Unit 1 — Limits & Continuity
Must Memorize
Definition of a Limit
The limit of \(f(x)\) as \(x \to a\) equals \(L\) if and only if both one-sided limits exist and are equal to \(L\).
\[\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L\]
⚡ Key: If left-hand limit ≠ right-hand limit, the two-sided limit does not exist (DNE).
Example
\(\displaystyle\lim_{x \to 0} \frac{\sin x}{x} = 1\) (fundamental trigonometric limit)
Must Memorize
Squeeze Theorem & L'Hôpital's Rule
If \(g(x) \le f(x) \le h(x)\) near \(a\) and \(\lim g = \lim h = L\), then \(\lim f = L\).
L'Hôpital: If \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\): \(\displaystyle\lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}\)
⚡ Key: L'Hôpital ONLY applies to indeterminate forms \(\tfrac{0}{0}\) or \(\tfrac{\infty}{\infty}\).
Must Memorize
Continuity
\(f\) is continuous at \(x = a\) iff three conditions hold:
1. \(f(a)\) is defined 2. \(\lim_{x\to a}f(x)\) exists 3. \(\lim_{x\to a}f(x) = f(a)\)
⚡ IVT: If \(f\) is continuous on \([a,b]\) and \(k\) is between \(f(a)\) and \(f(b)\), then \(\exists\, c \in (a,b)\) with \(f(c)=k\).
Unit 2–4 — Differentiation
Must Memorize
Derivative Definition & Rules
\[f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\]
Power: \(\frac{d}{dx}[x^n]=nx^{n-1}\) | Product: \((uv)'=u'v+uv'\)
Quotient: \(\left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}\) | Chain: \(\frac{d}{dx}[f(g(x))]=f'(g(x))\cdot g'(x)\)
⚡ Key Trig: \((\sin x)'=\cos x\), \((\cos x)'=-\sin x\), \((\tan x)'=\sec^2 x\)
⚡ Key Exp/Log: \((e^x)'=e^x\), \((\ln x)'=\frac{1}{x}\), \((a^x)'=a^x\ln a\)
Example
\(\dfrac{d}{dx}[\sin(x^2)] = \cos(x^2)\cdot 2x = 2x\cos(x^2)\)
Must Memorize
MVT & Related Rates
Mean Value Theorem: If \(f\) continuous on \([a,b]\), differentiable on \((a,b)\):
\[\exists\, c\in(a,b):\; f'(c)=\frac{f(b)-f(a)}{b-a}\]
⚡ Key: MVT guarantees an instantaneous rate = average rate. For related rates, differentiate both sides w.r.t. time \(t\).
Unit 5–7 — Integration
Must Memorize
Fundamental Theorem of Calculus
FTC Part 1: \(\frac{d}{dx}\int_a^{g(x)}f(t)\,dt = f(g(x))\cdot g'(x)\)
FTC Part 2: \(\int_a^b f(x)\,dx = F(b)-F(a)\) where \(F'=f\)
⚡ Net Change: \(\int_a^b f'(x)\,dx = f(b)-f(a)\) (net displacement, not total distance).
Must Memorize
Key Antiderivatives & U-Substitution
\(\int x^n dx=\frac{x^{n+1}}{n+1}+C\;(n\neq-1)\) | \(\int e^x dx=e^x+C\)
\(\int \frac{1}{x}dx=\ln|x|+C\) | \(\int \sin x\,dx=-\cos x+C\)
\(\int \cos x\,dx=\sin x+C\) | \(\int \sec^2 x\,dx=\tan x+C\)
⚡ U-sub: Let \(u=g(x)\), then \(du=g'(x)\,dx\). Replace all \(x\)s and \(dx\)s with \(u\) and \(du\).
BC Only — Series, Parametric & Polar
BC — Must Memorize
Taylor & Maclaurin Series
Taylor: \(f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n\)
Key Maclaurin series (center \(a=0\)):
\(e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}\) | \(\sin x=\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{(2n+1)!}\)
\(\cos x=\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(2n)!}\) | \(\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n,\;|x|<1\)
⚡ Ratio Test: \(L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|\). Converges if \(L<1\), diverges if \(L>1\).
BC — Must Memorize
Parametric & Polar Derivatives
Parametric: \(\frac{dy}{dx}=\frac{dy/dt}{dx/dt}\) | \(\frac{d^2y}{dx^2}=\frac{d(dy/dx)/dt}{dx/dt}\)
Arc length (parametric): \(L=\int_\alpha^\beta\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\,dt\)
Polar area: \(A=\frac{1}{2}\int_\alpha^\beta [r(\theta)]^2\,d\theta\)
BC — Must Memorize
Euler's Method & Logistic Growth
Euler: \(y_{n+1}=y_n+h\cdot f'(x_n,y_n)\)
Logistic: \(\frac{dP}{dt}=kP\!\left(1-\frac{P}{M}\right)\), carrying capacity \(M\)
\(P(t)=\frac{M}{1+Ae^{-kt}}\), where \(A=\frac{M-P_0}{P_0}\)
⚡ Key: Logistic growth rate is fastest when \(P = M/2\) (inflection point).