GNU General Public License v3.0 licensed. Source available on github.com/zifeo/EPFL.
Spring 2014: Analysis II
[TOC]
-
even (symetry y-axis)
$f(x)=f(-x)$ or odd (symetry origine)$f(x)=-f(-x)$ - (monotone) increasing or decreasing
-
composite function :
$g\cdot f(x)=g(f(x))$ - injective (one-to-one) : never takes twice the same value
- surjective (onto) : every vale in the codomain is hit at least once
- bijective : both one-to-one and onto, if bijective, an inverse function exist
- sequences
${a_n}_{n=1}^\infty$ converges or diverges - limit exists iff both sided limits exist.
-
continuity is when the limit exists at each point (caution on absolute sign, ++simplification does not change the discontinuties++)
- removable discontinuities (one point displaced, sided limits equivalent)
- jump discontinuities (sided limits differ)
- infinite discontinuities (one sided limit does not exist)
- removable discontinuities (one point displaced, sided limits equivalent)
-
asymptotes
- vertical
$f(x) \to \infty$ as$x \to a$ - horizontal
$f(x) \to a$ as$x \to \infty$ -
$\frac{ax-b}{x+c}$ has a h-asy at$a$ , a v-asy at$-c$ and a x-intercept at$x=\frac{b}{a}$ - slope of the oblique-asy is
$\lim_{x\to\infty} \frac{f(x)}{x}$
-
- vertical
$e^x = \lim_{n\to \infty} (1+\frac{x}{n})^n$
-
arithmetic series : converge to
$\sum_{n=1}^\infty n=\frac{n(n+1)}{2}$ -
geometric series : converge to
$\sum_{n=1}^\infty ar^{n-1}=\frac{a}{1-r}$ if$|r|< 1$ -
Riemann series :
$\sum_{n=1}^\infty \frac{1}{p^\alpha}$ converges if$\alpha > 1$ - sequences and series : if the series
$\sum a_n$ is convergent, then$\lim a_n=0$ . - finding the terms :
$S_n - S_o = a_n$ where$S_o$ is the one before$S_n$ . -
convergence tests
-
divergence test : if
$\lim a_n$ does not exist or is not equal to$0$ , then the series is divergent. -
limit test : (both positive) if
$\lim_{n\to \infty} \frac{a_n}{b_n}=c$ where$c$ is a finite number bigger than$0$ , then either both converge or diverge. - comparison test
-
alternating series test : if for all postive the next is smaller than the previous one and
$\lim_{n\to \infty} b_n=0$ , then the series converge. -
ratio test :
$\lim_{n\to \infty} |\frac{next}{previous}|$ gives less than$1$ , it is convergent, divergent if bigger than$1$ , else inconclusive. -
root test :
$\lim_{n\to \infty} \sqrt[n]{|a_n|}$ , same conclusion as ratio test. -
integral test : for continuous, positive, deacreasing function on
$[1,\infty)$ , the series is convergent iff$\int_1^\infty f(x)dx$ is convergent.
-
divergence test : if
-
absolute convergence : series
$\sum a_n$ is called absolutely convergent if$\sum |a_n|$ converges, it is conditionnaly convergent. -
power series :
$\sum_{n=0}^\infty c_n(x-a)^n$ is called centered at$a$ with radius of convergence$R$ and the series converges if$|x-a|< R$ (ratio test to find it) -
Taylor series (called Mclaurin series if centered at
$0$ ) :$\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$
-
derivatives
$f'(x)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$ -
critcal point :
$c$ such that$f'(c)=0$ ,$f'(c)=\infty$ (vertical tangent) or$f(c)$ does not exist -
closed interval method means looking at the critical numbers on
$[a,b]$ and at the endpoints of the interval -
extrema
$f'(x)=0$ -
$f''(x) > 0$ mimimum -
$f''(x) < 0$ maximum
-
-
concavity
- if
$\forall x \in I, f''(x) > 0$ , the function is concave upward on I :)- if
$\forall x \in I, f''(x) < 0$ , the function is concave downward on I :(
- if
- if
-
derivative of inverse function :
$(f^{-1})'(a)=\frac{1}{f'(f^{-1}(a))}$ or$(f^{-1})'(f(a))=\frac{1}{f'(a)}$ .
