Calculus

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Reference

Derivatives

\(\text{Sum: } \frac{d}{dx}(u+v)=\frac{du}{dx}+\frac{dv}{dx}\)

\(\text{Product: } \frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}\)

\(\text{Quotient: } \frac{d}{dx}\left(\frac{u}{v}\right) =\frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}\)

\(\text{Power: } \frac{d}{dx}(u^n)=nu^{n-1}\frac{du}{dx}\)

\(\text{Chain: } \frac{d}{dx}z(y(x))=\frac{dz}{dy}\frac{dy}{dx}\)

\(\text{Inverse: } \frac{dx}{dy}=\frac{1}{\frac{dy}{dx}}\)

\(\frac{d}{dx}\sin x=\cos x\)

\(\frac{d}{dx}\cos x=-\sin x\)

\(\frac{d}{dx}\tan x=\sec^2{x}=\frac{1}{\cos^2x}\)

\(\frac{d}{dx}\cot x=-\csc^2{x}=-\frac{1}{\sin^2x}\)

\(\frac{d}{dx}\sec x=\sec x \tan x\)

\(\frac{d}{dx}\csc x=-\csc x \cot x\)

\(\frac{d}{dx}\sin^{-1}x=\frac{1}{\sqrt{1-x^2}}\)

\(\frac{d}{dx}\tan^{-1}x=\frac{1}{1+x^2}\)

\(\frac{d}{dx}\sec^{-1}x=\frac{1}{|x|\sqrt{x^2-1}}\)

Limits and Continuity

\( \frac{\sin x}{x}\to1\)

\( \frac{1-\cos x}{x}\to0\)

\( \frac{1-\cos x}{x^2}\to \frac12\)

\( a_n \to 0 : |a_n| \lt \epsilon \text{ for all }n \gt N\)

\( a_n \to L : |a_n - L| \lt \epsilon \text{ for all }n \gt N\)

\( f(x) \to L: |f(x) - L| \lt \epsilon \text{ for }0 \lt |x-a| \lt \delta \)

\( f(x) \to f(a): \text{ Continuous at }a \text{ if } L=f(a) \)

\( \frac{f(x)-f(a)}{x-a}=f'(c): \text{ Mean Value Theorem} \)

\( \frac{f(x+\Delta x)-f(x)}{\Delta x}\to f'(x): \text{ Derivative at }x \)

\( \frac{f(x+\Delta x)-f(x-\Delta x)}{2\Delta x}\to f'(x): \text{ Centered } \)

\( \lim \frac{f(x)}{g(x)}=\lim \frac{f'(x)}{g'(x)}: \text{ l'Hôpital's Rule for }\frac00 \)

Maximum and Minimum

\( \text{Critical: } f'(x)=0 \text{ or no } f' \text{ or endpoint} \)

\( \text{Miniumum: } f'(x)=0 \text{ and } f''(x) \gt 0 \)

\( \text{Maxiumum: } f'(x)=0 \text{ and } f''(x) \lt 0 \)

\( \text{Inflection point: } f''(x) = 0 \)

\( \text{Newton's method: } x_n+1=x_n - \frac{f(x_n)}{f'(x_n)} \)

\( \text{Iteration } x_n+1=F(x_n) \text{ attracted to fixed point }x^* =F(x^*) \text{ if } |F'(x^*)| \lt 1 \)

\( \text{Stationary in 2D: } \frac{\partial f}{\partial x}=0, \frac{\partial f}{\partial y}=0 \)

\( \text{Miniumum: } f_{xx} \gt 0 \qquad f_{xx} f_{yy} \gt f_{xy}^2 \)

\( \text{Maxiumum: } f_{xx} \lt 0 \qquad f_{xx} f_{yy} \gt f_{xy}^2 \)

\( \text{Saddle point: } f_{xx} f_{yy} \gt f_{xy}^2 \)

\( \text{Newton in 2D: } \begin{cases} \\ g+g_x\Delta x+g_y\Delta y =0 \\ h+h_x\Delta x+h_y\Delta y =0 \\ \end{cases} \)

Algebra

\( \frac{a_1}{b_1} / \frac{\color{red}{a_2}}{\color{blue}{b_2}} = \frac{a_1}{b_1} \cdot \frac{\color{blue}{b_2}}{\color{red}{a_2}} \)

\( x^{-n}=\frac{1}{x^n} \)

\( \sqrt[n]=x^{\frac 1n} \)

\( x^2x^3=x^5 \)

\( (x^2)^3=x^6 \)

\( x^2 / x^3=x^{-1} \)

\( ax^2 + bx + c =0 \text{ has roots }x=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \)

