∫∑∂∞
Calculus BC
Self-Study Notes & Practice Problems
📚 Topics: Limits · Derivatives · Integrals · Series · Parametric · Polar
🧠 Mode: Tricky Questions · Multiple Choice
⭐ Level: Approachable but Sneaky
Progress:
0 / 20
📐 Unit 1 — Limits & Continuity
L'Hôpital:
0/0 or ∞/∞
→ diff top & bottom
Q1
L'HÔPITAL'S RULE
⭐⭐
If \(\lim_{x\to 0}\frac{\sin x}{x} = \frac{0}{0}\), apply L'Hôpital: \(\lim_{x\to 0}\frac{\cos x}{1} = 1\) ✓
Evaluate: \(\displaystyle\lim_{x \to 0} \dfrac{e^{2x} - 1 - 2x}{x^2}\)
my notes ✏️
Squeeze:
g≤f≤h & g,h→L
⟹ f→L
Q2
SQUEEZE THEOREM
⭐⭐
\(-1 \le \sin\theta \le 1\), so \(-x^2 \le x^2\sin(1/x) \le x^2\). As \(x\to0\), both bounds → 0, so the limit = 0.
Given \(-x^4 \le f(x) \le x^4\) for all \(x\), find \(\displaystyle\lim_{x \to 0} \frac{f(x)}{x^2}\).
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IVT:
continuous on [a,b]
⟹ hits every value
Q3
CONTINUITY / IVT
⭐⭐⭐
\(f(x)=\frac{x^2-4}{x-2}\) is NOT continuous at \(x=2\) (hole). But if we define \(f(2)=4\), it becomes continuous — a "removable discontinuity."
For what value of \(k\) is \(f\) continuous at \(x=1\)? \[f(x) = \begin{cases} kx^2 + 3 & x < 1 \\ 2k - x & x \ge 1 \end{cases}\]
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📈 Unit 2 — Derivatives
Chain Rule:
d/dx[f(g(x))]
= f'(g)·g'
Q4
IMPLICIT DIFFERENTIATION
⭐⭐⭐
Differentiating \(x^2 + y^2 = 25\) implicitly: \(2x + 2y\dfrac{dy}{dx} = 0 \Rightarrow \dfrac{dy}{dx} = -\dfrac{x}{y}\)
Given \(x^3 + y^3 = 6xy\), find \(\dfrac{dy}{dx}\) at the point \((3, 3)\).
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MVT:
f'(c) = [f(b)-f(a)]/(b-a)
some c exists!
Q5
MEAN VALUE THEOREM
⭐⭐
MVT says: if \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), there exists \(c\) such that \(f'(c) = \dfrac{f(b)-f(a)}{b-a}\). Think of it as: "average slope = instantaneous slope somewhere."
Let \(f(x) = x^3 - 2x\) on \([0, 2]\). By the MVT, there exists \(c \in (0,2)\) such that \(f'(c)\) equals the average rate of change. Find \(c\).
my notes ✏️
Related Rates:
DRAW → WRITE eq
→ diff both sides
Q6
RELATED RATES
⭐⭐⭐
A ladder 10 ft slides down a wall. Use \(x^2+y^2=100\). Differentiate: \(2x\tfrac{dx}{dt}+2y\tfrac{dy}{dt}=0\). Plug in and solve.
A spherical balloon is inflated so that its volume increases at \(100\text{ cm}^3/\text{s}\). How fast is the radius increasing when \(r = 5\text{ cm}\)?
\(V = \tfrac{4}{3}\pi r^3\)
my notes ✏️
Concavity:
f''>0 → concave UP (smile)
f''<0 → concave DOWN
Q7
CURVE SKETCHING / 2ND DERIV
⭐⭐⭐
\(f''(x) = 0\) gives inflection point candidates. But \(f''(c)=0\) alone doesn't guarantee an inflection — concavity must actually change!
Let \(f(x) = x^4 - 4x^3\). On which interval is \(f\) concave up?
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∫ Unit 3 — Integration
u-sub:
pick u = inside
du = derivative·dx
Q8
U-SUBSTITUTION
⭐⭐
\(\int x\cdot e^{x^2}dx\): let \(u=x^2\), \(du=2x\,dx\) → \(\tfrac{1}{2}\int e^u\,du = \tfrac{1}{2}e^{x^2}+C\)
Evaluate: \(\displaystyle\int \frac{3x^2}{x^3 + 1}\,dx\)
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IBP:
∫u dv = uv − ∫v du
LIATE: Log,Inv,Alg,Trig,Exp
Q9
INTEGRATION BY PARTS
⭐⭐⭐
\(\int x e^x dx\): u=x, dv=eˣdx → du=dx, v=eˣ → \(xe^x - \int e^x dx = xe^x - e^x + C\)
Evaluate: \(\displaystyle\int x^2 \ln x\,dx\)
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FTC Part 1:
d/dx[∫ₐˣ f(t)dt] = f(x)
Chain rule if upper ≠ x!
Q10
FUNDAMENTAL THEOREM OF CALC
⭐⭐⭐
\(\dfrac{d}{dx}\!\int_1^{x^2}\!\!\sin(t)\,dt = \sin(x^2)\cdot 2x\) — don't forget the chain rule on the upper bound!
Find \(\dfrac{d}{dx}\displaystyle\int_0^{x^3} \cos(t^2)\,dt\).
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Improper ∫:
replace ∞ with limit b→∞
Converges if finite!