-
Intermediate value theorem :
$f$ continuous on$[a,b]$ and$f(a)\not =f(b)$ , let$N$ be a number between$f(a)$ and$f(b)$ , then there exist a$c$ such that$f(c)=N$ -
Extreme value theorem : if
$f$ continuous on$[a,b]$ , then$f$ attains an absolute maximum and an absolute minimum on some number in that interval. - Bolzano-Weierstress theorem : every bounded sequence has a convergent subsequence
-
Fermat's theorem : if
$f$ has a local extremum at$c$ and$f'(c)$ exists, then$f'(c)=0$ . -
Rolle's theorem : if
$f$ continuous on$[a,b]$ and differentiable on$(a,b)$ and$f(a)=f(b)$ , then there is a number$c$ in$(a,b)$ such that$f'(c)=0$ -
Mean value theorem : if
$f$ continuous on$[a,b]$ and differentiable on$(a,b)$ , then there is a number$c$ in$(a,b)$ such that$f'(c)=\frac{f(b)-f(a)}{b-a}$ -
Cauchy's mean value theorem : let
$f$ ,$g$ be continuous on$[a,b]$ and differentiable in$(a,b)$ , then$\frac{f'(c)}{g'(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}$ . -
De l'Hospital's rule : suppose
$f$ and$g$ are differentiable and$g'(x)\not = 0$ on an open interval$I$ that contains$a$ . If the limit$f$ and the limit$g$ go both to$0$ or$\infty$ as$x\to a$ , then$\lim_{x\to a} \frac{f'(x)}{g'(x)}$
-
fundamental theorem of calculus : for
$f:[a,b]\to \mathbb R$ , continuous,$\frac{d}{dx}[\int_a^x f(s)ds]=f(x)$ $\int_a^b f(x)dx=\int_a^b F'(x)dx=F(b)-F(a)$
- if
$f$ is monotone or continuous, then it is integrable -
substitution
$\int f(g(x))g'(x) dx= F(g(x))+c$ (always take the biggest one, play with$u=\sqrt{x}\Rightarrow 2udu=dx$ , caution on sign) -
even function
$\int_{-a}^a f(x)= 2 \int_0^a f(x)dx$ -
odd function
$\int_{-a}^a f(x) dx = 0$ -
polynomial division :
$\frac{x^3-2x^2-4}{x^2+x+3}=(x-3)+\frac{1}{x^2+x+3}$ -
partial fractions :
$\frac{1}{x^2+2x-3}=\frac{1}{(x+3)(x-1)}=\frac{A}{x+3}+\frac{B}{x-1}$ where$A(x-1)=1$ and$B(x+3)=1$ -
by part
$\int f' g dx = fg-\int fg'dx$ or$\int fg dx=Fg-\int Fg'dx$ (twice by part should be done in the same derivation way) -
sin and cos
$\int \sin^n (x) \cos^m (x) dx$ - if
$n$ /$m$ is odd use$\sin^2+cos^2=1$ and substitution - if
$n$ /$m$ are even use half angle formulas ($2\sin^2 x=1-\cos 2x$ and$2\cos^2 x =1+\cos 2x$ )
- if
- inproper integrals : when bounds go to infinity or to a limit, the integral behaves as a Riemann series, use limit to compute it (can converges or not)
-
areas between curves :
$A=\int_a^b |f(x)-g(x)|dx$ (caution if lines cross, better separate) -
volumes
$V=\int_a^b A(x)dx$ -
volumes of cylindrical shells :
$V=2\pi\int_a^b x f(x) dx$ -
arc length :
$L=\int_a^b \sqrt{1+(f')^2}dx$ -
surface areas :
$S= 2\pi \int_a^b f(x) \sqrt{1+(f')^2}dx$
- notation :
$\vec{AB}=\vec v = < x_2-x_1,y_2-y_1,z_2-z_1 > $ - length :
