AP Calculus AB covers roughly one semester of college calculus. BC covers two semesters and includes all AB topics plus series (Taylor/Maclaurin), parametric/polar equations, and more integration techniques. BC topics are marked with BC throughout this guide.
Limits
Limit Definition and Notation
The limit lim[x→a] f(x) = L means f(x) gets arbitrarily close to L as x approaches a (but is not necessarily equal to f(a)). Limits describe behavior near a point, not at it.
One-sided limits: lim[x→a⁺] approaches from the right; lim[x→a⁻] from the left. A two-sided limit exists only if both one-sided limits exist and are equal.
Limits at infinity: lim[x→∞] f(x) describes the horizontal asymptote of f(x). For rational functions, compare degrees: if numerator degree < denominator, limit = 0; if equal, limit = ratio of leading coefficients; if numerator degree > denominator, limit = ±∞.
L'Hôpital's Rule BC emphasis
If a limit produces the indeterminate form 0/0 or ∞/∞, apply L'Hôpital's Rule: lim[x→a] f(x)/g(x) = lim[x→a] f'(x)/g'(x). Apply again if the result is still indeterminate. Only applies to quotient forms — convert other indeterminate forms (0·∞, ∞−∞, 1^∞, etc.) to fractions first.
Squeeze Theorem
If g(x) ≤ f(x) ≤ h(x) near a and lim g(x) = lim h(x) = L, then lim f(x) = L. Most commonly used to show lim[x→0] sin(x)/x = 1.
Continuity: f is continuous at a if (1) f(a) is defined, (2) the limit exists, and (3) lim = f(a). The Intermediate Value Theorem (IVT) follows from continuity: if f is continuous on [a,b] and f(a) < k < f(b), then there exists c in (a,b) where f(c) = k.
Derivatives
Definition
The derivative of f at x is f'(x) = lim[h→0] [f(x+h) − f(x)] / h. Geometrically, it's the slope of the tangent line. Practically, it measures instantaneous rate of change.
Differentiation Rules
| Rule | Formula | Example |
|---|---|---|
| Power Rule | d/dx[xⁿ] = nxⁿ⁻¹ | d/dx[x⁵] = 5x⁴ |
| Constant Multiple | d/dx[cf] = c·f' | d/dx[3x²] = 6x |
| Sum/Difference | d/dx[f ± g] = f' ± g' | Differentiate term by term |
| Product Rule | d/dx[fg] = f'g + fg' | "First times derivative of second + second times derivative of first" |
| Quotient Rule | d/dx[f/g] = (f'g − fg') / g² | "Lo d-Hi minus Hi d-Lo over Lo squared" |
| Chain Rule | d/dx[f(g(x))] = f'(g(x)) · g'(x) | Outer function derivative × inner function derivative |
| sin x | d/dx[sin x] = cos x | |
| cos x | d/dx[cos x] = −sin x | |
| tan x | d/dx[tan x] = sec²x | |
| eˣ | d/dx[eˣ] = eˣ | |
| ln x | d/dx[ln x] = 1/x | |
| aˣ | d/dx[aˣ] = aˣ ln a | |
| arcsin x | d/dx[arcsin x] = 1/√(1−x²) | |
| arctan x | d/dx[arctan x] = 1/(1+x²) |
Applications of Derivatives
Implicit differentiation: Differentiate both sides with respect to x. Whenever you differentiate a y term, multiply by dy/dx (chain rule). Then solve for dy/dx.
Related rates: Differentiate an equation with respect to time t. Plug in known rates and values, then solve for the unknown rate. Always draw a diagram and write the geometric relationship first.
Optimization: (1) Write the objective function. (2) Write the constraint. (3) Use the constraint to reduce the objective to one variable. (4) Find critical points (f'(x) = 0 or undefined). (5) Use the first or second derivative test to classify max/min. (6) Check endpoints if on a closed interval.
