Exponential Function Series Definition

An alternative definition of the exponential function through infinite series

Motivation My previous post about the definition of the exponential function has provided no connection between a well-known characterization (or an alternative definition) of the exponential function: $$\exp(s) = \lim_{n\to\infty} \sum_{k=1}^n \frac{s^k}{k!}.$$ The term to be summed is simpler than the one in the binomial expansion of $(1 + s / n)^n$. Solution We want this sum to be as small as possible as $n \to \infty$. $$\sum_{k=2}^n \left( 1 - \frac{n \cdot \dots \cdot (n + 1 - k)} {\underbrace{n \cdot \dots \cdot n}_{n^k}} \right) \, \frac{x^k}{k! [Read More]

Exponential Function Product Rule

A first definition of exponential and logarithmic functions

Motivation from compound interest Increase the number of times that the interest is compounded each year ($n$), so as to increase the final amount of money ($A$) $$A = P \left(1 + \frac{r}{n}\right)^{nt}.$$ $t$ and $r$ are the number of years and the interest rate per annum. As $n$ becomes large, we can approximate the amount by $$A = P e^{rt}.$$ The value of $A$ in the above formula is the amount compounded continuously. [Read More]

Existence of Four Triangle Centers

A vector proof

Settings Let $P$ be an arbitrary reference point. $\triangle ABC$ be a triangle. $\vec{a} = \overrightarrow{PA}$, $\vec{b} = \overrightarrow{PB}$, $\vec{c} = \overrightarrow{PC}$ Remark: $O$ is reserved for circumcenter. symbol name meaning $G$ centroid center of gravity $H$ orthocenter three “heights” are concurrent $I$ incenter center of inscribed circle $O$ circumcenter center of circumscribed circle Centroid Verify that $(\vec{a} + \vec{b} + \vec{c})/3$ satisfy the constraints. Orthocenter Let $H$ be the point of intersection of two altitudes $AA_H$ and $BB_H$ $\vec{h} = \overrightarrow{PH}$ \begin{align} (\vec{h} - \vec{a}) \cdot (\vec{b} - \vec{c}) &= 0 \\ (\vec{h} - \vec{b}) \cdot (\vec{c} - \vec{a}) &= 0 \end{align} Add these two equations together. [Read More]

Dot Products

My complicated derivation

Preface There’s a simpler derivation using the geometric defintion linearity from the definition orthogonality of the canonical basic vectors $\vec{i}$, $\vec{j}$ and $\vec{k}$. If I had read that, I wouldn’t have type this document in LaTeX. The following content was transcribed from a PDF file that I made three days ago. Content Recall: $\sum_{i=m}^{n} a_{i}=a_{m}+a_{m+1}+a_{m+2}+\cdots +a_{n-1}+a_{n},$ where $m$, $n$ are integers. Example: $\sum_{i=3}^6 i^2 = 3^2 + 4^2 + 5^2 + 6^2 = 86$ [Read More]

Functions for Arts

Summary of my two-day work

my HTML slides You may view the source code of this blog to see the source files. Goal To provide an introduction to the formal definition of functions for arts stream students. I aim to minimize the calculations use daily-life examples favor pictures over text. IT skills used Inkscape C-S-a for alignment. C-S-f for boundary and colors, and line patterns (i.e. arrows). C-S-d for document size. clipping process: prepare the object to be clipped. [Read More]

Distance From a Line

TikZ figures for a derivation from dot product

Background A secondary school student posted a coordinate geometry question on a Discord homework help server. A helper replied to that question with an illustration of the Perpendicular Distance Formula $$ d((x_1,y_1), L) = \frac{\lvert Ax_1 + By_1 + C \rvert}{\sqrt{A^2 + B^2}}, $$ where $L$ is the line $Ax + By + C = 0$ without proof. Goal To provide an illustrated derivation of this formula. Recall A basic property of the dot product: the “algebraic definition” is equivalent to the “geometric definition”. [Read More]

Elementary Log Inequality

TikZ figures for some graphical proofs of this inequality

Goal Show that $$ \ln (x) \le x - 1 $$ for all $x > 0$. Proof by inverse functions Think about their reflection along the line $y = x$ (i.e. their inverse function). Then we get $$e^x \ge x + 1$$ for all $x \in \Bbb{R}$, which is obviously true. Proof by definite integrals Here’s a second proof using definite integrals. Case 1: x > 1 The rectangle $[1,x] \times [0,1]$ contains the region under the graph of the reciprocal $y = 1/x$ in the domain $[1,x]$. [Read More]

A Plot for Euler's Constant

My first PGF Plot for step functions

I firsted tried with foreach, but that would create so much paths. I found them difficult to operate on later, for example, with PGF plots library fillbetween. const plot is a better solution. \documentclass{standalone} \usepackage{pgfplots} \pgfplotsset{compat=1.18} \usetikzlibrary{patterns} \usepgfplotslibrary{fillbetween} \begin{document} \newcommand\myN{8} \pgfplotsset{ axis lines=center, legend style={at={(1,1)},anchor=north east,fill=none}, title style={at={(0.5,1.05)}}, every axis x label/.style={ at={(ticklabel* cs:1)}, anchor=west, }, every axis y label/.style={ at={(ticklabel* cs:1)}, anchor=south, }, } \begin{tikzpicture} \begin{axis}[ title={sum of hatched region converges to Euler's constant}, xlabel={$x$}, ylabel={$y$}, xmin=0, xmax={\the\numexpr\myN+2}, ymin=0, ymax=1. [Read More]

Simpler Diagram for Trigonometric Functions With Unit Circle

Background In my previous post about unit circle and trigonometric functions, I included a graph with three trigonometry functions. I’m quite satisfied with my TikZ picture. Problem Unluckily, a secondary school student found that my diagram was too complicated. Solution \documentclass[tikz,border=2pt]{standalone} \usepackage{pgfplots} \pgfplotsset{compat=1.18} \usetikzlibrary{calc} \newcommand\mytheta{110} %angle theta \begin{document} \begin{tikzpicture}[scale=2] \coordinate[label=left:$O$] (O) at (0,0); \coordinate[label=above left:${A = (\cos\theta, \sin\theta)}$] (A) at (\mytheta:1); \coordinate[label=below left:${B = (\cos(-\theta), \sin(-\theta))}$] (B) at (-\mytheta:1); \coordinate[label=right:$E$] (E) at (1,0); \draw (O) circle (1); \draw (A) -- (O) node [midway, left] {$1$} -- (E); \draw (B) -- (O); \draw[-stealth] ($(O)! [Read More]

Tikz Illustration for Triangle Inequality

I thought the code would be easy, but it turned out that I spent one hour and half to draw this figure. \documentclass[border=2pt,preview]{standalone} \usepackage{tikz} \usepackage{caption} \newcommand{\sideA}{2} \newcommand{\sideB}{3} \newcommand{\sideC}{6} \colorlet{myred}{red!40!yellow} \colorlet{mygreen}{green!40} \colorlet{myblue}{blue!20} \begin{document} \begin{figure} \centering \begin{tikzpicture}[thick] \coordinate (A) at (0,0); \coordinate (B) at (\sideC,0); \draw[<->,>=stealth,myred] (A) -- ++ (-\sideA, 0) node [pos=0.5,below] {$\color{myred}{a}$}; \draw[<->,>=stealth,mygreen] (B) -- ++ (\sideB, 0) node [pos=0.5,below] {$\color{mygreen}{b}$}; \draw[<->,>=stealth,myblue] (A) -- (B) node [pos = 0.5, below]{$\color{myblue}{c}$}; \draw[dashed,myred] (A) circle (\sideA); \draw[dashed,mygreen] (B) circle (\sideB); \end{tikzpicture} \captionsetup{labelformat=empty} \caption{triangle inequality ${\color{myred}{a}} + {\color{mygreen}{b}} > {\color{myblue}{c}}$ with $\color{myred}{a = \sideA}$, $\color{mygreen}{b = \sideB}$, $\color{myblue}{c = \sideC}$} \end{figure} \end{document} The colors are customized for the Discord bot TeXit. [Read More]