$L=|\vec v|=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$ -
$(x,y)$ denotes a point,$< x,y >$ denotes a vector - basis vector
$\hat i= < 1,0,0 >\quad \hat j=< 0,1,0 > \quad \hat k = < 0,0,1>$ use$\vec v = \sum_{i=1}^n v_i \hat e_i$ in$\mathbb R^n$ - unit vector : given
$\vec u$ , the unit vector is$\vec v = \frac{\vec u}{|\vec u|}$
- euclidian space :
$\mathbb R^n$ must follow right hand rule -
dot product :
$\vec a\cdot \vec b = \sum_{i=1}^n a_i b_i = |\vec a||\vec b|\cos \theta$ - if product is null, then perpendicular
- if
$\vec a\cdot\vec b= \pm |\vec a||\vec b|$ , then$\vec a=c \vec b$ - if product is less than
$0$ , then the angle is acute, else the angle is obtuse - behaves like the multiplication
-
direction angle :
$\cos \alpha_i=\frac{a_i}{|\vec a|}$ and$\sum_{i=1}^n \cos^2 \alpha_i=1$ - distances :
$|xy|=\sqrt{\sum_{i=1}^n (x_i-y_i)^2}$ -
sphere of radius
$R$ at$x^c=(x_1^c,ldots ,x_n^c)$ :$R^2=\sum_{i=1}^n (x_i-x_i^c)^2$ -
scalar projection :
$|\vec c|=\vec n_a \cdot \vec b$ with$\vec n_a=\frac{\vec a}{|\vec a|}$ -
vector projection :
$\vec c = (\vec n_a \cdot \vec b)\vec n_a$ with$\vec n_a=\frac{\vec a}{|\vec a|}$ -
cross product :
$\vec a \times \vec b = < a_2b_3-a_3b_2, b_1a_3-a_1b_3, a_1b_2-a_2b_1$ - only
$\mathbb R^3$ - product perpendicular to
$a$ and$b$ - $\vec a \times \vec b = \begin{vmatrix} \hat i & \hat j & \hat k \ a_1&a_2&a_3 \ b_1&b_2&b_3 \end{vmatrix}$
$|\vec a \times \vec b|=|\vec a||\vec b|\sin \theta$ - if product is null,
$\vec a$ and$\vec b$ are parallel - the area of the parallelogram spanned is the lenght of the product
- behaves like the multiplication (but not symmetric nor comutative)
$\vec a \cdot (\vec b \times \vec c) = (\vec a \times \vec b)\cdot \vec c $ $\vec a \times (\vec b \times \vec c) = (\vec a \cdot \vec c)\vec b -(\vec a \cdot \vec b)\vec c$ $\vec a \times \vec b = -\vec b \times \vec a$
- only
- triple product (volume) : $\vec a \cdot (\vec b \times \vec c) = \begin{vmatrix} a_1&a_2&a_3 \ b_1&b_2&b_3\c_1&c_2&c_3 \end{vmatrix}$
-
lines :
$\vec r(t)=\vec r_0 + t \vec v$ or$x(t)=x_0+at\quad y(t)=y_0+bt\quad z(t)=z_0+ct$ and$\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}$ -
line segment :
$\vec r(t)=(1-t)\vec r_0+ t\vec r_1$ -
planes :
$\vec n \cdot (\vec r- \vec r_0)=0$ with$\vec r = < x(t),y(t),z(t) >$ and$\vec n = < a,b,c > $ $a(x-x_0)+b(y-y_0)+c(z-z_0)=0$ - can be simplified to
$ax+by+cz+d=0$ by expanding - normal distance with a point :
$D=\frac{|ax_1+by_1+cz_1+d|}{\vec n}$ with$(x_1, y_1, z_1)=P$
- notation :
$\vec r(t)=f(t)\hat i +g(t)\hat j+h(t)\hat k$ - continous if
$\lim_{t\to a} \vec(t)=\vec r(a)$ - derivative
$\vec r'(t)=\lim_{\alpha \to 0}\frac{\vec r(t+\alpha)-\vec r(t)}{\alpha}$ -
