Your FREE AP Calculus BC Flashcards 2026 – 200+ Cards

AP Calculus BC Flashcards

Derivative of sin x

$\cos x$

Derivative of cos x

$-\sin x$

Derivative of eˣ

$e^x$

Derivative of ln x

$\dfrac{1}{x}$

Power rule

$\dfrac{d}{dx} x^n = n x^{n-1}$

Geometric series sum

$\dfrac{a}{1 - r}$ when $|r| < 1$

Ratio Test

Converges if $\lim |a_{n+1}/a_n| < 1$; diverges if > 1; inconclusive if = 1.

Polar area

$\dfrac{1}{2}\displaystyle\int_\alpha^\beta [r(\theta)]^2\, d\theta$

Arc length (y = f(x))

$\displaystyle\int_a^b \sqrt{1 + (f'(x))^2}\, dx$

Quadratic formula

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Limit (definition)

The value a function approaches as its input approaches a number; it can exist where the function is undefined.

Two-sided limit exists when

The left-hand limit and right-hand limit are equal.

Continuous at x = c

The limit at c exists, $f(c)$ is defined, and $\lim_{x\to c} f(x) = f(c)$.

Removable discontinuity

A hole in the graph: the limit exists but does not equal the function value (or the value is missing).

Jump discontinuity

The left and right limits exist but are not equal.

Infinite discontinuity

The function approaches ±∞ near the point (a vertical asymptote).

Intermediate Value Theorem

If f is continuous on $[a,b]$, it takes every value between $f(a)$ and $f(b)$.

Squeeze (Sandwich) Theorem

If $g(x) \le f(x) \le h(x)$ and $g, h$ share a limit L at c, then $f \to L$ too.

Limit of sin(x)/x as x→0

$1$

Limit of (1 − cos x)/x as x→0

$0$

Rational function limit as x→∞

Compare degrees: same → ratio of leading coefficients; bigger top → ±∞; bigger bottom → 0.

Indeterminate form 0/0

Resolve by factoring, conjugates, or L'Hospital's Rule.

Vertical asymptote

Where the denominator is 0 (and the numerator isn't), giving an infinite limit.

Horizontal asymptote

The value the function approaches as x → ±∞ (an end-behavior limit).

Derivative (limit definition)

$f'(a) = \lim_{h\to 0} \dfrac{f(a+h) - f(a)}{h}$

Derivative as alt definition

$f'(a) = \lim_{x\to a} \dfrac{f(x) - f(a)}{x - a}$

Derivative meaning

Instantaneous rate of change; the slope of the tangent line.

Differentiability implies

Continuity. (But continuity does NOT imply differentiability.)

Where f is not differentiable

At corners, cusps, vertical tangents, and discontinuities.

Product rule

$(fg)' = f'g + fg'$

Quotient rule

$\left(\dfrac{f}{g}\right)' = \dfrac{f'g - fg'}{g^2}$

Derivative of tan x

$\sec^2 x$

Derivative of sec x

$\sec x \tan x$

Derivative of cot x

$-\csc^2 x$

Derivative of csc x

$-\csc x \cot x$

Derivative of aˣ

$a^x \ln a$

Derivative of a constant

$0$

Derivative of x

$1$

Constant multiple rule

$(c f)' = c f'$

Sum/difference rule

$(f \pm g)' = f' \pm g'$

Chain rule

$\dfrac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

Chain rule example

$\dfrac{d}{dx}\sin(5x^3) = \cos(5x^3)\cdot 15x^2$

Implicit differentiation

Differentiate both sides, attach dy/dx to every y-term, then solve for dy/dx.

Implicit example: x² + y² = 25

$\dfrac{dy}{dx} = -\dfrac{x}{y}$

Derivative of arcsin x

$\dfrac{1}{\sqrt{1 - x^2}}$

Derivative of arccos x

$-\dfrac{1}{\sqrt{1 - x^2}}$

Derivative of arctan x

$\dfrac{1}{1 + x^2}$

Derivative of an inverse function

$(f^{-1})'(b) = \dfrac{1}{f'(a)}$ where $f(a) = b$.