Mean Value Theorem (MVT): If f is continuous on [a,b] and differentiable on (a,b), there exists c in (a,b) where f'(c) = [f(b) − f(a)] / (b − a). The instantaneous rate of change equals the average rate of change somewhere in the interval.
Integrals
Riemann Sums and the Definite Integral
A Riemann sum approximates the area under a curve by dividing it into rectangles. Left, right, and midpoint sums use the left, right, and midpoint of each subinterval as the rectangle height. The definite integral ∫[a to b] f(x) dx is the limit of Riemann sums as the subinterval width approaches zero.
Fundamental Theorem of Calculus
FTC Part 1: If F(x) = ∫[a to x] f(t) dt, then F'(x) = f(x). Differentiation and integration are inverse operations.
FTC Part 2: ∫[a to b] f(x) dx = F(b) − F(a), where F is any antiderivative of f. Use this to evaluate definite integrals.
Basic Antiderivatives and u-Substitution
| Function | Antiderivative |
|---|---|
| xⁿ (n ≠ −1) | xⁿ⁺¹/(n+1) + C |
| 1/x | ln|x| + C |
| eˣ | eˣ + C |
| sin x | −cos x + C |
| cos x | sin x + C |
| sec²x | tan x + C |
u-substitution: Let u = the "inside" function. Find du, express everything in terms of u, integrate, then substitute back. For definite integrals, either change the limits of integration (convert to u-values) or substitute back before evaluating.
Area between curves: ∫[a to b] [f(x) − g(x)] dx where f(x) ≥ g(x) on [a,b]. Find intersection points first to set limits. If the curves cross, split the integral.
BC-Only Topics BC
Series Convergence Tests
| Test | When to Use | Conclusion |
|---|---|---|
| Divergence Test | Always try first | If lim aₙ ≠ 0, series diverges. If lim aₙ = 0, inconclusive. |
| Integral Test | aₙ = f(n) where f is positive, continuous, decreasing | Series and integral converge or diverge together |
| p-Series Test | Σ 1/nᵖ form | Converges if p > 1; diverges if p ≤ 1 |
| Comparison Test | Compare to known series | If aₙ ≤ bₙ and Σbₙ converges, Σaₙ converges; if aₙ ≥ bₙ and Σbₙ diverges, Σaₙ diverges |
| Limit Comparison Test | When direct comparison is unclear | lim aₙ/bₙ = L (0 < L < ∞) → same behavior as Σbₙ |
| Ratio Test | Factorials, exponentials | L = lim|aₙ₊₁/aₙ|: L < 1 converges, L > 1 diverges, L = 1 inconclusive |
| Alternating Series Test | Alternating signs (−1)ⁿ | Converges if terms decrease to 0; remainder |R| ≤ first omitted term |
Common Taylor Series
These should be memorized:
- eˣ = Σ xⁿ/n! = 1 + x + x²/2! + x³/3! + ... (converges for all x)
- sin x = Σ (−1)ⁿ x²ⁿ⁺¹/(2n+1)! = x − x³/3! + x⁵/5! − ...
- cos x = Σ (−1)ⁿ x²ⁿ/(2n)! = 1 − x²/2! + x⁴/4! − ...
- 1/(1−x) = Σ xⁿ = 1 + x + x² + x³ + ... (converges for |x| < 1)
- ln(1+x) = Σ (−1)ⁿ⁺¹ xⁿ/n = x − x²/2 + x³/3 − ... (converges for −1 < x ≤ 1)
✅ Key Takeaways
- Master the differentiation rules table — the chain rule appears in nearly every AB/BC free-response problem.
- The Fundamental Theorem of Calculus connects derivatives and integrals; Part 2 is your primary tool for evaluating definite integrals.
- For optimization and related rates, write the equation first, then differentiate — never differentiate until you've set up the problem algebraically.
- On BC series questions, try the Divergence Test first — if the limit of terms ≠ 0, you're done.
- Memorize the five common Taylor series (eˣ, sin x, cos x, 1/(1-x), ln(1+x)) — they appear frequently on BC FRQs.