AP Calculus BC — Series and BC-Only Topics flashcards

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The BC-only content of AP Calculus BC: integration by parts, partial fractions, improper integrals, Euler's method, logistic growth, arc length, parametric, vector and polar calculus, and every Unit 10 series test, Taylor series and error bound.

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Learn deck · part of AP Calculus BC · written against: College Board AP Calculus BC Course and Exam Description (BC-only topics and Unit 10 series). Updated 2026-10-10.

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What is the integration-by-parts formula, and how do you usually choose $u$ ⁠?

$\int u dv = uv - \int v du$ ⁠. Choose $u$ as the factor that gets simpler when differentiated; the LIATE order (log, inverse trig, algebraic, trig, exponential) ranks good choices.
Integration and Accumulation of Change › Integration by Parts and Partial Fractions
How do you evaluate $\int x e^x dx$ ⁠?

Let $u = x$ and $dv = e^x dx$ ⁠: $x e^x - e^x + C$ ⁠.
Integration and Accumulation of Change › Integration by Parts and Partial Fractions
What is the update rule in Euler's method with step size $h$ ⁠?

$y_{n+1} = y_n + h \cdot f\left(x_n, y_n\right)$ ⁠, where $f(x, y) = \frac{dy}{dx}$ ⁠: follow the tangent slope for one step, then recompute the slope.
Differential Equations › Euler's Method and Logistic Models
Use Euler's method with $h = 0.1$ and two steps to estimate $y(0.2)$ if $\frac{dy}{dx} = x + y$ and $y(0) = 1$ ⁠.

Step 1: slope 1, so $y(0.1) \approx 1.1$ ⁠. Step 2: slope $0.1 + 1.1 = 1.2$ ⁠, so $y(0.2) \approx 1.1 + 0.12 = 1.22$ ⁠.
Differential Equations › Euler's Method and Logistic Models
What is the arc length of $y = f(x)$ from $x = a$ to $x = b$ ⁠?

$\int_a^b \sqrt{1 + \left(f'(x)\right)^2} dx$ ⁠, which requires $f'$ to be continuous on $[a, b]$ ⁠.
Applications of Integration › Arc Length
What is the arc length of $y = \frac{2}{3}x^{3/2}$ from $x = 0$ to $x = 3$ ⁠?

$f'(x) = \sqrt{x}$ ⁠, so the length is $\int_0^3 \sqrt{1 + x} dx = \frac{2}{3}\left(8 - 1\right) =$ $\frac{14}{3}$ ⁠.
Applications of Integration › Arc Length
How do you find $\frac{dy}{dx}$ for a curve given by $x(t)$ and $y(t)$ ⁠?

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ ⁠, valid where $\frac{dx}{dt} \neq 0$ ⁠.
Parametric Polar and Vector-Valued Functions › Parametric Equations
How do you find $\frac{d^2y}{dx^2}$ for a parametric curve, and what is the trap?

Differentiate $\frac{dy}{dx}$ with respect to $t$ ⁠, then divide by $\frac{dx}{dt}$ ⁠.

Watch out: it is not $\frac{d^2y/dt^2}{d^2x/dt^2}$ ⁠.
Parametric Polar and Vector-Valued Functions › Parametric Equations
The sequence $a_n = \frac{n}{2n + 1}$ converges. Does the series $\sum a_n$ converge?

No. The sequence tends to $\frac{1}{2}$ ⁠, and because the terms do not approach 0, the series diverges by the nth-term test.
Infinite Sequences and Series › Sequences and Geometric Series
What does the nth-term test say, and what can it never prove?

If $\lim a_n \neq 0$ (or does not exist), $\sum a_n$ diverges. If $\lim a_n = 0$ the test is inconclusive; it can never prove convergence.
Infinite Sequences and Series › Sequences and Geometric Series

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Topic outline follows the BC-only topics of the College Board AP Calculus AB and BC Course and Exam Description (effective Fall 2020), Units 6–10 (the AP name is used nominatively; no course or exam text reproduced).
OpenStax Calculus Volumes 1 and 2, consulted as references; no text copied.

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