AP Calculus BC Last-Minute Review (Night Before)

The exam is tomorrow. Do not open new material. Spend 30-45 minutes skimming this page, then close your notes and sleep.

AB essentials: derivatives

Function	Derivative
$x^n$	$n x^{n-1}$
$e^x$	$e^x$
$\ln x$	$1/x$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$
$\sec x$	$\sec x \tan x$
$\arcsin x$	$\frac{1}{\sqrt{1-x^2}}$
$\arctan x$	$\frac{1}{1+x^2}$
$a^x$	$a^x \ln a$

Chain rule: $(f(g(x)))' = f'(g(x)) \cdot g'(x)$ . Product rule: $(uv)' = u'v + uv'$ . Quotient rule: $(u/v)' = \frac{u'v - uv'}{v^2}$ .

AB essentials: integrals

Function	Antiderivative
$x^n$ ( $n \neq -1$ )	$\frac{x^{n+1}}{n+1} + C$
$1/x$	$\ln \\|x\\| + C$
$e^x$	$e^x + C$
$\sin x$	$-\cos x + C$
$\cos x$	$\sin x + C$
$\sec^2 x$	$\tan x + C$
$\sec x \tan x$	$\sec x + C$
$1/(1+x^2)$	$\arctan x + C$
$1/\sqrt{1-x^2}$	$\arcsin x + C$

BC essentials: parametric & polar

Concept	Formula
Parametric $\frac{dy}{dx}$	$\frac{dy/dt}{dx/dt}$
Parametric speed	$\sqrt{(dx/dt)^2 + (dy/dt)^2}$
Parametric arc length	$\int_a^b \sqrt{(dx/dt)^2 + (dy/dt)^2} \, dt$
Polar area	$A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 \, d\theta$
Polar area (between curves)	$\frac{1}{2} \int (r_{\text{out}}^2 - r_{\text{in}}^2) \, d\theta$

BC essentials: Maclaurin series (MEMORIZE)

Series	Interval
$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$	All $x$
$\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$	All $x$
$\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$	All $x$
$\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n$	$\\|x\\| < 1$
$\ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n}$	$-1 < x \leq 1$

BC essentials: convergence tests

When you see $\sum a_n$ , ask:

Test	When to use	Condition	Result
Geometric	Form is $\sum ar^n$	$\\|r\\| < 1$	Converges to $\frac{a}{1-r}$
$p$ -series	Form is $\sum \frac{1}{n^p}$	$p > 1$	Converges; $p \leq 1$ diverges
Ratio test	Factorials, exponentials, powers	$L = \lim \left\\|\frac{a_{n+1}}{a_n}\right\\|$	$L < 1$ ⟹ converge; $L > 1$ ⟹ diverge
Alternating	$\sum (-1)^n a_n$ , $a_n$ decreasing	$\lim a_n = 0$	Converges (but test convergence of $\\|a_n\\|$ separately)
Integral test	$a_n = f(n)$ , $f$ decreasing	$\int_1^{\infty} f(x) \, dx$ converges	Series converges
Comparison	Compare to known series	$0 \leq a_n \leq b_n$	If $\sum b_n$ converges, so does $\sum a_n$

💡 IOC Strategy: Use ratio test to find radius; then test endpoints separately (plug in, use alternating or $p$ -test).

Top 12 exam traps

Polar area: Write $\frac{1}{2} \int r^2 \, d\theta$ , not $\int r^2 \, d\theta$ .
Parametric speed: $\sqrt{(dx/dt)^2 + (dy/dt)^2}$ , never $(dx/dt + dy/dt)$ or $|(dx/dt) + (dy/dt)|$ .
Parametric $dy/dx$ : $\frac{dy/dt}{dx/dt}$ , not $\frac{dy}{dx}$ (which is what you're trying to find).
IOC endpoints: Test them separately. Endpoint convergence can differ from interior convergence.
Ratio test $L = 1$ : Inconclusive — you must use another test.
Lagrange error: Find $M = \max |f^{(n+1)}|$ on the interval; don't guess a bound.
Chain rule with FTC: $\frac{d}{dx} \int_a^{g(x)} f(t) \, dt = f(g(x)) \cdot g'(x)$ — don't drop $g'(x)$ .
Implicit differentiation: Every $y$ becomes $\frac{dy}{dx}$ via chain rule; don't forget this step.
Distance vs displacement: Distance uses $|v(t)|$ ; displacement is $\int v(t) \, dt$ (can be negative).
MVT hypotheses: Continuous on $[a,b]$ AND differentiable on $(a,b)$ — must state both.
Integration by parts: Use LIATE priority when choosing $u$ . Often $u = \text{polynomial}$ or $u = \ln x$ .
Series notation: Write limits and indices clearly; e.g., $\sum_{n=1}^{\infty}$ , not just $\sum$ .

Integration by parts: LIATE

Priority for choosing $u$ (so $dv$ is what remains):

Logarithm ( $\ln x$ )
Inverse trig ( $\arcsin x$ , $\arctan x$ )
Algebraic ( $x^2$ , $x$ )
Trigonometric ( $\sin x$ , $\cos x$ )
Exponential ( $e^x$ )

Example: $\int x \sin x \, dx$ . Choose $u = x$ (Algebraic) and $dv = \sin x \, dx$ .

Theorems you may cite

IVT: $f$ continuous on $[a,b]$ , $k$ between $f(a), f(b)$ ⟹ ∃ $c$ with $f(c) = k$ .
MVT: $f$ continuous on $[a,b]$ , differentiable on $(a,b)$ ⟹ ∃ $c$ with $f'(c) = \frac{f(b)-f(a)}{b-a}$ .
EVT: $f$ continuous on $[a,b]$ ⟹ $f$ has absolute max and min.
FTC Part 1: $\frac{d}{dx} \int_a^x f(t) \, dt = f(x)$ .
FTC Part 2: $\int_a^b f'(x) \, dx = f(b) - f(a)$ .

Calculator essentials (check batteries tonight!)

Numerical derivative: nDeriv(f(x), x, value)
Definite integral: fnInt(f(x), x, a, b)
Find intersection: Graph both functions, 2nd → CALC → intersect.
Find zero: Graph, 2nd → CALC → zero.
Solve $f(x) = k$ : Use nSolve(f(x) - k, x, guess) or graph and intersect.

Score boundaries

Approximate ranges out of 108 total:

5: ~70+ points
4: ~58–69 points
3: ~42–57 points
2: ~30–41 points
1: below ~30 points

You need ~65% to score a 5. Leave some questions blank and still earn the top score.

Morning-of checklist

☐ 8 hours of sleep (tonight and last night).
☐ Real breakfast (protein + slow carbs; no sugar crash).
☐ 2 sharpened #2 pencils, blue/black pens.
☐ Approved calculator + spare batteries (replace them now).
☐ Photo ID + AP exam ID label.
☐ Water bottle, snack for the break.
☐ Watch (no alarm) if the room lacks a clock.
☐ Arrive 30 minutes early.

During the exam: strategies

MCQ section:

Mark any question that takes >90 seconds; come back to it.
Use process of elimination aggressively; wrong answers often have sign errors or computation mistakes.
Skim all questions first; build confidence with the easy ones.

FRQ section:

Read all 6 questions before writing anything.
Start with the FRQ you're most confident on.
Always write the integral or derivative setup before the final answer.
Never leave an FRQ blank. Partial credit on setup + work > zero points.
Box your final answer or circle it clearly.

One final thought

You've done the work. The preparation is complete. Your job tomorrow is to show what you know — write clearly, cite theorems by name, and trust the process.

Take a deep breath. You've got this. 🎯

Final resource: Need a quick review of a specific topic? Browse AP Calculus BC topics →. Or revisit FRQ patterns →.

Good luck tomorrow. See you on the other side. ✓