\( x^2 + 2bx + c =0 \text{ has roots }x=-b \pm \sqrt{b^2-c} \)

\( \text{Completing the square: }ax^2+bx+c=a(x+\frac{b}{2a})^2+c-\frac{b^2}{4a} \)

\( \text{Partial fractions }\frac{cx+d}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b} \)

\( \text{Mistakes:}\frac{a}{b+c} \ne \frac ab + \frac ac \qquad \sqrt{x^2+a^s} \ne x+a \)

Fundamental Theorem of Calculus

\( \frac{d}{dx} \int_a^x v(t)dt=v(x) \)

\( \frac{d}{dx} \int_{a(x)}^{b(x)} v(t)dt = v(b(x))\frac{db}{dx}-v(a(x))\frac{da}{dx} \)

\( \int_0^b y(x)dx=\lim_{\Delta x \to 0} \Delta x[y(\Delta x)+ y(2\Delta x)+\cdots+y(b)] \)

Circle, Line, and Plane

\( x=r\cos \omega t \;,\; y=r\sin \omega t \;,\; \text{ speed }\omega r\)

\( y=mx+b \text{ or } y-y_0=m(x-x_0)\)

\( \text{Plane } ax+by+cz=d \text{ or } a(x-x_0)+b(y-y_0)+c(z-z_0)=0 \)

\( \text{Normal vector } a \mathbf{i}+ b \mathbf{j}+c \mathbf{k} \)

\( \text{Distance to }(0,0,0): |d|/\sqrt{a^2+b^2+c^2} \)

\( \text{Line }(x,y,z)=(x_0,y_0,z_0)+t(v_1,v_2,v_3) \)

\( \text{No parameter: }\frac{x-x_0}{v_1}=\frac{y-y_0}{v_2}=\frac{z-z_0}{v_3} \)

\( \text{Projection: }\mathbf{p}=\frac{\mathbf{b}\cdot\mathbf{a}}{\mathbf{a}\cdot\mathbf{a}}\mathbf{a} \qquad |\mathbf{p}|=|\mathbf{b}|\cos \theta \)

Trigonometric Identities

\( \sin^2 x+\cos^2 x=1 \)

\( \tan^2 x+1=\sec^2 x \)

\( 1+\cot^2 x=\csc^2 x \)

\( \sin 2x = 2 \sin x \cos x \text{ (double angle)} \)

\( \cos 2x = \cos^2 x - \sin^2 x = 2 \cos^2 x -1=1-2\sin^2 x \)

\( \sin (s \pm t) = \sin s \cos t \pm \cos s \sin t \)

\( \cos (s \pm t) = \cos s \cos t \mp \sin s \sin t \)

\( \tan (s + t) = \frac{\tan s + \tan t}{1-\tan s \tan t} \)

\( c^2=a^2+b^2-2ab\cos\theta \text{ (Law of cosines) } \)

\( \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \text{ (Law of sines) } \)

\( a\cos\theta + b\sin\theta = \sqrt{a^2+b^2}\cos(\theta-\tan^{-1}\frac ba) \)

\( \cos(-x)=\cos x \qquad \sin(-x)=-\sin x \)

\( \sin(\frac{\pi}{2} \pm x)=\cos x \qquad \cos(\frac{\pi}{2} \pm x)=\mp \cos x \)

\( \sin(\pi \pm x)=\mp \sin x \qquad \cos(\pi \pm x)=- \cos x \)

Trigonometric Integrals

\( \int\sin^2 x\,dx=\frac{x-\sin x \cos x}{2} = \int\frac{1-\cos 2x}{2}\,dx=\frac x2-\frac{\sin 2x}{4} \)

\( \int\cos^2 x\,dx=\frac{x+\sin x \cos x}{2} = \int\frac{1+\cos 2x}{2}\,dx=\frac x2+\frac{\sin 2x}{4} \)

\( \int\tan^2x\,dx=\tan x-x \)

\( \int\cot^2x\,dx=-\cot x-x \)

\( \int\sin^n\,dx=-\frac{\sin^{n-1}x\cos x}{n}+\frac{n-1}{n}\int\sin^{n-2}x\,dx \)

\( \int\cos^n\,dx=+\frac{\cos^{n-1}x\sin x}{n}+\frac{n-1}{n}\int\cos^{n-2}x\,dx \)

\( \int\tan^n\,dx=\frac{\tan^{n-1}x}{n-1}-\int\tan^{n-2}x\,dx \)

\( \int\tan x\,dx=-\ln|\cos x| \)

\( \int\cot x\,dx=+\ln|\sin x| \)

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