AP Calculus BC β€’ Self-Study Notes
Calc BC Quiz πŸ“
20 Core Problems · Key Topics · Tricky Traps Included ✏️
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β‘  Limits & Continuity
Q1 Limits Easy
Evaluate: \(\displaystyle\lim_{x \to 0} \frac{\sin(3x)}{5x}\)
SAME-ANGLE TRICK β€” \(\lim_{x\to0}\frac{\sin(kx)}{kx}=1\), so match the angle! Multiply top & bottom to match.
Q2 Continuity Medium
For \(f(x) = \begin{cases} x^2 + k & x < 2 \\ 3x - 1 & x \geq 2 \end{cases}\) to be continuous at \(x = 2\), what is \(k\)?
LEFT = RIGHT β€” continuity means \(\lim^- = \lim^+ = f(a)\). Set the pieces equal at the meeting point!
β‘‘ Differentiation
Q3 Chain Rule Easy
If \(f(x) = \sin(x^3 + 1)\), find \(f'(x)\).
OUTSIDE-IN β€” Chain rule: differentiate the outer function first, keep inner same, then multiply by inner's derivative.
Q4 Implicit Diff. Medium
Find \(\dfrac{dy}{dx}\) if \(x^2 + y^2 = 25\).
(Tricky: don't forget \(y\) is a function of \(x\)!)
TREAT-Y-AS-f(x) β€” Every time you differentiate a \(y\) term, tack on \(\dfrac{dy}{dx}\) right after. Then solve for it!
Q5 L'HΓ΄pital's Rule Medium
Evaluate \(\displaystyle\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}\)
(Check the indeterminate form first!)
0/0 β†’ LOPITAL β€” When you get \(\tfrac{0}{0}\) or \(\tfrac{\infty}{\infty}\), differentiate top AND bottom separately, then re-evaluate.
β‘’ Integration Techniques
Q6 U-Substitution Easy
Evaluate \(\displaystyle\int 2x\cos(x^2)\, dx\)
SPOT-THE-INSIDE β€” Look for a function and its derivative sitting together. Let \(u\) = the "inside" function.
Q7 Integration by Parts Hard
Evaluate \(\displaystyle\int x\,e^x\, dx\)
LIATE β€” Choose \(u\) by priority: Log β†’ Inverse trig β†’ Algebraic β†’ Trig β†’ Exponential. Formula: \(\int u\,dv = uv - \int v\,du\)
Q8 Partial Fractions Hard
Evaluate \(\displaystyle\int \frac{1}{x^2 - 1}\, dx\)
(Tricky trap: factor the denominator first!)
FACTOR-SPLIT-SOLVE β€” Factor denominator β†’ write as \(\tfrac{A}{x-1}+\tfrac{B}{x+1}\) β†’ multiply out β†’ match coefficients.
β‘£ Differential Equations
Q9 Separable ODE Medium
Solve \(\dfrac{dy}{dx} = \dfrac{x}{y}\) with \(y(0) = 3\). Find \(y\) when \(x=4\).
SEPARATE-INTEGRATE-SOLVE β€” Move all \(y\)'s left, all \(x\)'s right β†’ integrate both sides β†’ plug in initial condition for \(C\).
Q10 Euler's Method Medium
Use Euler's Method with step \(h=1\) to approximate \(y(2)\) given \(\dfrac{dy}{dx} = x + y\) and \(y(0) = 1\).
NEW = OLD + SLOPE Γ— STEP β€” Formula: \(y_{n+1} = y_n + f(x_n, y_n)\cdot h\). Repeat for each step like a relay race!
β‘€ Infinite Series
Q11 Geometric Series Easy
Find the sum: \(\displaystyle\sum_{n=0}^{\infty} \left(\frac{2}{3}\right)^n\)
S = a/(1-r) β€” Geometric series converges only if \(|r| < 1\). First term \(a\), ratio \(r\). Memorize this one formula!
Q12 Ratio Test Hard
Determine convergence of \(\displaystyle\sum_{n=1}^{\infty} \frac{n!}{n^n}\) using the Ratio Test.
(Common trap: \(n^n\) grows much faster than \(n!\)
RATIO: L < 1 = CONVERGE β€” Ratio Test: \(L = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|\). If \(L<1\): converge. \(L>1\): diverge. \(L=1\): inconclusive!
Q13 Taylor Series Hard
The Maclaurin series for \(e^x\) is \(\displaystyle\sum_{n=0}^{\infty}\frac{x^n}{n!}\). Use this to write the Maclaurin series for \(e^{-x^2}\).
SUBSTITUTE-DON'T-REDERIVE β€” Never rederive from scratch! Just replace \(x\) with the new expression everywhere. Fast and foolproof.
Q14 Interval of Convergence Hard
Find the interval of convergence of \(\displaystyle\sum_{n=1}^{\infty}\frac{(x-2)^n}{n}\).
(Trap: always check the endpoints separately!)
RATIO β†’ RADIUS β†’ ENDPOINTS β€” Use Ratio Test to find \(R\), write \(|x-c|<R\), then manually test each endpoint by plugging back in!
β‘₯ Applications of Integration
Q15 Area Between Curves Medium
Find the area between \(y = x^2\) and \(y = x\) for \(0 \leq x \leq 1\).
TOP MINUS BOTTOM β€” Area = \(\int_a^b [\text{top} - \text{bottom}]\,dx\). Always check which function is on top in your interval!
Q16 Volumes (Disk Method) Hard
Find the volume when \(y = \sqrt{x}\), \(0 \leq x \leq 4\) is rotated about the \(x\)-axis.
DISK = Ο€ rΒ² β€” Volume by disk: \(V = \pi\int_a^b [f(x)]^2\,dx\). Think of stacking infinitely thin circular disks!
⑦ Parametric & Polar
Q17 Parametric Derivatives Medium
If \(x = t^2\) and \(y = t^3\), find \(\dfrac{dy}{dx}\) in terms of \(t\).
dy/dx = (dy/dt)Γ·(dx/dt) β€” Parametric slope = divide the two derivatives. Never differentiate \(y\) with respect to \(x\) directly!
Q18 Polar Area Hard
Find the area enclosed by \(r = 2\cos\theta\) (one full loop).
(Trap: what are the correct limits of integration?)
POLAR AREA = ½∫rΒ²dΞΈ β€” Formula: \(A = \frac{1}{2}\int_\alpha^\beta r^2\,d\theta\). For \(r=2\cos\theta\), one loop is traced from \(\theta = -\frac{\pi}{2}\) to \(\frac{\pi}{2}\).
⑧ BC Bonus β€” Tricky Traps!
Q19 Improper Integral Hard
Evaluate \(\displaystyle\int_1^{\infty} \frac{1}{x^2}\,dx\)
(Classic trap: does this converge or diverge?)
p-TEST: p>1 CONVERGE β€” For \(\int_1^\infty \frac{1}{x^p}\,dx\): converges if \(p>1\), diverges if \(p\leq 1\). Replace \(\infty\) with \(b\), take the limit!
Q20 FTC Part II Medium
If \(g(x) = \displaystyle\int_2^{x^3} \sin(t)\, dt\), find \(g'(x)\).
(Trap: the upper limit is NOT just \(x\)!)
FTC2 + CHAIN RULE β€” \(\frac{d}{dx}\int_a^{u(x)} f(t)\,dt = f(u(x))\cdot u'(x)\). Plug the upper limit into \(f\), then multiply by its derivative!
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