📐 AP Calculus BC · 20 Questions

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Limits & Continuity

1
L'Hôpital's Rule Medium
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Memory Key 0/0 or ∞/∞ → DIFFERENTIATE top & bottom separately (not quotient rule!)
📘 Worked Example
\(\displaystyle\lim_{x\to 0}\frac{\sin x}{x}\) gives \(\tfrac{0}{0}\) form → apply L'Hôpital: \(\displaystyle\lim_{x\to 0}\frac{\cos x}{1} = 1\)
Evaluate the limit:
\[\lim_{x \to 0} \frac{e^{3x} - 1 - 3x}{x^2}\]
2
Continuity Easy
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Memory Key Continuous at a: EXIST · EQUAL · EQUAL (limit exists, limit = f(a))
Which value of \(k\) makes \(f\) continuous at \(x = 2\)?
\[f(x) = \begin{cases} kx^2 - 1 & x < 2 \\ 3x + 1 & x \geq 2 \end{cases}\]

Differentiation

3
Chain Rule Easy
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Memory Key Chain: OUTSIDE' × INSIDE' — peel like an onion, outside-in
Find \(f'(x)\) if \(f(x) = \sin^3(4x)\).
4
Implicit Differentiation Medium
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Memory Key Implicit: diff BOTH sides, attach dy/dx to every y-term (chain rule on y)
📘 Worked Example
\(x^2 + y^2 = 25\): differentiate → \(2x + 2y\dfrac{dy}{dx} = 0\) → \(\dfrac{dy}{dx} = -\dfrac{x}{y}\)
Given \(x^3 + y^3 = 6xy\), find \(\dfrac{dy}{dx}\).
5
Related Rates Hard
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Memory Key Related Rates: DRAW → LABEL → EQUATION → DIFFERENTIATE with respect to t
A ladder 10 ft long rests against a vertical wall. The bottom slides away at 2 ft/s. How fast is the top sliding down when the bottom is 6 ft from the wall?

Integration

6
Integration by Parts Medium
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Memory Key LIATE: Log · Inverse trig · Algebraic · Trig · Exponential → pick u in this order
📘 Formula
\(\displaystyle\int u\,dv = uv - \int v\,du\)
Evaluate: \(\displaystyle\int x\,e^x\,dx\)
7
U-Substitution Easy
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Memory Key u-sub: spot INNER FUNCTION → set u = inside → du replaces dx chain
Evaluate: \(\displaystyle\int \frac{2x}{x^2+1}\,dx\)
8
Definite Integral · Area Medium
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Memory Key Area between curves: ∫(TOP − BOTTOM)dx — always top minus bottom!
Find the area of the region enclosed by \(y = x^2\) and \(y = 2x\).

Series & Convergence

9
Ratio Test Hard
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Memory Key Ratio Test: L = lim|a_{n+1}/a_n| → L<1 converge · L>1 diverge · L=1 inconclusive
Determine convergence of \(\displaystyle\sum_{n=1}^{\infty}\frac{n!}{n^n}\).
10
Taylor Series Hard
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Memory Key Memorize: e^x · sin x · cos x · 1/(1-x) — BUILD from these, don't re-derive!
📘 Must-Know Series
\(e^x = \displaystyle\sum_{n=0}^{\infty}\dfrac{x^n}{n!} = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots\)
The coefficient of \(x^4\) in the Maclaurin series for \(e^{2x}\) is:
11
Alternating Series Test Medium
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Memory Key AST: terms DECREASE & approach 0 → converges. Error ≤ first omitted term.
Which series converges by the Alternating Series Test?

Differential Equations

12
Separable ODE Easy
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Memory Key Separable: GET y's LEFT · x's RIGHT → integrate both sides → solve for y
Solve: \(\dfrac{dy}{dx} = \dfrac{x}{y}\), with \(y(0) = 3\).
13
Logistic Growth Hard
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Memory Key Logistic: dP/dt = kP(1 − P/M) → fastest growth at P = M/2 (half capacity!)
A population \(P\) satisfies \(\dfrac{dP}{dt} = 0.2P\!\left(1-\dfrac{P}{500}\right)\). The growth rate is maximum when:

Parametric & Polar

14
Parametric Derivatives Medium
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Memory Key Parametric slope: dy/dx = (dy/dt) ÷ (dx/dt) — divide, don't chain!
For \(x = t^2 - 1,\ y = t^3 - 3t\), find \(\dfrac{dy}{dx}\) at \(t = 2\).
15
Polar Area Hard
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Memory Key Polar Area = ½∫r² dθ — the ½ is from the "pie slice" sector formula!
The area enclosed by \(r = 2\cos\theta\) is:

Applications of Derivatives & Integrals

16
MVT · FTC Medium
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Memory Key FTC Part 2: d/dx ∫[a to g(x)] f(t)dt = f(g(x))·g'(x) — chain rule on upper limit!
If \(F(x) = \displaystyle\int_{1}^{x^2} \sqrt{t^3+1}\,dt\), then \(F'(x) =\)
17
Volume · Disk Method Medium
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Memory Key Disk/Washer: V = π∫(R²−r²)dx — outer² minus inner², times π
Volume when \(y = \sqrt{x}\) on \([0,4]\) is revolved about the x-axis:
18
Arc Length Hard
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Memory Key Arc Length = ∫√(1+(dy/dx)²) dx — square root with 1 plus slope squared
The arc length of \(y = \dfrac{x^2}{2} - \dfrac{\ln x}{4}\) from \(x=1\) to \(x=2\) is:
(Hint: notice \(1 + (y')^2\) becomes a perfect square.)

Tricky Traps 🚨

19
Absolute vs. Conditional Convergence Hard
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Memory Key Absolute: ∑|aₙ| converges → absolutely. Conditional: converges but ∑|aₙ| diverges.
\(\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\) is:
20
Improper Integral Hard
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Memory Key Improper: replace ∞ with limit variable b → take lim as b→∞ AFTER integrating
📘 Strategy
\(\displaystyle\int_1^{\infty}\frac{1}{x^p}\,dx\) converges iff \(p > 1\); value \(= \dfrac{1}{p-1}\)
Evaluate: \(\displaystyle\int_{0}^{\infty} x\,e^{-x^2}\,dx\)
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