⚡LEADING TERM RULE: Same degree → ratio of leading coefficients. Higher degree bottom → 0. Higher degree top → ±∞.
Unit 2 · Differentiation
Q 03
Chain RuleEasy
If \(f(x) = \sin(x^2 + 1)\), find \(f'(x)\).
⚡CHAIN: derivative of outside × derivative of inside. \(\frac{d}{dx}[\sin(u)] = \cos(u)\cdot u'\)
Q 04
Implicit DifferentiationMedium
Given \(x^2 + y^2 = 25\), find \(\dfrac{dy}{dx}\).
⚡IMPLICIT: Differentiate both sides. Every \(y\)-term gets a \(\frac{dy}{dx}\) multiplied (chain rule on y). Then solve for \(\frac{dy}{dx}\).
Q 05
Related RatesMedium
A spherical balloon is being inflated so that its volume increases at \(10 \text{ cm}^3/\text{s}\). How fast is the radius increasing when \(r = 5\text{ cm}\)? (Volume of sphere: \(V = \tfrac{4}{3}\pi r^3\))
⚡RELATED RATES: Write equation → differentiate both sides w.r.t. time t → plug in known values → solve for unknown rate.
Unit 3 · Integration
Q 06
u-SubstitutionEasy
Evaluate: \(\displaystyle\int 2x\cos(x^2)\,dx\)
⚡U-SUB SPOT: See a function and its derivative sitting next to each other? Let \(u\) = the inside function.
Q 07
Integration by PartsMedium
Evaluate: \(\displaystyle\int x e^x\,dx\)
⚡LIATE ORDER: Pick u from: Log · Inverse trig · Algebraic · Trig · Exponential. Here \(u=x\) (Algebraic), \(dv=e^x dx\).
⚡PARTIAL FRAC: Factor denominator → write \(\frac{A}{x-1}+\frac{B}{x+1}\) → multiply both sides by full denominator → solve for A, B.
Q 09
Improper IntegralsMedium
Determine whether \(\displaystyle\int_1^{\infty} \frac{1}{x^2}\,dx\) converges or diverges. If it converges, find its value.
⚡p-TEST: \(\int_1^\infty \frac{1}{x^p}dx\) → converges if \(p>1\), diverges if \(p\leq 1\). Here \(p=2>1\) → converges.
Unit 4 · Differential Equations
Q 10
Separable DEsEasy
Solve the differential equation: \(\dfrac{dy}{dx} = 2xy\), given \(y(0) = 3\).
⚡SEPARABLE: Get all y's on left, all x's on right → integrate both sides → apply initial condition to find C.
Q 11
Logistic GrowthMedium
A population follows the logistic model \(\dfrac{dP}{dt} = 0.4P\!\left(1 - \dfrac{P}{500}\right)\). What is the carrying capacity, and at what population value is the growth rate maximum?
⚡LOGISTIC: \(\frac{dP}{dt}=kP(1-\frac{P}{M})\) → carrying capacity \(= M\). Max growth at \(P = \frac{M}{2}\).
Unit 5 · Infinite Series
Q 12
Geometric SeriesEasy
Find the sum: \(\displaystyle\sum_{n=0}^{\infty} \left(\frac{2}{3}\right)^n\)
⚡GEOMETRIC SUM: \(\sum_{n=0}^{\infty} r^n = \dfrac{1}{1-r}\) when \(|r|<1\). First identify \(r\).
Q 13
Ratio TestMedium
Determine convergence of \(\displaystyle\sum_{n=1}^{\infty} \frac{n!}{n^n}\) using the Ratio Test.
⚡RATIO TEST: \(L = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|\). If \(L<1\) → converges. If \(L>1\) → diverges. If \(L=1\) → inconclusive.
Q 14
Taylor SeriesMedium
Which of the following is the Maclaurin series for \(e^x\)?
⚡MEMORIZE 3 BIG ONES: \(e^x = \sum\frac{x^n}{n!}\) · \(\sin x = \sum\frac{(-1)^n x^{2n+1}}{(2n+1)!}\) · \(\cos x = \sum\frac{(-1)^n x^{2n}}{(2n)!}\)
Q 15
Alternating SeriesMedium
Does \(\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^n}{n}\) converge or diverge?
⚡AST CHECK: (1) \(b_n > 0\) ✓ (2) \(b_n\) is decreasing ✓ (3) \(\lim b_n = 0\) ✓ → Converges by Alternating Series Test.
Unit 6 · Parametric, Polar & Vectors
Q 16
Parametric DerivativesMedium
Given \(x = t^2\), \(y = t^3\), find \(\dfrac{dy}{dx}\).
⚡PARAMETRIC dy/dx: \(\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}\). Differentiate x and y separately w.r.t. t, then divide.
Q 17
Polar AreaHard
Find the area enclosed by the polar curve \(r = 2\cos\theta\) for \(0 \leq \theta \leq \pi\).
⚡POLAR AREA: \(A = \dfrac{1}{2}\int_{\alpha}^{\beta} r^2\,d\theta\). Don't forget the \(\frac{1}{2}\) out front!
Unit 7 · Applications of Integration
Q 18
Arc LengthMedium
The arc length of \(y = f(x)\) from \(x=a\) to \(x=b\) is given by which formula?
⚡ARC LENGTH: \(L = \int_a^b \sqrt{1+[f'(x)]^2}\,dx\). The \(+1\) is always there — it's from the Pythagorean theorem in disguise.
Q 19
Volume (Disk/Washer)Medium
The region bounded by \(y = \sqrt{x}\), \(x=4\), and \(y=0\) is revolved around the \(x\)-axis. Find the volume.
⚡DISK METHOD: Rotate around x-axis → \(V = \pi\int_a^b [f(x)]^2\,dx\). Always square the radius and multiply by \(\pi\).
Q 20
Fundamental Theorem of CalculusHard
If \(g(x) = \displaystyle\int_0^{x^2} \!\cos(t)\,dt\), find \(g'(x)\).
⚡FTC Part 2 + Chain Rule: \(\frac{d}{dx}\int_a^{u(x)}f(t)\,dt = f(u(x))\cdot u'(x)\). Plug in upper limit → multiply by its derivative.