Logarithmic differentiation

Take ln of both sides, then differentiate implicitly — useful for variable exponents like xˣ.

Derivative of ln|x|

$\dfrac{1}{x}$

Derivative of xˣ

$x^x(\ln x + 1)$ (by logarithmic differentiation).

Velocity

$v(t) = s'(t)$, the derivative of position.

Acceleration

$a(t) = v'(t) = s''(t)$.

Speed

$|v(t)|$, the absolute value of velocity.

Particle speeding up

Velocity and acceleration have the SAME sign.

Particle slowing down

Velocity and acceleration have OPPOSITE signs.

Related rates method

Differentiate the linking equation with respect to time t, then substitute known values and rates.

L'Hospital's Rule

For $0/0$ or $\infty/\infty$, $\lim \dfrac{f}{g} = \lim \dfrac{f'}{g'}$.

Linearization (tangent-line approx)

$L(x) = f(a) + f'(a)(x - a)$.

Differential dy

$dy = f'(x)\, dx$.

Critical point

Where f'(x) = 0 or is undefined — a candidate for a relative extremum.

f increasing when

$f'(x) > 0$.

f decreasing when

$f'(x) < 0$.

First-derivative test

f' changes − to + → relative minimum; + to − → relative maximum.

Concave up when

$f''(x) > 0$.

Concave down when

$f''(x) < 0$.

Inflection point

Where concavity changes — f'' changes sign.

Second-derivative test

At a critical point: f'' > 0 → minimum; f'' < 0 → maximum; f'' = 0 → inconclusive.

Mean Value Theorem

$f'(c) = \dfrac{f(b) - f(a)}{b - a}$ for some c in (a, b).

Rolle's Theorem

If $f(a) = f(b)$, there is a c in (a, b) with $f'(c) = 0$.

Extreme Value Theorem

A continuous function on a closed interval attains an absolute max and min.

Absolute extrema on [a, b]

Compare f at the critical points and at the two endpoints.

Optimization steps

Write the quantity, reduce to one variable, set the derivative to 0, then verify the extremum.

Antiderivative

A function whose derivative is the integrand; the indefinite integral, written + C.

Definite integral meaning

Net accumulated change — the signed area under a rate curve.

FTC Part 1 (evaluation)

$\displaystyle\int_a^b f(x)\, dx = F(b) - F(a)$.

FTC Part 2 (derivative of integral)

$\dfrac{d}{dx}\displaystyle\int_a^x f(t)\, dt = f(x)$.

FTC Part 2 with chain rule

$\dfrac{d}{dx}\displaystyle\int_a^{g(x)} f(t)\, dt = f(g(x))\cdot g'(x)$.

Power rule for integrals

$\displaystyle\int x^n\, dx = \dfrac{x^{n+1}}{n+1} + C,\ n \ne -1$.

Integral of 1/x

$\ln|x| + C$.

Integral of eˣ

$e^x + C$.

Integral of sin x

$-\cos x + C$.

Integral of cos x

$\sin x + C$.

Integral of sec² x

$\tan x + C$.

u-substitution

Reverses the chain rule: substitute u for an inner function so the integral becomes a standard form.

Integration by parts (BC)

$\displaystyle\int u\, dv = uv - \int v\, du$.

LIATE rule (BC)

Order for choosing u in by-parts: Logs, Inverse trig, Algebraic, Trig, Exponential.

Partial fractions (BC)

Split a rational function into simpler fractions that can be integrated individually.

Improper integral (BC)

An integral with an infinite limit or unbounded integrand; evaluated as a limit.

Left Riemann sum (increasing f)

Underestimates the integral.

Right Riemann sum (increasing f)

Overestimates the integral.

