of Taylor Series (which we first saw in Module 9).
Module 20 - Taylor Series Revisited
Our last module. We take the work we did in Modules 9 and 19, put it all together and extend it. We find or derive Taylor Series for several very difficult functions and get some astounding results along the way. This is really some of the coolest math you will ever see.