AP Calculus BC

What this covers

• Integration by parts (including repeated application and the tabular shortcut) and partial fraction decomposition for improper rational integrands
• Improper integrals: rewriting with limits, testing convergence, and connecting them to the integral test for series
• Parametric and vector-valued motion — dy/dx and d²y/dx² from dx/dt and dy/dt, speed, total distance travelled, and arc length
• Polar curves: sketching, area of a single region and area between two curves, and finding intersection points that the equations hide
• Euler's method and logistic differential equations, including reading the carrying capacity and the point of fastest growth
• Series work: geometric, p-series, nth-term, integral, comparison, limit comparison, ratio and alternating series tests; radius and interval of convergence; Taylor and Maclaurin polynomials; alternating series error bound and Lagrange error bound

Where learners get stuck

Treating the interval of convergence as finished once the ratio test gives an inequality, without checking the two endpoints separately

The ratio test is inconclusive when the limit equals 1, which is exactly what happens at endpoints. Students learn the test as a single procedure and never register that it hands the problem back to them at the boundary, so they lose the point for endpoint behaviour almost every time.

Writing arc length or speed for parametric curves as if x were the independent variable

AB habits carry over. Learners reach for √(1+(dy/dx)²) instead of √((dx/dt)²+(dy/dt)²), because they have never had to think about a curve traced in time rather than graphed over an interval.

Confusing the alternating series error bound with the Lagrange error bound, and confusing both with the value of the remainder

Both produce a number that bounds an error, and both appear in the same unit. Students memorise 'next term' and 'max derivative over factorial' as interchangeable, then apply the alternating bound to a series that is not alternating, or forget that Lagrange requires a bound on the (n+1)th derivative on a specific interval.

Setting two polar equations equal to each other and assuming that finds every intersection

The same point has many (r, θ) representations, and the pole can lie on both curves at different θ values. The algebra feels complete, so nothing signals that a graph is still required.

What a session looks like

A session usually opens with one problem you attempt aloud while Evelyn listens for where the reasoning stalls — often a missing justification rather than a missing computation. From there you work through two or three related problems, with the tutor asking you to name the convergence test and state why its conditions hold before you use it, or to set up a polar area integral and defend the limits before integrating. Free-response practice includes saying the justification sentence out loud, since BC readers award points for stated reasoning. You can bring in homework, a returned test, or ask for drilling on a single unit such as series or parametrics.

Helpful to know first

• Comfort with the AB content: limits, derivative rules including chain and implicit, the Fundamental Theorem of Calculus, u-substitution, related rates and optimisation
• Separable differential equations and slope fields
• Solid trigonometry — unit circle values, identities such as sin²θ = (1 − cos2θ)/2 for polar area, and inverse trig derivatives
• Sequences, factorials, and sigma notation from precalculus
• Ability to use a graphing calculator for numerical integration, derivative at a point, and graphing in parametric and polar modes

Questions

What is on BC that is not on AB?

Integration by parts, partial fractions, improper integrals, arc length, logistic differential equations, Euler's method, parametric and vector-valued functions, polar coordinates, and the entire infinite series unit. Roughly 40% of the BC exam is material AB never touches, and series is the largest single piece of it.

My child is strong in AB but lost in series — can you focus only on that?

Yes. Sessions can stay on sequences and series for as long as needed: choosing a convergence test and justifying it, radius and interval of convergence with endpoint checks, building Maclaurin series from the known ones for eˣ, sin x, cos x and 1/(1−x), and error bounds.

Does this help with the AB subscore?

The BC exam reports an AB subscore drawn from the shared content, so work on limits, derivatives, integration and applications feeds directly into it. Sessions can be split between BC-only topics and shoring up AB material if the subscore is the concern.

How much of a session is calculator work?

As much as you want. BC has calculator and no-calculator sections, so sessions can practise both — deciding when a numerical integral is acceptable, how to report a decimal answer to three places, and how to do the same problem by hand when the calculator is not allowed.