tangent unit vector
$\tau(t)=\frac{r'(t)}{|r'(t)|}$ - integral
$\int_a^b \vec r(t) dt=\vec R(b)-\vec R(a)$ -
arc length
$L=\int_a^b \sqrt{(f')^2+(g')^2+(h')^2}dt=\int_a^b |\vec r'(t)|dt$ - curve is smooth if
$r'(t)$ is continuous and$r'(t)\not = 0$ -
curvature
$\kappa (s)=\frac{1}{R}=|\frac{d\tau}{ds}|=\frac{|\tau'(t)|}{|\vec r'(t)|}=\frac{|\vec r' \times \vec r''|}{|\vec r'|^3}=\frac{|f''(x)|}{(1+|f'(x)|^2)^{3/2}}$ -
normal vector
$\vec N(t)=\frac{\tau'(t)}{|\tau'(t)|}$ -
binormal vector
$\vec B(t) = \vec T \times \vec N$ -
3 planes : normal (
$T$ ), osculating ($B$ ), recifying ($N$ ) - level(set) curves : shape of the function giving a constant
- functions of n variables :
$f:\mathbb R^n \to \mathbb R$ - limits must exist and be unique (same one) for ALL paths
- continuous if
$\lim_{(x,y)\to (a,b)}f(x,y)=f(a,b)$ - can use substituion in limits (polar coordinates)
- continuous if
-
partial derivative :
$f_x(x,y)=\frac{\partial f}{\partial x}\quad f_y(x,y)=\frac{\partial f}{\partial y}$ -
directional derivative :
$D_{\vec v} f=\lim_{h\to 0}\frac{f(x+ah,y+bh)-f(x,y)}{h}=\nabla f \cdot \vec v$ where$\vec v = < a,b >$ and$|\vec v|=1$ -
gradient :
$\nabla f=< \frac{\partial f}{\partial x},\frac{\partial f}{\partial y} >$ -
tangent plane/linear approximation :
$l(\vec x)=f(\vec x_0)+\nabla f(\vec x_0)\cdot (\vec x-\vec x_0)$ - differentiable at
$\vec x_0$ if$l(\vec x)$ exists such that$f(\vec x)=l(\vec x)+r(\vec x)$ and$\lim_{\vec x\to \vec x_0} \frac{r(\vec x)}{|\vec x-\vec x_0|}=0$ or if partial derivatives are contineous -
direction of maximal increase
$\vec v^+=\frac{\nabla f}{|\nabla f|}$ -
direction of maximal decrease
$\vec v^-=-\frac{\nabla f}{|\nabla f|}$ -
direction of no chance
$\vec v$ is perpendicular to$\nabla f$ -
normal vector/plan :
$\vec n = < \nabla f, -1 >$ or$\vec n = <-\nabla f, 1 >$
- if
$f_{xy}$ and$f_{yx}$ are continuous then$f_{xy}=f_{yx}$ -
second derivative :
$D_{\vec w \vec v}f=\vec w^T \vec H \vec v$ -
Hessian : $\vec H = \begin{bmatrix} f_{xx}&f_{xy}\ f_{yx}&f_{yy} \end{bmatrix}$ or
$\vec H_{ij}=\frac{\partial^2 f}{\partial x_i \partial x_j}$ -
quadratic approximation :
$q(\vec x)=f(\vec x_0)+\nabla f(\vec x_0)\cdot (\vec x-\vec x_0)+\frac{1}{2}\vec (x-\vec x_0)^T \vec H (\vec x_0)(\vec x-\vec x_0)$ -
extrema
$\nabla f(\vec x)=0$ - if
$f_{xx} > 0$ and$|H|=f_{xx}f_{yy}-f_{xy}^2 > 0$ , it is a minimum - if
$f_{xx} < 0$ and$|H|=f_{xx}f_{yy}-f_{xy}^2 > 0$ , it is a maximum - if
$|H|=f_{xx}f_{yy}-f_{xy}^2 < 0$ , it is a saddle point - do not forget to check if boundaries (lines, circles) are smaller/bigger than critical points
- if
-
eigenvalues
$Q(\vec h)=\frac{1}{2}\vec h^T \vec H \vec h$ -
$Q(h) > 0$ if all eigenvalues of$\vec H$ are positive -
