I’ve recently finished Khan Academy’s AP Calculus AB. I started the course mid-March and finished it a few days ago, so it took roughly 3 months of following along. I’d like to offer as much of my insight as possible to the people considering this course and give an overview of how much work it actually is.
If you’ve just finished a precalculus course and want to know what awaits, give this a read-through to calibrate your expectations and a few pointers on concepts that might take some proactive research outside of course material.
Prerequisites
In order to start introductory calculus, you’ll need a working understanding of trigonometry. You probably won’t be using anything exotic, but knowing the right triangle ratios, the unit circle, sinusoidal equation, and the Pythagorean identity are the bare minimum. Khan Academy’s unit on limits involves taking limits of expressions with trig functions, and these will show up repeatedly as you move into derivatives/antiderivatives. The laws of sines/cosines aren’t strictly necessary, so you could feasibly skip that if you were in a hurry.
As for the algebraic side of things, you’ll probably do factor polynomials and rearrange equations often, but nothing requires learning it above a functional level. Calculus AB starts out decently light on computation and manipulations become more tedious towards the end, but nothing has required a really obscure, competitive math kind of strategy.
Limits
Limits are the first thing you encounter in a standard AB course, and they are appropriately straightforward. You learn the intuitive concept, the epsilon-delta definition, compose them with arithmetic, and work with unbounded graphs. This could feasibly be covered in a week.
This is taught to lead to the concept of a derivative, which only requires the basic understanding of limits (not arithmetic composition). Every section in this unit can be understood by watching a single video for each, and I’ve found that 2x speed is maintainable for every video in AB.
Derivatives
After you finish limits, you get the basic instantaneous velocity problem as a motivation for slope at a point. Khan Academy spends four units on differential calculus, but I agree with the pacing. You start with the basic product/quotient rules, then move to chain rule, and everything follows naturally after that. After the chain rule was taught, the rest of the derivative material consisted of elaborations and applications of it.
The course didn’t prove the rules (product/quotient/chain), but they were very memorizable. Clear patterns are observable, and you eventually internalize them after doing enough problems. In order to get 100% mastery in each unit, I had to go through a unit test at the end of each one. If I didn’t understand a concept, it was exposed quickly and I’d have to start over on the test to maximize my mastery points. Doing all four units is probably achievable within 1.5-2.5 weeks if you’re doing about 90 minutes of work every day.
Integration
Sal Khan did a great job with integration. After doing Riemann sums and learning the antiderivative, the Fundamental Theorem of Calculus was extremely intuitive to grasp. u-substitution as a reverse of the power rule and integration by parts as a reverse of the product rule were as expected. The only spot where something was left unclear was multiplying both sides by dx, despite it being clarified as “not a fraction” in the differential units. You’ll probably have to look outside your course material to understand it, but with enough problem volume, you can achieve accurate answers just by pattern recognition.
AB ends with an “Applications of Integration” unit, and it does a decent job of solidifying the introduction unit’s knowledge. The most notable content in this unit is rotating an area around a line, but everything is conceptually trivial (washer method especially). For two units, the introduction is mostly volume-heavy, and the application practice is sometimes tedious. With the same daily time investment of 90 minutes, I’d estimate 1-2 weeks for finishing.
Differential equations
Not sure if this is in the official AB curriculum, but Khan Academy includes this in between the integration units. It’s a very short unit, so it’s almost definitely a shallow look into the area. You look at ODEs, approximating with Euler’s method, solving separable equations, and logistic models for population. Possible to complete in 1-2 days.
Conclusion
“Calculus” has somewhat of an intimidating connotation in some conversations, but the AB curriculum is very beginner-friendly. No insights take long durations of mulling over videos to unlock, and I don’t envision anyone struggling over the Khan Academy problems. If you finished everything in the prerequisites section, finishing the course and fully understanding it within a few months of time is almost guaranteed. 2x speed on everything and watching roughly one video per distinct topic are practices I’ve found useful for expediting the process.
This does a good job of giving tools to work with, but if the goal is competing at an integration bee, practicing competition questions or solving the integral of the day is a good use of time. Overall, the AB course was predictable if not slightly underwhelming. That’s all for today, and until next time, I am out.