Trapezoidal rule

Averages the left and right sums; accurate for nearly linear data.

Midpoint Riemann sum

Uses the height at the midpoint of each subinterval.

Accumulation function

$g(x) = \displaystyle\int_a^x f(t)\, dt$ gives net change of f's antiderivative from a to x.

Net change theorem

$\displaystyle\int_a^b f'(x)\, dx = f(b) - f(a)$ — integrating a rate gives total change.

Average value of f on [a, b]

$\dfrac{1}{b - a}\displaystyle\int_a^b f(x)\, dx$.

Integral of 1/(1 + x²)

$\arctan x + C$.

Differential equation

An equation involving a derivative; a solution is a function satisfying it.

Slope field

A grid of short segments showing the slope dy/dx of solutions at many points.

Separable equation method

Separate variables (y on one side, x on the other), integrate both sides, apply the initial condition.

General vs particular solution

General has + C; a particular solution uses an initial condition to find C.

Euler's method (BC)

$y_{\text{new}} = y_{\text{old}} + \dfrac{dy}{dx}\cdot \Delta x$.

Euler's method idea (BC)

A repeated tangent-line step; smaller step size Δx gives better accuracy.

Exponential growth/decay model

$\dfrac{dy}{dt} = ky \Rightarrow y = y_0 e^{kt}$.

Logistic model (BC)

$\dfrac{dP}{dt} = kP\left(1 - \dfrac{P}{M}\right)$, with carrying capacity M.

Logistic fastest growth (BC)

At $P = M/2$, half the carrying capacity.

Logistic limit (BC)

$P \to M$ (the carrying capacity) as $t \to \infty$.

Newton's law of cooling

$\dfrac{dT}{dt} = k(T - T_s)$; T approaches the surrounding temperature $T_s$.

Area between two curves

$\displaystyle\int_a^b (\text{top} - \text{bottom})\, dx$.

Disk method (volume)

$\pi\displaystyle\int_a^b [R(x)]^2\, dx$.

Washer method (volume)

$\pi\displaystyle\int_a^b \big([R(x)]^2 - [r(x)]^2\big)\, dx$.

Shell method (volume)

$2\pi\displaystyle\int_a^b (\text{radius})(\text{height})\, dx$.

Volume by known cross-sections

$\displaystyle\int_a^b A(x)\, dx$, the integral of the cross-sectional area.

Average value of a function

$\dfrac{1}{b - a}\displaystyle\int_a^b f(x)\, dx$.

Arc length of y = f(x) (BC)

$\displaystyle\int_a^b \sqrt{1 + (f'(x))^2}\, dx$.

Total distance traveled

$\displaystyle\int_a^b |v(t)|\, dt$ (integrate speed, not velocity).

Displacement

$\displaystyle\int_a^b v(t)\, dt$ (integrate velocity).

Accumulated amount from a rate

Integrate the rate over the interval to get the total accumulated quantity.

Parametric slope (BC)

$\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$.

Parametric second derivative (BC)

$\dfrac{d^2 y}{dx^2} = \dfrac{d/dt(dy/dx)}{dx/dt}$.

Parametric speed (BC)

$\sqrt{\left(\dfrac{dx}{dt}\right)^2 + \left(\dfrac{dy}{dt}\right)^2}$.

Parametric arc length (BC)

$\displaystyle\int_a^b \sqrt{(dx/dt)^2 + (dy/dt)^2}\, dt$.

Vector-valued function (BC)

Position $(x(t), y(t))$; velocity $(x'(t), y'(t))$; acceleration $(x''(t), y''(t))$.

Polar to rectangular (BC)

$x = r\cos\theta,\ y = r\sin\theta$.

Rectangular to polar (BC)

$r^2 = x^2 + y^2,\ \tan\theta = y/x$.

Polar area (BC)

$\dfrac{1}{2}\displaystyle\int_\alpha^\beta [r(\theta)]^2\, d\theta$.