Q11
IMPROPER INTEGRALS
⭐⭐⭐
\(\int_1^\infty \tfrac{1}{x^2}dx = \lim_{b\to\infty}\left[-\tfrac{1}{x}\right]_1^b = 0-(-1) = 1\) ← converges!
Determine whether \(\displaystyle\int_1^{\infty} \dfrac{1}{x}\,dx\) converges or diverges. If it converges, find its value.
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🌀 Unit 4 — Differential Equations
Sep. of Vars:
dy/y = k·dx
→ ln|y| = kx + C
Q12
SEPARABLE DIFF. EQUATIONS
⭐⭐
\(\tfrac{dy}{dx} = 2xy\): separate → \(\tfrac{dy}{y} = 2x\,dx\) → \(\ln|y|=x^2+C\) → \(y = Ae^{x^2}\)
Solve the differential equation \(\dfrac{dy}{dx} = y\sin x\) with initial condition \(y(0) = 2\).
my notes ✏️
Euler's Method:
y_new = y_old + h·f'
small h = more accurate
Q13
EULER'S METHOD
⭐⭐⭐
If \(\tfrac{dy}{dx}=x+y\), \(y(0)=1\), step \(h=0.1\):
\(y(0.1) \approx 1 + 0.1\cdot(0+1) = 1.1\)
Use Euler's Method with \(h = 0.5\) to approximate \(y(1)\) if \(\dfrac{dy}{dx} = x - y\) and \(y(0) = 1\).
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Σ Unit 5 — Infinite Series (BC Only!)
Ratio Test:
L=lim|a_n+1/a_n|
L<1→conv, L>1→div
Q14
SERIES CONVERGENCE — RATIO TEST
⭐⭐⭐
\(\sum \tfrac{n!}{2^n}\): Ratio = \(\tfrac{(n+1)!}{2^{n+1}}\cdot\tfrac{2^n}{n!} = \tfrac{n+1}{2}\to\infty\) → Diverges!
Use the Ratio Test on \(\displaystyle\sum_{n=1}^{\infty} \dfrac{n!}{n^n}\). What is \(\lim_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|\)?
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Taylor Series:
f(x)=Σ f⁽ⁿ⁾(a)/n! · (x−a)ⁿ
Memorize: eˣ, sin, cos!
Q15
TAYLOR / MACLAURIN SERIES
⭐⭐⭐
\(e^x = \sum_{n=0}^\infty \dfrac{x^n}{n!} = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!}+\cdots\)
The Maclaurin series for \(\sin x\) is \(x - \tfrac{x^3}{3!} + \tfrac{x^5}{5!} - \cdots\). Use the first two nonzero terms to approximate \(\sin(0.1)\). Which answer is closest?
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Alternating Series:
|error| ≤ |first omitted term|
Simple & powerful!
Q16
ALTERNATING SERIES / ERROR BOUND
⭐⭐⭐
\(\sum_{n=1}^\infty \tfrac{(-1)^{n+1}}{n} = 1 - \tfrac{1}{2}+\tfrac{1}{3}-\cdots\). Stopped after \(n=4\): error \(\le \tfrac{1}{5} = 0.2\).
\(\displaystyle\sum_{n=1}^\infty \dfrac{(-1)^{n+1}}{n^2}\) is approximated by the first 4 terms. What is the maximum error of this approximation?
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🌀 Unit 6 — Parametric, Polar & Vectors
Parametric dy/dx:
= (dy/dt)/(dx/dt)
d²y/dx²: diff dy/dx by t!
Q17
PARAMETRIC DERIVATIVES
⭐⭐⭐
\(x=t^2, y=t^3\): \(\tfrac{dy}{dx}=\tfrac{3t^2}{2t}=\tfrac{3t}{2}\). At \(t=2\): slope = 3.
A curve is given parametrically by \(x = t^2 - 1\) and \(y = t^3 - 3t\). Find all values of \(t\) where the tangent line is horizontal.
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Polar Area:
A = ½∫r²dθ
Don't forget the ½!
Q18
POLAR AREA
⭐⭐⭐
Area enclosed by \(r=2\sin\theta\), \(0\le\theta\le\pi\):
\(A = \tfrac{1}{2}\int_0^\pi (2\sin\theta)^2\,d\theta = 2\int_0^\pi\sin^2\theta\,d\theta = \pi\)
Find the area enclosed by one petal of \(r = \cos(2\theta)\).
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Arc Length (param):
∫√[(dx/dt)²+(dy/dt)²] dt
Speed = magnitude of velocity!
Q19
ARC LENGTH / SPEED (VECTORS)
⭐⭐⭐
Velocity \(\vec{v}(t) = \langle x'(t), y'(t)\rangle\). Speed \(= |\vec{v}| = \sqrt{(x')^2+(y')^2}\). Position \(= \int\vec{v}\,dt\).
A particle moves with \(x'(t) = 3\cos t\) and \(y'(t) = 4\sin t\). What is the speed of the particle at \(t = \dfrac{\pi}{2}\)?
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Power Series ROC:
Use Ratio Test on aₙ
Then test endpoints!
Q20
POWER SERIES — RADIUS OF CONVERGENCE
⭐⭐⭐⭐
\(\sum \tfrac{x^n}{n}\): Ratio Test gives \(|x|\lt 1\). Test endpoints: \(x=1\) → harmonic series (diverges), \(x=-1\) → alternating harmonic (converges). IOC: \([-1, 1)\).
Find the radius of convergence of \(\displaystyle\sum_{n=0}^{\infty} \dfrac{(x-2)^n}{3^n}\).
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🎓
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