$Q(h) < 0$ if all eigenvalues of$\vec H$ are negative - if all eigenvalues are positive, it is a local minima
- if all eigenvalues are negative, it is a local maxima
- if eigenvalues change sign, it is a saddle point
-
-
$C^1$ is the class of functions for which$\nabla f$ exists and is continuous -
$C^2$ is the class of functions for which$\vec H$ exists and is symmetric -
Taylor polynomial of order
$2$ :$T_{x_0}^1(\vec x)=f(\vec x_0)+(\vec x - \vec x_0)\cdot \nabla f(\vec x_0)+\frac{1}{2}(\vec x - \vec x_0)^T \vec H (\vec x - \vec x_0)$ - Taylor
$T_{x_0}^j$ exist if$C^j$ exist :$T_{x_0}^j(\vec x)=\sum_{k=0}^j\frac{1}{k!}((\vec x-\vec x_0)\cdot\nabla)^k f(\vec x_0)$ -
chain rule :
$\frac{df}{dt}=\nabla f \cdot \vec x'$ and$\frac{df}{dt_i}=\nabla\cdot \frac{d\vec x}{dt_i}$ -
implicit function theorem :
$f'(x)=-\frac{f_x}{f_y}$ -
constrained optimization : find
$\vec x_0$ such that$\nabla f=\lambda \nabla g$ knowing$g(x,y)=k$ - Lagrange multiplier
$\lambda$ - multiple constrained
$\nabla f = \sum_{i=1}^k \mu_i \nabla g_i$ with$i=1\ldots k \quad g_i(x)=c_i$
- Lagrange multiplier
-
Fubine theorem : if continuous,
$\int_a^b \int_c^d f(x,y) dx dy=\int_c^d \int_a^b f(x,y) dy dx$ - variable separation :
$f(x,y)=g(x)h(y)\quad \int_a^b \int_c^d f(x,y)dx dy=\int_a^b g(x)dx \int_c^d h(y)dy$ -
variable bounds :
$V=\int_a^b \int_{f(x)}^{g(x)} f(x,y) dx dy$ -
$(x,y)=[a,b]\times [f(x), g(x)]$ :$\int_{\mathbb D} h(x,y) dx dy = \int_a^b \int_{f(x)}^{g(x)} h(x,y)dxdy$ -
$(x,y)=[f(y), g(y)]\times [c,d]$ :$\int_{\mathbb D} h(x,y) dx dy = \int_c^d \int_{f(y)}^{g(y)} h(x,y)dxdy$ $\int_{\mathbb D} f(\vec x)d\vec x=\int_{\mathbb D_1} f(\vec x)d\vec x+\int_{\mathbb D_2} f(\vec x)d\vec x$ $\int_E f(\vec x)d \vec x=\int_a^b \int_{g_1(x)}^{g_2(x)} \int_{u_1(x,y)}^{u_2(x,y)} f(x,y,z) dz dy dx$ - caution on boundaries :
$x=\sqrt{4-y^2}$ is only a half circle - caution on axes (y-axis first ?)
-
surface area :
$S=\int_{\mathbb D} \sqrt{1+(f_x)^2+(f_y)^2}d\mathbb A$ - Jacobian : $J=|\frac{\partial(x,y)}{\partial(u,v)}|=\begin{vmatrix} \frac{dx}{du}&\frac{dx}{dv} \ \frac{dy}{du}&\frac{dy}{dv}\end{vmatrix}$
-
substitution :
$\int_{\mathbb D}f(\vec x) d\vec x = \int_S f(T(\vec u)) J d\vec u$ - polar coordinates :
$x = r \cos \theta\quad y = r\sin\theta\quad J=r$ and$\int_{\mathbb D}f(\vec x)d\vec x = \int_a^b \int_{g(\theta)}^{f(\theta)} f(r,\theta)r dr d\theta$ - cylindrial coordinates :
$x = r \cos \theta\quad y = r\sin\theta\quad z=z \quad J=r$ - spherical coordinates :
$x = \rho \sin \phi \cos \theta\quad y = \rho \sin\phi\sin\theta\quad z=\rho\cos\phi \quad J=\rho^2\sin \phi$ where$0 \le \theta \le 2\pi$ and$0\le \phi\le\pi$ (starting on the positive y-axis side) - affine transformations : (exemple : barycentric coordinates)