Area between polar curves (BC)

$\dfrac{1}{2}\displaystyle\int_\alpha^\beta (r_{\text{out}}^2 - r_{\text{in}}^2)\, d\theta$.

Polar slope dy/dx (BC)

Convert to $x = r\cos\theta,\ y = r\sin\theta$, then use the parametric slope in θ.

Cardioid (BC)

A polar curve like $r = 1 + \cos\theta$ — a heart shape symmetric about the x-axis.

Circle r = 2cos θ (BC)

A circle of radius 1 centered at (1, 0).

Velocity vector magnitude (BC)

$|v(t)| = \sqrt{(x'(t))^2 + (y'(t))^2}$ — the particle's speed.

Position from velocity vector (BC)

Integrate each component of the velocity vector and add the initial position.

Sequence converges when

Its terms approach a single finite limit as n → ∞.

Series converges when

Its sequence of partial sums approaches a finite limit.

Geometric series (BC)

$\sum ar^n$ converges to $\dfrac{a}{1 - r}$ iff $|r| < 1$.

p-series (BC)

$\sum \dfrac{1}{n^p}$ converges iff $p > 1$.

Harmonic series (BC)

$\sum \dfrac{1}{n}$ — diverges (the p = 1 case), even though terms → 0.

nth-Term (Divergence) Test (BC)

If $a_n \not\to 0$, the series diverges. (Can't prove convergence.)

Ratio Test (BC)

$L = \lim |a_{n+1}/a_n|$: L < 1 converges, L > 1 diverges, L = 1 inconclusive.

Root Test (BC)

$L = \lim \sqrt[n]{|a_n|}$: L < 1 converges, L > 1 diverges, L = 1 inconclusive.

Integral Test (BC)

If f is positive, decreasing, continuous, $\sum a_n$ and $\int f$ converge or diverge together.

Comparison Test (BC)

Compare a positive series term-by-term to a known convergent or divergent series.

Limit Comparison Test (BC)

If $\lim a_n/b_n$ is a positive finite number, both series share the same fate.

Alternating Series Test (BC)

If terms alternate sign, decrease in size, and approach 0, the series converges.

Absolute convergence (BC)

$\sum |a_n|$ converges; this implies $\sum a_n$ converges.

Conditional convergence (BC)

$\sum a_n$ converges but $\sum |a_n|$ diverges (e.g. alternating harmonic).

Alternating series error bound (BC)

The error is at most the absolute value of the first omitted term.

Power series (BC)

$\sum c_n (x - a)^n$, centered at a.

Radius of convergence (BC)

Distance from the center within which a power series converges; found via the Ratio Test.

Interval of convergence (BC)

The set of x for which the power series converges — test endpoints separately.

Taylor series (BC)

$\sum_{n=0}^{\infty} \dfrac{f^{(n)}(a)}{n!}(x - a)^n$.

Maclaurin series (BC)

A Taylor series centered at a = 0.

Maclaurin series for eˣ (BC)

$\sum_{n=0}^{\infty} \dfrac{x^n}{n!} = 1 + x + \dfrac{x^2}{2!} + \cdots$

Maclaurin series for sin x (BC)

$x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \cdots$

Maclaurin series for cos x (BC)

$1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \cdots$

Maclaurin series for 1/(1 − x) (BC)

$\sum_{n=0}^{\infty} x^n = 1 + x + x^2 + \cdots,\ |x| < 1$

Taylor polynomial (BC)

A finite truncation of a Taylor series that approximates f near the center.

Lagrange error bound (BC)

Bounds how far a Taylor polynomial can be from the true value of f.

Telescoping series (BC)

A series whose partial sums collapse because consecutive terms cancel.

0.444... as a series (BC)

Geometric with a = 4/10, r = 1/10 → $\dfrac{4/10}{1 - 1/10} = \dfrac{4}{9}$.

