Skip to content

Single Variable

Derivative

A function of a real variable \(y=f(x)\) is differentiable at a point \(a\) of its domain, if its domain contains an open interval \(I\) containing \(a\), the limit:

\(L=\lim_{h\rightarrow0}\frac{f(a+h)-f(a)}{h}\)

L'Hôpital's Rule

Used to find the limit value of an expression

The limits of the numerator (分子) and denominator (分母) are both 0 or infinite.

Taylor's Formular

\(f(x)=\sum_{i=0}^\infty\frac{f^{i}(x_0)}{i!}(x-x_0)^i\)

\(e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+o(x^3)\)

Indefinite Integral

\(\int f(x)dx=F(x)+C\)

Definite Integral

The definition of definite integral is based on finding the area of a curved-sided trapezoid (曲边梯形). Therefore, the definition of definite integral is used to find the area, i.e. to get a number.

Newton Leibniz Formula

Given \(\int f(x)dx=\Phi(x)+C\)

\(\int_a^bf(x)dx=\Phi(b)-\Phi(a)\)

Multivariable

Double Integral

How to calculate?

  1. find the domain of each variable
  2. step by step, each step for a variable

z.B. \(\sigma\): y=cx, x=a, x=b, Ox. c>0, b>a>0, \(\iint_\sigma(x+y)d\sigma\)

evidently, \(a\leq x\leq b\), \(0\leq y\leq cx\)

\(\iint_\sigma(x+y)d\sigma\)

\(=\int_a^bdx\int_0^{cx}(x+y)dy\)

\(=\int_a^b[xy+\frac{y^2}{2}]_0^{cx}dx\)

\(=\int_a^b(\frac{2c+c^2}{2})x^2dx\)

\(=\frac{1}{6}(2c+c^2)(b^3-a^3)\)

If introduce new prarms u,v s.t. \(x=x(u,v)\), \(y=y(u,v)\)

then \(\iint_D f(x,y)dxdy=\iint_{D'}f[x(u,v),y(u,v)]|\frac{\partial(x,y)}{\partial(u,v)}|dudv\)

Jacobian \(J=|\frac{\partial(x,y)}{\partial(u,v)}|=x'_uy'_v-x'_vy'_u\)

Line Integral

\(\int_Cf(x,y)ds=\int_a^bf(x(t),y(t))\sqrt{x'(t)^2+y'(t)^2}dt\)

z.B. \(L\) is a part of unit circle in the first quadrant (第一象限), \(\int_Lxyds\)

Parametric equation of \(L\) is: \(x=\cos t\), \(y=\sin t\), \(0\leq t\leq\frac{\pi}{2}\)

Then \(\int_Lxyds=\int_0^{\frac{\pi}{2}}\cos t\sin t\sqrt{(-\sin t)^2+\cos^2t}dt\)

\(=\int_0^{\frac{\pi}{2}}\cos t\sin tdt=\frac{1}{2}\)

or

\(y=\sqrt{1-x^2}\), \(0\leq x\leq 1\)

Then \(\int_Lxyds=\int_0^1x\sqrt{1-x^2}\times\sqrt{1+\frac{x^2}{1-x^2}}dx\)

\(=\int_0^1xdx=\frac{1}{2}\)

Numerical Series

Convergence 收敛

Divergence 发散

p- series \(\sum_n\frac{1}{n^p}\): when p>1, convergent; when p<=1, divergent.

z.B. \(\sum_{n=1}^\infty\frac{1}{n(n+1)(n+2)}\)

\(a_n=\frac{1}{n(n+1)(n+2)}=\frac{n+2-n}{2n(n+1)(n+2)}\)

\(=\frac{1}{2n(n+1)}-\frac{1}{2(n+1)(n+2)}=b_n-b_{n+1}\)

\(\lim_{n\rightarrow\infty}b_n=\lim_{n\rightarrow\infty}\frac{1}{2n(n+1)}=0=b\)

\(\therefore \sum_{n=1}^\infty a_n=b_1-b=\frac{1}{4}\)