AP Calculus AB Last-Minute Review (Night Before)

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

• Formula Sheet
• Common Traps
• Calculator Tips

The exam is tomorrow. This is not the time to learn new content — it's the time to skim, reset, and sleep. Spend 30-45 minutes on this page, then put your notes away.

Must-know derivatives

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

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

Must-know integrals

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

Theorems you may have to cite by name

• Intermediate Value Theorem (IVT): $f$ continuous on $[a, b]$, $k$ between $f(a)$ and $f(b)$ ⟹ ∃ $c \\in (a, b)$ with $f(c) = k$.
• Mean Value Theorem (MVT): $f$ continuous on $[a, b]$ AND differentiable on $(a, b)$ ⟹ ∃ $c$ with $f'(c) = \\frac{f(b)-f(a)}{b-a}$.
• Extreme Value Theorem (EVT): $f$ continuous on $[a, b]$ ⟹ $f$ attains an absolute max and min on $[a, b]$.
• Fundamental Theorem of Calculus (FTC):
  • Part 1: $\\frac{d}{dx} \\int_a^x f(t) \\, dt = f(x)$.
  • Part 2: $\\int_a^b f'(x) \\, dx = f(b) - f(a)$.

Application formulas to know cold

• Average value: $\\bar f = \\frac{1}{b-a} \\int_a^b f(x) \\, dx$.
• Area between curves: $\\int_a^b (\\text{top} - \\text{bottom}) \\, dx$.
• Volume by disks: $V = \\pi \\int_a^b [R(x)]^2 \\, dx$.
• Volume by washers: $V = \\pi \\int_a^b \\left([R(x)]^2 - [r(x)]^2\\right) dx$.
• Volume by cross sections (squares): $V = \\int_a^b [s(x)]^2 \\, dx$.
• Distance traveled: $\\int_a^b |v(t)| \\, dt$.
• Displacement: $\\int_a^b v(t) \\, dt$.
• Speed: $|v(t)|$.

Top 10 traps that cost real points

1. Forgetting "+ C" on indefinite integrals.
2. Distance vs displacement — distance uses $|v(t)|$.
3. Chain rule with FTC: $\\frac{d}{dx} \\int_a^{g(x)} f(t) \\, dt = f(g(x)) \\cdot g'(x)$. Don't forget $g'(x)$.
4. MVT hypotheses — must say continuous on $[a,b]$ and differentiable on $(a,b)$.
5. Justifying max/min — must cite First or Second Derivative Test.
6. Units missing on real-world FRQ answers (liters, meters, seconds, etc.).
7. Riemann sum from a table — read the question carefully: left, right, midpoint, or trapezoid?
8. Implicit differentiation — every $y$ becomes $\\frac{dy}{dx}$ via the chain rule.
9. Speeding up vs slowing down — speeding up: $v$ and $a$ same sign.
10. Decimal rounding — at least 3 places on calculator section.

Calculator quick reference

• Numerical derivative at a point: nDeriv(f(x), x, value)
• Definite integral: fnInt(f(x), x, a, b)
• Find intersection: Graph both, then 2nd → CALC → intersect.
• Find a zero: Graph, then 2nd → CALC → zero.
• Solve $f(x) = k$: Either graph $f(x) - k$ and find zero, or use nSolve.

Battery check tonight. Bring spares.

Score boundaries (recent years)

Approximate ranges out of 108 total points:

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

You only need to answer ~64-65% correctly to score a 5. You can leave several questions blank and still earn the top score. Don't panic if a problem stumps you.

Morning-of checklist

• ☐ 8 hours of sleep.
• ☐ Real breakfast (protein + slow carbs, not just sugar).
• ☐ 2 sharpened #2 pencils, blue/black pens.
• ☐ Approved calculator + spare batteries.
• ☐ Photo ID + AP ID label sheet.
• ☐ Water bottle, snack for the break.
• ☐ Watch (without alarm) if your room doesn't have a clock.
• ☐ Arrive 30 minutes early.

During the exam

1. MCQ: Mark and skip anything that takes >90 seconds. Come back.
2. No-calc MCQ: Use process of elimination aggressively. The wrong answers are usually computational sign-flips.
3. FRQ: Read all 6 questions first. Start with the one you're most confident on.
4. FRQ: Always write the integral or derivative expression before the answer.
5. Don't leave anything blank on FRQs. Set up the integral, define your variables — partial credit adds up.

One last thing

You've prepared. The work is done. Trust it. Show up rested, breathe between sections, and remember that the rubric wants to give you points — your job is to write clearly enough that it can.

Good luck. You've got this. 🎯

Need more focused review? Browse the AP Calculus AB topic library → or revisit the FRQ practice guide →.