Building new series (BC)

Differentiate, integrate, or substitute into a known Maclaurin series to get a new one.

Convergence test order (BC)

Check nth-Term first, recognize geometric/p-series, then Ratio, Comparison, Alternating.

Tangent line equation

$y - f(a) = f'(a)(x - a)$.

Normal line slope

The negative reciprocal of the tangent slope, $-1/f'(a)$.

Second derivative meaning

Rate of change of the rate — concavity (and acceleration for motion).

Global vs local extremum

A local (relative) extremum is best nearby; a global (absolute) one is best on the whole domain.

Definite integral of an odd function over [−a, a]

$0$ (by symmetry).

Definite integral with equal limits

$\displaystyle\int_a^a f(x)\, dx = 0$.

Reversing integral limits

$\displaystyle\int_a^b f = -\displaystyle\int_b^a f$.

Riemann sum (definition)

Approximates a definite integral by summing areas of rectangles.

Integral of a constant k

$kx + C$.

Fundamental period of sin/cos

$2\pi$.

Continuity of polynomials

Polynomials are continuous and differentiable everywhere.

Tangent vs secant line

A tangent touches at one point (instantaneous slope); a secant connects two (average slope).

Implicit second derivative

Differentiate dy/dx implicitly again, substituting the first derivative where it appears.

When to use related rates

When several quantities change with time and are tied by one equation.

Average rate of change

$\dfrac{f(b) - f(a)}{b - a}$ — the slope of the secant line.

Instantaneous rate of change

The derivative at a point — the limit of average rates as the interval shrinks.

Solving an initial value problem

Integrate to a general solution, then use the initial condition to find C.

Polar symmetry test (BC)

Replace θ with −θ (x-axis), π − θ (y-axis), or θ + π (origin) and check if the equation is unchanged.

Parametric to rectangular (BC)

Eliminate the parameter t by solving one equation for t and substituting.

Why test power-series endpoints (BC)

The Ratio Test gives L = 1 at the endpoints, so it is inconclusive there.

Maclaurin series for arctan x (BC)

$x - \dfrac{x^3}{3} + \dfrac{x^5}{5} - \cdots,\ |x| \le 1$.

Maclaurin series for ln(1 + x) (BC)

$x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \cdots,\ -1 < x \le 1$.

Partial sum

The sum of the first n terms of a series; its limit (if finite) is the series sum.

Bounded monotonic sequence (BC)

A sequence that is bounded and monotonic converges.

nth term of a geometric sequence

$a_n = a_1 r^{n-1}$.

Why nth-Term Test can't prove convergence (BC)

Terms going to 0 is necessary but not sufficient — e.g. the harmonic series.

Comparison with a p-series (BC)

Compare a messy positive series to 1/nᵖ to decide convergence quickly.

Factorial growth in the Ratio Test (BC)

Factorials and nth powers make the Ratio Test the natural first choice.

Centering a Taylor series (BC)

The center a is where you take the derivatives; choose it near the x-values you'll evaluate.

Improper integral convergence (BC)

Converges if the defining limit is a finite number; otherwise it diverges.

Average value vs Mean Value Theorem for Integrals

There is a c with $f(c)$ equal to the average value of f on [a, b].

Position, velocity, acceleration chain

Differentiate position → velocity → acceleration; integrate to reverse.

Endpoint extrema

Absolute extrema can occur at the endpoints of a closed interval, not just critical points.

Why BC includes an AB subscore

BC contains all AB content, so AB-content questions yield a separate AB subscore.

Definite integral of a rate gives

The net change in the quantity over the interval (the Net Change Theorem).

Parametric particle at rest (BC)

Occurs when both dx/dt and dy/dt are 0 at the same t.

Maclaurin series radius for 1/(1 − x) (BC)

Radius 1; converges for $|x| < 1$.

Alternating harmonic series (BC)

$\sum (-1)^{n+1}/n$ converges (conditionally) to $\ln 2$.