AP Calculus BC Last-Minute Review (Night Before)

title: "AP Calculus BC Last-Minute Review (Night Before)" description: "The night-before AP Calculus BC checklist: must-know formulas, Maclaurin series, convergence tests, common traps, calculator tips, and morning advice. Skim in 45 minutes." date: "2026-01-15" examDate: "May AP Exam" topics:

• Formula Reference
• Convergence Tests
• Common Traps
• Exam Day Tips

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

1. Polar area: Write $\frac{1}{2} \int r^2 \, d\theta$, not $\int r^2 \, d\theta$.
2. Parametric speed: $\sqrt{(dx/dt)^2 + (dy/dt)^2}$, never $(dx/dt + dy/dt)$ or $|(dx/dt) + (dy/dt)|$.
3. Parametric $dy/dx$: $\frac{dy/dt}{dx/dt}$, not $\frac{dy}{dx}$ (which is what you're trying to find).
4. IOC endpoints: Test them separately. Endpoint convergence can differ from interior convergence.
5. Ratio test $L = 1$: Inconclusive — you must use another test.
6. Lagrange error: Find $M = \max |f^{(n+1)}|$ on the interval; don't guess a bound.
7. Chain rule with FTC: $\frac{d}{dx} \int_a^{g(x)} f(t) \, dt = f(g(x)) \cdot g'(x)$ — don't drop $g'(x)$.
8. Implicit differentiation: Every $y$ becomes $\frac{dy}{dx}$ via chain rule; don't forget this step.
9. Distance vs displacement: Distance uses $|v(t)|$; displacement is $\int v(t) \, dt$ (can be negative).
10. MVT hypotheses: Continuous on $[a,b]$ AND differentiable on $(a,b)$ — must state both.
11. Integration by parts: Use LIATE priority when choosing $u$. Often $u = \text{polynomial}$ or $u = \ln x$.
12. 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):

1. Logarithm ($\ln x$)
2. Inverse trig ($\arcsin x$, $\arctan x$)
3. Algebraic ($x^2$, $x$)
4. Trigonometric ($\sin x$, $\cos x$)
5. 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. ✓