$$\frac{dy}{dx}=\cfrac{\frac{dy}{dt}}{\frac{dx}{dt}}=\cfrac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}$$
$$\frac{d^2y}{dx^2}=\cfrac{\frac{d}{dt}\Big(\frac{dy}{dx}\Big)}{\frac{dx}{dt}}$$
Explicit Arclength
$$\int_a^b\sqrt{1+\Big(f'(x)\Big)^2}\,dx$$Parametric Arclength
$$\int_{t_1}^{t_2} \sqrt{\Big(\frac{dx}{dt}\Big)^2+\Big(\frac{dy}{dt}\Big)^2}\,dt$$$$\text{Speed}=\sqrt{\Big(\frac{dx}{dt}\Big)^2+\Big(\frac{dy}{dt}\Big)^2}$$
$$\text{Distance}=$$$$\int_{t_1}^{t_2} \sqrt{\Big(\frac{dx}{dt}\Big)^2+\Big(\frac{dy}{dt}\Big)^2}\,dt$$
Polar Coordinates
$$x=r\cos{\theta}$$$$y=r\sin{\theta}$$Polar Area
$$\frac{1}{2}\int_{\alpha}^{\beta}\Big(r(\theta)\Big)^2\,d\theta$$$$e^x$$
$$1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots$$$$+\frac{x^n}{n!}+\dots$$
$$\cos{x}$$
$$1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\dots$$$$+(-1)^n\frac{x^{2n}}{(2n)!}+\dots$$
$$\sin{x}$$
$$x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\dots$$$$+(-1)^n\frac{x^{2n+1}}{(2n+1)!}+\dots$$
$$\frac{1}{1-x}$$$$|x|\lt1$$
$$1+x+x^2+x^3+\dots$$$$+x^n+\dots$$
Lagrange Error Bound
$$\text{Error}=|f(x)-P_n(x)|$$$$\le \frac{M}{(n+1)!}|x-c|^{n+1}$$Alt Series Error Bound
$$\text{Error}=|S-S_n|$$$$\le t_{n+1}$$