AP Calculus BC Formula Sheet — Complete Reference (2026)
The AP Calculus BC exam provides a formula reference sheet — but most students don't know exactly what's on it or how to use it efficiently. Here's the complete breakdown.
What's Included on the AP Calculus BC Formula Sheet
College Board provides a formula sheet for AP Calculus BC covering:
- Derivatives of common functions
- Integration formulas
- Series (Taylor/Maclaurin)
- Parametric and polar formulas
The sheet is available for both sections of the exam (calculator and no-calculator).
Derivatives
Basic Derivatives (on the sheet)
| Function | Derivative |
|---|---|
| d/dx[x^n] | nx^(n-1) |
| d/dx[sin x] | cos x |
| d/dx[cos x] | -sin x |
| d/dx[tan x] | sec²x |
| d/dx[cot x] | -csc²x |
| d/dx[sec x] | sec x tan x |
| d/dx[csc x] | -csc x cot x |
| d/dx[e^x] | e^x |
| d/dx[a^x] | a^x ln a |
| d/dx[ln x] | 1/x |
| d/dx[log_a x] | 1/(x ln a) |
Inverse Trig Derivatives (on the sheet)
| Function | Derivative |
|---|---|
| d/dx[arcsin x] | 1/√(1-x²) |
| d/dx[arccos x] | -1/√(1-x²) |
| d/dx[arctan x] | 1/(1+x²) |
These are frequently tested — know them even though they're provided.
Rules (know these, not always on sheet)
- Chain rule: d/dx[f(g(x))] = f'(g(x))·g'(x)
- Product rule: d/dx[uv] = u'v + uv'
- Quotient rule: d/dx[u/v] = (u'v - uv')/v²
Integrals
Basic Integration Formulas (on the sheet)
| Integral | Result |
|---|---|
| ∫x^n dx | x^(n+1)/(n+1) + C, n ≠ -1 |
| ∫(1/x) dx | ln |
| ∫e^x dx | e^x + C |
| ∫a^x dx | a^x/ln a + C |
| ∫sin x dx | -cos x + C |
| ∫cos x dx | sin x + C |
| ∫tan x dx | -ln |
| ∫1/√(1-x²) dx | arcsin x + C |
| ∫1/(1+x²) dx | arctan x + C |
BC-Specific Integration (on the sheet)
| Integral | Result |
|---|---|
| ∫by parts: ∫u dv | uv - ∫v du |
| ∫1/(a²+x²) dx | (1/a)arctan(x/a) + C |
| ∫1/√(a²-x²) dx | arcsin(x/a) + C |
Integration by parts is BC-only and appears on most FRQ. Memorize: LIATE (Logarithm, Inverse trig, Algebraic, Trig, Exponential) — choose u from earlier in the list.
Series (BC Only)
This section is on the formula sheet and is critical for AP Calculus BC.
Convergence Tests
| Test | When to use |
|---|---|
| nth term test | First check — if lim a_n ≠ 0, series diverges |
| Geometric series | r < 1 converges to a/(1-r) |
| p-series: Σ(1/n^p) | Converges if p > 1 |
| Ratio test | Power series and factorials |
| Integral test | Decreasing, positive functions |
| Comparison / Limit comparison | Comparing to known series |
| Alternating series test | Alternating signs, terms → 0 |
Taylor and Maclaurin Series (on the sheet)
| Function | Series | Centered at |
|---|---|---|
| e^x | 1 + x + x²/2! + x³/3! + ... | x = 0 |
| sin x | x - x³/3! + x⁵/5! - ... | x = 0 |
| cos x | 1 - x²/2! + x⁴/4! - ... | x = 0 |
| 1/(1-x) | 1 + x + x² + x³ + ... | x = 0, |
| ln(1+x) | x - x²/2 + x³/3 - ... | x = 0, -1 < x ≤ 1 |
These are provided on the formula sheet. Still memorize them — you need to recognize them instantly on MC, and writing them from memory in FRQ is faster.
General Taylor series centered at x = a: f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + ...
Lagrange Error Bound (on the sheet)
|R_n(x)| ≤ M·|x-a|^(n+1)/(n+1)!
where M is the maximum value of |f^(n+1)| on the interval. This is provided but you must know how to apply it.
Parametric Equations
All parametric formulas are on the formula sheet.
| Formula | Use |
|---|---|
| dy/dx = (dy/dt)/(dx/dt) | Slope of parametric curve |
| d²y/dx² = (d/dt[dy/dx])/(dx/dt) | Second derivative (concavity) |
| Length = ∫√[(dx/dt)² + (dy/dt)²] dt | Arc length |
| Speed = √[(dx/dt)² + (dy/dt)²] | Speed of a particle |
Polar Equations
| Formula | Use |
|---|---|
| x = r cosθ, y = r sinθ | Converting polar to rectangular |
| r² = x² + y² | Distance |
| Area = ½∫r² dθ | Area enclosed by polar curve |
| Area between curves = ½∫(r₁² - r₂²) dθ | Area between two polar curves |
Common mistake: Forgetting the ½ in the polar area formula.
What's NOT on the Formula Sheet
These must be memorized:
- L'Hôpital's Rule — lim f/g = lim f'/g' (when form is 0/0 or ∞/∞)
- Intermediate Value Theorem — if f is continuous on [a,b], it takes all values between f(a) and f(b)
- Mean Value Theorem — f'(c) = [f(b)-f(a)]/(b-a) for some c in (a,b)
- Fundamental Theorem of Calculus:
- Part 1: d/dx[∫ₐˣ f(t)dt] = f(x)
- Part 2: ∫ₐᵇ f(x)dx = F(b) - F(a)
- Average value of a function: (1/(b-a))∫ₐᵇ f(x)dx
Score Cutoffs for AP Calculus BC
Use our AP Calculus BC Score Calculator to estimate your score.
| AP Score | Composite Range |
|---|---|
| 5 | 68–108 |
| 4 | 54–67 |
| 3 | 40–53 |
| 2 | 26–39 |
| 1 | 0–25 |
Common AP Calc BC Exam Tasks
- Everything from AB plus BC-only topics — integration by parts, partial fractions, improper integrals, parametric derivatives, polar area, and series convergence tests.
- Taylor/Maclaurin series — write series centered at a point, find the interval of convergence, and use the error bound for alternating series. These are high-value BC FRQ questions.
- Parametric and polar — find $dy/dx$ from parametric equations using $dy/dt \div dx/dt$. Find polar area with $\frac{1}{2}\int r^2 d\theta$. Know when to switch between forms.
- Logistic differential equations — know the form $dP/dt = kP(1-P/M)$, identify carrying capacity $M$, and describe long-run behavior.
Frequently Asked Questions
Does AP Calculus BC provide a formula sheet?
No. Neither AP Calc AB nor BC provides a formula sheet. All formulas — including the Taylor series and polar area formula — must be memorized.
What is the AP Calc BC AB subscore?
AP Calc BC reports two scores: your overall BC score (1–5) and an AB subscore reflecting your performance on the AB-content questions within the BC exam. Strong AB preparation directly raises your subscore.