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20 Core Questions · AP BC Level

Calculus BC
Mastery Quiz

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§1 Limits & Continuity
Q 01
Limits — L'Hôpital's Rule
KEYWORD: 0/0 or ∞/∞ → differentiate top & bottom separately
📖 Key Example
\( \lim_{x\to 0}\dfrac{\sin x}{x} \) gives \(\tfrac{0}{0}\) → apply L'Hôpital: \(\dfrac{\cos x}{1}\Big|_{x=0}=1\)
Evaluate  \(\displaystyle\lim_{x \to 0} \dfrac{e^{3x}-1-3x}{x^2}\)
Q 02
Limits — Squeeze / Trig
KEYWORD: sin(Δ)/Δ → 1 as Δ→0; cos(Δ)−1/Δ² → −1/2
Which value does  \(\displaystyle\lim_{x\to 0^+} x\ln x\)  approach?
§2 Differentiation
Q 03
Derivatives — Implicit
KEYWORD: differentiate both sides w.r.t. x; \(\frac{d}{dx}[y^n]=ny^{n-1}\frac{dy}{dx}\)
📖 Key Example
\( x^2+y^2=25 \) → \(2x+2y\frac{dy}{dx}=0\) → \(\frac{dy}{dx}=-\frac{x}{y}\)
If \(x^3+y^3=6xy\), find \(\dfrac{dy}{dx}\).
Q 04
Derivatives — Related Rates
KEYWORD: DRAW diagram → write equation → differentiate w.r.t. t → plug in
A ladder 10 ft long leans against a 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?
Q 05
Derivatives — Chain Rule Trap
KEYWORD: \(\frac{d}{dx}[f(g(x))]=f'(g(x))\cdot g'(x)\) — don't forget the inner derivative!
If \(f(x)=\sin^2(3x^2)\), then \(f'(x)=\)
§3 Integration
Q 06
Integration — u-substitution
KEYWORD: spot f(g(x))·g'(x) → let u=g(x), du=g'(x)dx
📖 Key Example
\(\int 2x\cos(x^2)\,dx\): let \(u=x^2\), \(du=2x\,dx\) → \(\int\cos u\,du=\sin u+C=\sin(x^2)+C\)
Evaluate  \(\displaystyle\int \dfrac{x}{\sqrt{1-x^4}}\,dx\)
Q 07
Integration — Parts
KEYWORD: LIATE picks u — Logarithm, Inverse trig, Algebraic, Trig, Exponential
\(\displaystyle\int x e^x\,dx =\)
Q 08
Integration — Area Between Curves
KEYWORD: Area = \(\int_a^b[\text{top}-\text{bottom}]\,dx\) — always top minus bottom!
Find the area enclosed by \(y=x^2\) and \(y=x+2\).
Q 09
Integration — Improper Integrals
KEYWORD: replace ∞ with limit variable b → take \(\lim_{b\to\infty}\)
Does  \(\displaystyle\int_1^{\infty}\dfrac{1}{x^2}\,dx\)  converge? If so, what is its value?
§4 Differential Equations
Q 10
Diff. Eq. — Separation of Variables
KEYWORD: separate → \(\frac{dy}{g(y)}=f(x)\,dx\) → integrate both sides → solve for y
📖 Key Example
\(\frac{dy}{dx}=ky\) → \(\frac{dy}{y}=k\,dx\) → \(\ln|y|=kx+C\) → \(y=Ae^{kx}\)
Solve  \(\dfrac{dy}{dx}=\dfrac{x}{y}\)  with \(y(0)=3\).
Q 11
Diff. Eq. — Logistic Growth
KEYWORD: \(\frac{dP}{dt}=kP(1-\frac{P}{L})\) → max growth rate at \(P=\frac{L}{2}\)
A population follows \(\dfrac{dP}{dt}=0.4P\!\left(1-\dfrac{P}{500}\right)\). The population grows fastest when \(P=\)
§5 Parametric & Polar
Q 12
Parametric — Arc Length
KEYWORD: Arc Length \(=\int_a^b\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt\)
A curve is given by \(x=t^2,\; y=t^3\) for \(0\le t\le 1\). The arc length integral is:
Q 13
Polar — Area
KEYWORD: Polar area \(=\frac{1}{2}\int_\alpha^\beta r^2\,d\theta\)
Find the area enclosed by the polar curve \(r=2\cos\theta\).
§6 Infinite Series
Q 14
Series — Geometric
KEYWORD: Geometric sum \(=\frac{a}{1-r}\) only when \(|r|<1\)
Find the sum:  \(\displaystyle\sum_{n=0}^{\infty}\left(\dfrac{2}{3}\right)^n\)
Q 15
Series — Ratio Test
KEYWORD: Ratio test: \(L=\lim\left|\frac{a_{n+1}}{a_n}\right|\) → L<1 conv, L>1 div, L=1 inconclusive
Use the Ratio Test to determine the convergence of  \(\displaystyle\sum_{n=1}^{\infty}\dfrac{n!}{3^n}\).
Q 16
Series — Radius of Convergence
KEYWORD: Apply Ratio Test → solve \(|x|\cdot L_\text{coeff}<1\) for \(|x|\)
Find the radius of convergence of  \(\displaystyle\sum_{n=1}^{\infty}\dfrac{(x-2)^n}{n\cdot 4^n}\).
Q 17
Series — Taylor / Maclaurin
KEYWORD: \(e^x=\sum\frac{x^n}{n!}\), \(\sin x=\sum\frac{(-1)^n x^{2n+1}}{(2n+1)!}\), \(\cos x=\sum\frac{(-1)^n x^{2n}}{(2n)!}\)
📖 Key Example
\(\cos x = 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots\) so \(\cos(x^2)=1-\frac{x^4}{2}+\frac{x^8}{24}-\cdots\)
The Maclaurin series for \(xe^{x^2}\) begins with:
Q 18
Series — Alternating Series Test
KEYWORD: AST: terms must decrease → 0; error ≤ first omitted term
The series  \(\displaystyle\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}}{n}=1-\tfrac{1}{2}+\tfrac{1}{3}-\cdots\)  converges by the Alternating Series Test. The error from using the first 4 terms is at most:
§7 FTC & Applications
Q 19
FTC Part 1 — Derivative of Integral
KEYWORD: \(\frac{d}{dx}\int_a^{g(x)}f(t)\,dt=f(g(x))\cdot g'(x)\) — chain rule applies!
📖 Key Example
\(\frac{d}{dx}\int_0^{x^3}\!\sin t\,dt=\sin(x^3)\cdot 3x^2\)
If \(F(x)=\displaystyle\int_0^{x^2}\!\!\sqrt{1+t^3}\,dt\), find \(F'(x)\).
Q 20
Volume — Washer Method
KEYWORD: Washer \(V=\pi\int[R^2-r^2]\,dx\) — outer² minus inner²; Disk is just \(\pi\int R^2\,dx\)
The region bounded by \(y=\sqrt{x}\) and \(y=x^2\) is revolved about the \(x\)-axis. The volume is: