微积分

学习地址

• 微积分的本质 by 3blue1brown
• math is fun: Calculus

Current Progress: Differential Equations
中断，先去听 3blue1brown 的课，或者 mooc

todo todo

1. Magic Hexagon for Trig Identities
2. Factorial !
3. Combinations and Permutations
4. Gamma Function

Confusions

1. Did it just drop out of the sky?

$$f(x)=c_0+c_1(x-a)+c_2(x-a{)}^2+c_3(x-a{)}^3+\dots$$

2. confuse all pages: Fourier Series
3. todo todo

Derivative

We write dx instead of "Δx heads towards 0".

$\frac{\mathrm{d}\sin (x)}{\mathrm{d}x}$ and $si{n}^{\prime }(x)$ both mean "The derivative of sin(x)"

$$\frac{\mathrm{d}\sin (x)}{\mathrm{d}x}=cos(x)$$

or

$$si{n}^{\prime }(x)=cos(x)$$

Derivative Rules TABLE

Common Functions	Function	Derivative
Constant	$c$	$0$
Line	$x$	$1$
	$ax$	$a$
Square	$x^2$	$2x$
Square Root	$\sqrt{x}$	$\frac{1}{2}{x}^{-\frac{1}{2}}$
Exponential	$e^x$	$e^x$
	$ax$	$\ln (a)ax$
Logarithms	$ln(x)$	$1/x$
	${\log }_a(x)$	$1/(xln(a))$
Trigonometry (x is in radians)	$sin(x)$	$cos(x)$
	$cos(x)$	$-sin(x)$
	$tan(x)$	$sec2(x)$
Inverse Trigonometry	$si{n}^{-1}(x)$	$1/\sqrt{(}1-x^2)$
	$co{s}^{-1}(x)$	$-1/\sqrt{(}1-x^2)$
	$ta{n}^{-1}(x)$	$1/(1+x^2)$

Rules	Function	Derivative
Multiplication by constant	$cf$	$c{f}^{\prime }$
Power Rule	$x^n$	$n{x}^{n-1}$
Sum Rule	$f+g$	$f^{\prime }+g^{\prime }$
Difference Rule	$f-g$	$f^{\prime }-g^{\prime }$
Product Rule	$fg$	$f{g}^{\prime }+f^{\prime }g$
Quotient Rule	$f/g$	$\frac{f^{\prime }g-g^{\prime }f}{g^2}$
Reciprocal Rule	$1/f$	$-f^{\prime }/f^2$

Chain Rule

Notation	Chain Rule
Using $\frac{\mathrm{d}}{\mathrm{d}x}$	$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}y}{\mathrm{d}u}\frac{\mathrm{d}u}{\mathrm{d}x}$
Using ${}^{\prime }$ (meaning derivative of)	$f(g(x))=f^{\prime }(g(x))g^{\prime }(x)$
As "Composition of Functions"	$f°g=(f^{\prime }°g)\times g^{\prime }$

Composition of Functions

The result of $f()$ is sent through $g()$
It is written: $(g°f)(x)$
Which means: $g(f(x))$

L'Hôpital's Rule

L'Hôpital

L'Hôpital is pronounced "lopital". He was a French mathematician from the 1600s.

$$\underset{x\to c}{lim}\frac{f(x)}{g(x)}=\underset{x\to c}{lim}\frac{f^{\prime }(x)}{g^{\prime }(x)}$$

Concave Upward and Downward

Derivatives:

• When the slope continually increases, the function is concave upward.
• When the slope continually decreases, the function is concave downward.

second derivative:

• When the second derivative is positive, the function is concave upward.
• When the second derivative is negative, the function is concave downward.

Example

the function $x^2$

Slope Stays the Same

Example: y = 2x + 1
2x + 1 is a straight line.

It is concave upward.
It is also concave downward.

It is not strictly concave upward.
And it is not strictly concave downward.

Differentiable

Differentiable means that the derivative exists

Taylor Series

$$f(x)=c_0+c_1(x-a)+c_2(x-a{)}^2+c_3(x-a{)}^3+\dots$$$$f(x)=f(a)+\frac{f^{\prime }(a)}{1!}(x-a)+\frac{f^{″}(a)}{2!}(x-a{)}^2+\frac{f^{‴}(a)}{3!}(x-a{)}^3+\dots$$

Maclaurin Series

A Maclaurin Series is a Taylor Series where a=0, so all the examples we have been using so far can also be called Maclaurin Series.

	Taylor Series expansion	As Sigma Notation
$e^x$	$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots$	$\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}$
$sinx$	$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots$	$\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{(2n+1)!}{x}^{2n+1}$
$cosx$	$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots$	$\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{(2n)!}{x}^{2n}$
$\frac{1}{1-x}$ for $|x|<1$	$\frac{1}{1-x}=1+x+x^2+x^3+x^4+\cdots$	$\sum _{n=0}^{\mathrm{\infty }}{x}^n$

Integration

Integration Rules TABLE

Common Functions	Function	Integral
Constant	$\int adx$	ax + C$
Variable	$\int xdx$	$x^2/2+C$
Square	$\int x^{2}dx$	$x^3/3+C$
Reciprocal	$\int (1/x)dx$	$ln|x|+C$
Exponential	$\int e^{x}dx$	$e^x+C$
	$\int a^{x}dx$	$a^x/ln(a)+C$
	$\int ln(x)dx$	$xln(x)-x+C$
Trigonometry (x in radians)	$\int cos(x)dx$	$sin(x)+C$
	$\int sin(x)dx$	$-cos(x)+C$
	$\int se{c}^2(x)dx$	$tan(x)+C$

Rules	Function	Integral
Multiplication by constant	$\int cf(x)dx$	$c\int f(x)dx$
Power Rule (n≠−1)	$\int xndx$	$\frac{x^{n+1}}{n+1}+C$
Sum Rule	$\int (f+g)dx$	$\int fdx+\int gdx$
Difference Rule	$\int (f-g)dx$	$\int fdx-\int gdx$

Integration by Parts

$$\int uvdx=u\int vdx-\int u^{\prime }(\int vdx)dx$$

how to say $\int uvdx$ : (u integral v) minus integral of (derivative u, integral v)

come from where?

1. $(uv{)}^{\prime }=u{v}^{\prime }+u^{\prime }v$
2. $\int (uv{)}^{\prime }dx=\int u{v}^{\prime }dx+\int u^{\prime }vdx$
3. $uv=\int u{v}^{\prime }dx+\int u^{\prime }vdx$
4. $\int u{v}^{\prime }dx=uv-\int u^{\prime }vdx$
5. replace v' with w and v with $\int wdx$

$$\int uwdx=u\int wdx-\int u^{\prime }(\int wdx)dx$$

Choose u and v carefully

A helpful rule of thumb is I LATE. Choose u based on which of these comes first:

• I: Inverse trigonometric functions such as $si{n}^{-1}(x)$ , $co{s}^{-1}(x)$ , $ta{n}^{-1}(x)$
• L: Logarithmic functions such as $ln(x)$ , $log(x)$
• A: Algebraic functions such as $x^2$ , $x^3$
• T: Trigonometric functions such as $sin(x)$ , $cos(x)$ , $tan(x)$
• E: Exponential functions such as $e^x$ , $3^x$

Definite Integrals

$$\underset{a}{\overset{b}{\int }}uvdx=[u\int vdx-\int u^{\prime }(\int vdx)dx{]}_a^b$$$$\underset{a}{\overset{b}{\int }}uvdx=[u\int vdx{]}_a^b-\underset{a}{\overset{b}{\int }}{u}^{\prime }(\int vdx)dx$$

Integration by Substitution

Notation

It is usual to show the indefinite integral (without the +C) inside square brackets, with the limits a and b after, like this:
$\underset{1}{\overset{2}{\int }}2xdx={\left[x^2\right]}_1^2$

Properties

$$\underset{a}{\overset{b}{\int }}f(x)+g(x)dx=\underset{a}{\overset{b}{\int }}f(x)dx+\underset{a}{\overset{b}{\int }}g(x)dx$$$$\underset{a}{\overset{b}{\int }}f(x)dx=-\underset{b}{\overset{a}{\int }}f(x)dx$$$$\underset{a}{\overset{a}{\int }}f(x)dx=0$$$$\underset{a}{\overset{b}{\int }}f(x)dx=\underset{a}{\overset{c}{\int }}f(x)dx+\underset{c}{\overset{b}{\int }}f(x)dx$$

Definite vs Indefinite Integrals

The Definite Integral between a and b is the Indefinite Integral at b minus the Indefinite Integral at a.

Arc Length

Using Calculus to find the length of a curve.

The Arc Length Formula

$S=\underset{a}{\overset{b}{\int }}\sqrt{1+(f^{\prime }(x){)}^2}dx$

Integral Approximations

We can estimate the area under a curve by slicing a function up. There are many ways of finding the area of each slice such as:

• Left Rectangular Approximation Method (LRAM)
• Right Rectangular Approximation Method (RRAM)
• Midpoint Rectangular Approximation Method (MRAM)
• Trapezoidal Rule
• Simpson's Rule

Riemann Sums

the previous 4 methods are also called Riemann Sums after the mathematician Bernhard Riemann.

Maximum Error

• For Midpoint: $\left|E\right|=\frac{K(b-a{)}^3}{24n^2}$
• For Trapezoidal: $\left|E\right|=\frac{K(b-a{)}^3}{12n^2}$
• For Simpson's: $\left|E\right|=\frac{M(b-a{)}^5}{180n^4}$

where

• |E| is the absolute value of the maximum error (could be plus or minus)
• a is the start of the interval
• b is the end of the interval
• n is the number of slices
• K is the greatest second derivative over the interval.
• M is the greatest fourth derivative over the interval.
  (By "greatest" we mean the maximum absolute value.)

Fourier Series

confusion

Differential Equations

Compound Interest

where

• FV = Future Value
• PV = Present Value
• r = annual interest rate
• n = number of periods

Effective Annual Rate = (1+(r/n))n − 1

Where:

• r = Nominal Rate (the rate they mention)
• n = number of periods that are compounded (example: for monthly n=12)

APR means "Annual Percentage Rate": it shows how much you will actually be paying for the year (including compounding, fees, etc).

Examples

Example 1: "1% per month" actually works out to be 12.683% APR (if no fees).

Example 2: "6% interest with monthly compounding" works out to be 6.168% APR (if no fees).

confusion