$\vec x = \vec v_1 \beta_1 + \vec v_2 \beta_2 \quad 0 \le \beta_i \le 1 \quad \beta_1+\beta_2=1$ - rotation matrix : $A=\begin{pmatrix} \cos \theta && -\sin \theta \ \sin \theta && \cos \theta \end{pmatrix}$
- polar coordinates :
-
modeling :
$P(0)=P_0\quad P(t)=P_0 e^{kt}$ -
model for world population :
$P'=kP(1-\frac{P}{M})$ -
critical points & equilibrium points
$\frac{dp}{dt}=0$ - direction fields :
$f(v,t)=v'$ -
separable equations :
$\frac{du}{dx}=g(x)f(u) \Rightarrow \int\frac{1}{f(u)}dx=\int g(x)dx$ -
linear equations : all continuous
$\frac{du}{dx}+P(x)u=Q(x) \Rightarrow u(x)=\frac{1}{I(x)}\int I(x)Q(x)dx +C$ where$I=e^{\int P(x) dx}$ -
Picard's theorem : if
$f(u,t)$ is continuous in$t$ and Lipschitz in$u$ (means L exist such that$|f(u,t)-f(v,t)|\le L|u-v|$ ), then a unique solution exist$u(t)=u(0)+\int_a^b f(u(s), s) ds$ -
stability : if
$|u(0)-u_0|\le \epsilon$ , it is stable if there exists$\gamma$ such that$|u(t)-u_0| < \gamma\forall t$ -
linear systems :
$\frac{d\vec u}{dt}=\vec A \vec u$ where $\vec A = \begin{pmatrix}a&b \ c&d\end{pmatrix}$ (can be the Jacobian)- if eigenvalues are positive, it is unstable
- if eigenvalues are negative, it is stable
- if the product of eigenvalues is negative, it is a saddle point
-
predator prey models : prey
$R'=kR$ and predator$W'=-rW$ (predator has a negative slope) -
predator prey solutions :
$R'=kR-aRW\quad W'=-rW+bRW \Rightarrow R=\frac{r}{b}\quad W=\frac{k}{a}$ - prey over predator is non linear :
$k\ln W-aW=-r\ln R +bR+C$ -
money rate change model :
$x'=\frac{M-x}{M}$
$P(x)\frac{d^2u}{dx^2}+Q(x)\frac{du}{dx}+R(x)u=G(x)$ - if
$G(x)=0$ , it is homogeneous - if
$u_1(x)$ and$u_2(x)$ are linearly independent, then all solutions are expressed as$u(x)=c_1u_1(x)+c_2u_2(x)$ - take the homogeneous, guess the solution
$u(x)=\alpha e^{rx}$ , find the characteristic equation$ar^2+br+c=0$ - if delta is postive,
$r_1$ and$r_2$ are real,$u(x)=c_1 e^{r_1x}+c_2 e^{r_2x}$ - if delta is negative, solutions are complex,
$u(x)=c_1 e^{r_1x}+c_2 e^{r_2x}=e^{\alpha x}(c_1\cos \beta x+c_2 \sin \beta x)$ - if delta is null, solutions are the same and real
$u(x)=c_1 e^{rx}+c_2 x e^{rx}$ - given a particular solution and the general homogeneous then the general solution inhomogeneous is
$u(x)=u_p(x)+u_h(x)$ - entered the guessed (polynomial, exponential, trigonometric) general inhomogeneous solution in the equation and solve
$x^3+y^3=(x+y)(x^2-xy+y^2)$ $\sqrt{b}-\sqrt{a}=\frac{b-a}{\sqrt{b}+\sqrt{a}}$ $\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\cdots+\frac{1}{2^n}=1-\frac{1}{2^n}$ -
$\sqrt{1+x}=1+\frac{x}{2}$ when$x << 1$
