📐 Concept Reference — Memorize These
Unit 1–2 · Limits & Continuity
A limit describes the value a function approaches, not necessarily the value it reaches.
$$\lim_{x \to c} f(x) = L \iff \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = L$$
Squeeze Theorem: If $g(x) \le f(x) \le h(x)$ near $c$ and $\lim g = \lim h = L$, then $\lim f = L$.
$$\lim_{x \to 0} \frac{\sin x}{x} = 1 \qquad \lim_{x \to 0} \frac{1-\cos x}{x} = 0$$
Unit 3–5 · Differentiation
The derivative measures the instantaneous rate of change. Key rules to master:
Power: $\frac{d}{dx}[x^n]=nx^{n-1}$ | Chain: $\frac{d}{dx}[f(g(x))]=f'(g(x))\cdot g'(x)$
Product: $(fg)'=f'g+fg'$ | Quotient: $\left(\frac{f}{g}\right)'=\frac{f'g-fg'}{g^2}$
$\frac{d}{dx}[\sin x]=\cos x$ | $\frac{d}{dx}[\cos x]=-\sin x$ | $\frac{d}{dx}[e^x]=e^x$
$\frac{d}{dx}[\ln x]=\frac{1}{x}$ | $\frac{d}{dx}[\tan x]=\sec^2 x$
Unit 6 · Integration (AB & BC)
Fundamental Theorem of Calculus links differentiation and integration:
FTC Part 1: $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$
FTC Part 2: $\int_a^b f(x)\,dx = F(b)-F(a)$
$\int x^n\,dx = \frac{x^{n+1}}{n+1}+C$ (n≠-1) | $\int e^x\,dx=e^x+C$ | $\int \frac{1}{x}\,dx=\ln|x|+C$
Unit 7–8 · Differential Equations & Applications
Separable: $\frac{dy}{dx}=g(x)h(y) \Rightarrow \int\frac{dy}{h(y)}=\int g(x)\,dx$
Exponential growth: $y=y_0 e^{kt}$ | Logistic: $\frac{dP}{dt}=kP\!\left(1-\frac{P}{M}\right)$
Unit 9–10 · Series (BC Only)
A power series converges within its radius of convergence $R$. Key series:
Geometric: $\sum_{n=0}^\infty ar^n = \frac{a}{1-r},\;|r|<1$
$e^x=\sum_{n=0}^\infty\frac{x^n}{n!}$ | $\sin x=\sum_{n=0}^\infty\frac{(-1)^n x^{2n+1}}{(2n+1)!}$
$\ln(1+x)=\sum_{n=1}^\infty\frac{(-1)^{n+1}x^n}{n},\;|x|\le1,x\ne-1$
Taylor: $f(x)=\sum_{n=0}^\infty\frac{f^{(n)}(a)}{n!}(x-a)^n$
📝 20 Exam-Style Questions
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