AP Calculus AB & BC
For most students, this is the first math class where being good at following the steps stops being enough.
Limits and derivatives don’t behave like anything your student has likely seen before, and the pace doesn’t slow down for anybody. Sometimes the calculus really is the problem, and we work on the calculus. Sometimes the calculus is fine and something underneath it is causing the problem, like a weak geometry foundation, a factoring habit built on a misconception, or a trig identity nobody ever actually taught them. I don’t assume which one it is. I find out, and then we fix that.
Where strong math students hit the wall
A student who hits a wall in AP Calculus usually got here by being good at math. It’s usually one of a handful of specific things — and since each one has a different fix, the first thing we do is figure out which it is:
- The math becomes more conceptual: Limits, derivatives, and integrals involve a different kind of thinking than algebra. Students who got by on pattern-matching and tricks/shortcuts tend to hit a wall here.
- The course keeps building on itself: Limits lead into derivatives, derivatives into applications, and derivatives and accumulation into integrals. When an early idea never fully settles, later topics do not replace it — they pile on top of it.
- The relentless pace: AP Calculus synthesizes three years of prior math while introducing completely foreign mechanics over just seven months. The course marches toward a fixed May exam date, leaving little room to pause and let these new ideas fully digest.
- The algebra and trig underneath: Often, the actual calculus step, like taking the derivative or setting up the integral, is completely correct. But because these problems take multiple steps to solve, weak factoring, messy fraction management, or forgotten unit circle trig can turn great calculus into wrong answers.
- The illusion of mastery: A student can often follow along perfectly when a teacher evaluates a complex limit or sets up an optimization problem on the board, only to freeze when they are on their own. Watching someone else do calculus feels logical, but generating that initial setup independently is a “haven’t truly owned it yet” problem, and it is very solvable.
- Rigid FRQ requirements: The free-response section demands strict notational precision and specific theorem justifications (like the Mean Value Theorem). Capable kids lose easy points for omitting a dx or improperly phrasing a conclusion, which has nothing to do with their actual understanding of the math.
- The compounding speed of BC: BC covers everything AB does and then just keeps going. If a foundational topic like integration techniques never quite sets, the next unit builds directly on top of it anyway.
The entire course, AB through BC
The College Board organizes AP Calculus AB into eight units. BC covers all eight and adds two more.
Limits & Continuity
What it means for a function to behave smoothly, and how we handle a value a function only approaches
Differentiation: Definition & Fundamental Properties
What a derivative actually is — an instantaneous rate of change — plus the core rules: power, product, and quotient
Differentiation: Composite, Implicit & Inverse Functions
The chain rule in depth, implicit differentiation, and the derivatives of inverse functions
Contextual Applications of Differentiation
Motion, related rates, and linear approximation — derivatives doing real work
Analytical Applications of Differentiation
Reading a graph through its derivatives: increasing/decreasing, concavity, and optimization
Integration & Accumulation of Change
Riemann sums, the definite integral, and the Fundamental Theorem that ties integration and differentiation together
Differential Equations
Slope fields, modeling growth and decay, and solving separable equations
Applications of Integration
Area between curves, volumes of revolution, average value, and net change over time
BC adds two units
Parametric, Polar & Vector-Valued Functions
Curves defined by a parameter, polar coordinates, and vector-valued functions — and doing calculus on all three
Infinite Sequences & Series
Convergence tests, plus Taylor and Maclaurin series — approximating a function as an infinite polynomial
Taking AP Precalculus first? That’s one of the three courses I teach — and since AP Calculus is built directly on top of it, getting precalculus solid now is critical for success in calculus.
Calculus learned as a tool, taught by someone who used it
I didn’t learn calculus in a math class and stop there. I learned it in physics and then in civil engineering, where a derivative isn’t a box on a worksheet — it’s the load on a beam, and getting it wrong is how a bridge comes down.
- ~20 years in education, and a former Department Chair of Mathematics & Computer Science.
- Degrees in physics and civil engineering — I learned calculus as a tool for solving physics and engineering problems, not just as pure math. So “when would I ever use this?” gets an actual example instead of a shrug.
- Trained to diagnose misconceptions, assess true understanding, and design the instruction that fixes both — not merely solve the problem in front of us.
Dr. Hurley is a fantastic teacher that knew how to inspire his students and get them inspired to participate in normally difficult classes like computer science and calculus. … I wish I could have him again at Harvard!
IvorCalculus & Computer ScienceHarvard · now at Georgetown Law