$\sin x=\frac{e^{ix}-e^{-ix}}{2i}$ $\cos x=\frac{e^{ix}+e^{-ix}}{2}$ $2\sin^2 t = 1-\cos 2t$ $2\cos^2 x = 1+\cos 2x$ $\sin 2x = 2\sin x \cos x$ $\sin(a+b)=\sin a\cos b+\cos a\sin b$ $\cos(a+b)=\cos a\cos b-\sin a\sin b$ $\tan^2 x +1=\frac{1}{\cos^2 x}$ $\cot^2 x +1=\frac{1}{\sin^2 x}$ $\sin 0 = \frac{1}{2}\sqrt{0} = \cos \pi/2 = 0$ -
$\sin \pi/6 = \frac{1}{2}\sqrt{1} = \cos \pi/3 = \frac{1}{2}$ $\sin \pi/4 =\frac{1}{2}\sqrt{2} = \cos \pi/4 =\frac{\sqrt{2}}{2}$
$\sin \pi/3 = \frac{1}{2}\sqrt{3} = \cos \pi/6=\frac{\sqrt{3}}{2}$ $\sin \pi/2 = \frac{1}{2}\sqrt{4} = \cos 0 = 1$
- 3D
- Ellipse
$(\frac{x}{a})^2+(\frac{y}{b})^2+(\frac{z}{c})^2=1$ - Hyperboloid 1 sheet
$(\frac{x}{a})^2+(\frac{y}{b})^2-(\frac{z}{c})^2=1$ (nuclear reactor) - Hyperboloid 2 sheet
$(\frac{x}{a})^2-(\frac{y}{b})^2-(\frac{z}{c})^2=1$ (two bowl) - Cone
$(\frac{x}{a})^2+(\frac{y}{b})^2-(\frac{z}{c})^2=0$ - Elliptic paraboloid
$(\frac{x}{a})^2+(\frac{y}{b})^2=\frac{z}{c}$ (bowl) - Hyperbolic Paraboloid
$(\frac{x}{a})^2-(\frac{y}{b})^2=\frac{z}{c}$ (saddle)
- Ellipse
- 2D
- Ellipse
$(\frac{x}{a})^2+(\frac{y}{b})^2=1$ - Hyperboles
$(\frac{x}{a})^2-(\frac{y}{b})^2=1$
- Ellipse
$\tan'x=\sec^2x$ $\csc'x = -\csc x\cot x$ $\sec'x = \sec x\tan x$ $\cot'x = -\csc^2x$ $\sin'^{-1}x = \frac{1}{\sqrt{1-x^2}}$ $\cos'^{-1}x = -\frac{1}{\sqrt{1-x^2}}$ $\tan'^{-1}x = \frac{1}{1+x^2}$ $\csc'^{-1}x = -\frac{1}{x\sqrt{x^2-1}}$ $\sec'^{-1}x = \frac{1}{x\sqrt{x^2-1}}$ $\cot'^{-1}x = -\frac{1}{1+x^2}$
$\int \ln x dx = x \ln x - x$ $\int x \ln xdx= \frac{1}{4}x^2(2\ln x -1)$ $\int \frac{1}{x\log x}dx=\log (\log x)$ $\int \frac{1}{x\log^2 x}dx=-\frac{1}{\log x}$ $\int \frac{1}{x}dx = \ln |x|$ $\int a^x dx = \frac{1}{\ln a} a^x$ $\int \tan x dx = \ln |\sec x| $ $\int \frac{a}{a^2+x^2}dx = \tan^{-1}\frac{x}{a}$ $\int \frac{a}{a^2-x^2}dx = \frac{1}{2}\ln\left|\frac{x+a}{x-a}\right|$ $\int \frac{1}{\sqrt{a^2-x^2}} dx = \sin^{-1} \frac{x}{a}$ $\int \frac{1}{\sqrt{x^2-a^2}} dx = \cosh^{-1} \frac{x}{a}$ $\int \frac{1}{\sqrt{x^2+a^2}} dx = \sinh^{-1} \frac{x}{a}$ $\int \sin^{-1} x dx=\sqrt{1-x^2}+x\sin^{-1} x$ $\int \cos^{-1} x dx=-\sqrt{1-x^2}+x\cos^{-1} x$ $\int \tan^{-1} x dx=-\frac{1}{2}\ln(x^2+1)+x\tan^{-1} x$ $\int \sin x \cos xdx = -\frac{1}{2}\cos^2 x$
$\frac{1}{1-ax}=\sum_{n=0}^\infty (ax)^n=1+ax+ax^2+ax^3+\cdots$ $\frac{1}{1+x}=\sum_{n=0}^\infty (-x)^n=1-x+x^2-x^3+\cdots$ $\frac{1}{(1-x)^2}=\sum_{n=0}^\infty (n+1)x^n=1+2x+3x^2+4x^3+\cdots$ $\sin x=\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$ $\cos x=\sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots$ $\tan^{-1} x=\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+1}=x-\frac{x^3}{3}+\frac{x^5}{5}-\cdots$ $ln(1+x)=\sum_{n=1}^\infty (-1)^n \frac{x^n}{n}=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots$