Hopefully the short and sweet is best. Here's what I want out of a first semester class regardless of background. I think depending on the individual student they will be ready to engage with these ideas at different levels of rigor and complexity.
- Calculus I: Concept of Differentiation and Integration, connection through simple ODE's. Students should be able to use these ideas to solve real problems, and constantly be moving back and forth between approximate and exact solutions, discussing accuracy of models and solutions.
Next, depending on the context the first is a more traditional approach(service class) or an alternative more focused on modeling with calculus.
Calculus II: Representing functions with series and doing calculus on these. Basically, a complexifying ideas from first semester through tasks that necessitate representing functions with Taylor series. Light connections to work in ODE and PDE problems to motivate what would likely be a class or two in Diff Eq's. Finally, students should experience some work with vectors and higher dimensional problems.
Calculus II(alternative): Introduce Taylor polynomials and series through solving differential equations problems, spend half the course here focusing on both solving problems and dealing with the rigor of series conversion. Spend the second half focused on vector calculus and solving some good problems in 3D. I prefer this, and would love if students had an introductory understanding of solving physical problems so important to the emergence of calculus, particularly something of Newton's laws and their use in modeling things like a wave.
In any of these I would want a student to use a computer well to help solve the problems. Python is the best from my perspective in terms of doing calculus with students.
This means there are only two concepts I suppose: Integration and Differentiation. These concepts are related through the FTC, which describes their inverse character. Students need to understand this first.
Second, we can't differentiate or integrate most functions that we would want to. As such, we need approximation techniques which come in the form of Taylor series.
Third, all of these problems above can occur in dimensions higher than 2, and students should understand some of the operational complexity that comes when moving to a higher dimension with the derivative and integral, but also recognize their utility in situations much like those from first semester.