College Board · AP Exam Style
AP Calculus
AB / BC
Master every essential topic — from limits to series — with exam-style problems designed to match the rigor of the real AP exam.
20
Questions
45
Minutes
10
Topics
5
Choices Each
Essential Review
Core Concepts & Key Formulas
Limits
Definition: limx→a f(x) = L
L'Hôpital's Rule: If 0/0 or ∞/∞,
lim f(x)/g(x) = lim f′(x)/g′(x)
Squeeze Theorem: g(x) ≤ f(x) ≤ h(x)
L'Hôpital's Rule: If 0/0 or ∞/∞,
lim f(x)/g(x) = lim f′(x)/g′(x)
Squeeze Theorem: g(x) ≤ f(x) ≤ h(x)
Derivatives
Power Rule: d/dx[xⁿ] = nxⁿ⁻¹
Chain Rule: [f(g(x))]′ = f′(g(x))·g′(x)
Product: (uv)′ = u′v + uv′
Quotient: (u/v)′ = (u′v − uv′)/v²
Chain Rule: [f(g(x))]′ = f′(g(x))·g′(x)
Product: (uv)′ = u′v + uv′
Quotient: (u/v)′ = (u′v − uv′)/v²
Integrals
FTC I: d/dx[∫ₐˣ f(t)dt] = f(x)
FTC II: ∫ₐᵇ f(x)dx = F(b) − F(a)
u-substitution: ∫f(g(x))g′(x)dx
= ∫f(u)du
FTC II: ∫ₐᵇ f(x)dx = F(b) − F(a)
u-substitution: ∫f(g(x))g′(x)dx
= ∫f(u)du
Series (BC)
Geometric: Σarⁿ = a/(1−r), |r|<1
Taylor: f(x) = Σ f⁽ⁿ⁾(a)/n! · (x−a)ⁿ
Ratio Test: lim|aₙ₊₁/aₙ|<1 ⟹ converge
Taylor: f(x) = Σ f⁽ⁿ⁾(a)/n! · (x−a)ⁿ
Ratio Test: lim|aₙ₊₁/aₙ|<1 ⟹ converge
Applications
Area: ∫ₐᵇ [f(x)−g(x)]dx
Volume (disk): π∫ₐᵇ [f(x)]²dx
MVT: f′(c) = [f(b)−f(a)]/(b−a)
Arc Length: ∫√(1+[f′(x)]²)dx
Volume (disk): π∫ₐᵇ [f(x)]²dx
MVT: f′(c) = [f(b)−f(a)]/(b−a)
Arc Length: ∫√(1+[f′(x)]²)dx
Diff. Equations
Separable: dy/dx = g(x)h(y)
→ ∫(1/h(y))dy = ∫g(x)dx
Logistic: dP/dt = kP(1−P/M)
Solution: P = M/(1+Ae⁻ᵏᵗ)
→ ∫(1/h(y))dy = ∫g(x)dx
Logistic: dP/dt = kP(1−P/M)
Solution: P = M/(1+Ae⁻ᵏᵗ)
★ Must-Memorize Formula Sheet
- d/dx[sin x] = cos x | d/dx[cos x] = −sin x
- d/dx[tan x] = sec²x | d/dx[sec x] = sec x tan x
- d/dx[eˣ] = eˣ | d/dx[ln x] = 1/x
- d/dx[arcsin x] = 1/√(1−x²)
- d/dx[arctan x] = 1/(1+x²)
- ∫sin x dx = −cos x + C
- ∫eˣ dx = eˣ + C | ∫1/x dx = ln|x| + C
- ∫sec²x dx = tan x + C
- IVT: f continuous on [a,b], then f hits all values between f(a) & f(b)
- EVT: f continuous on [a,b] ⟹ f has absolute max & min on [a,b]
- Maclaurin: eˣ = Σxⁿ/n!, sin x = Σ(−1)ⁿx²ⁿ⁺¹/(2n+1)!
- Parametric: dy/dx = (dy/dt)/(dx/dt); d²y/dx² = [d(dy/dx)/dt]/(dx/dt)
📐 Worked Example
Find: limx→0 (sin 3x) / (5x)
Apply limx→0 (sin kx)/x = k:
limx→0 sin(3x)/(5x) = (3/5) · limx→0 sin(3x)/(3x) = 3/5 · 1 = 3/5
limx→0 sin(3x)/(5x) = (3/5) · limx→0 sin(3x)/(3x) = 3/5 · 1 = 3/5
Find: ∫(x² + 3x − 2) dx
Apply power rule term-by-term:
= x³/3 + 3x²/2 − 2x + C
= x³/3 + 3x²/2 − 2x + C
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