Lagrange Multiplier - 01

在微積分課程裡我們有學到如何利用 Lagrange multiplier 來解 constraint optimization 問題. 這邊要介紹課本裡沒教的 Lagrangian function. Goal: 我們想要解以下這個問題 $$ \min_{x} f(x), \quad \text{subject to } \quad g(x)=0. $$ Observation 微積分課本告訴我們

Lagrange Multiplier - 02

這裡我們再多討論一點 Lagrangian function. 我們先看最簡單的一維問題, 求一個有限制式的函數最小值問題: $$ \min_{x} f(x), \quad \text{subject to } \quad g(x)=0. $$ 我們引進 Lagrangian function $$ L(x, \lambda) = f(x) + \lambda g(x) $$ 並且知道

Lagrange Multiplier - 03

這裡我們討論一下 Lagrange multiplier. 我們知道, 如果想要解以下這個有限制式的最佳化問題 $$ \min_{x} f(x), \quad \text{subject to } \quad g(x)=k, $$ 一個方式是引進 Lagrange multiplier, $\lambda$, 然後可以列出以下兩個式子 $$ \partial_x f +

Sec.15.8 - Roller Derby

Applied Project in Sec.15.8, Calculus by Stewart Chinese version 滾動競賽 Suppose that a solid ball(a marble), a hollow ball (a squash ball), a solid cylinder (a steel bar), and a hollow cylinder (a lead pipe) roll down a slope. Which of these objects reaches the bottom first? To answer this question, we consider a ball or cylinder with mass $m$, radius $r$, and moment of inertia $I$

Sec.15.8 - 滾動競賽

Applied Project in Sec.15.8, Calculus by Stewart 英文版請見 Roller Derby 假設有一實心圓球(如一顆彈珠)、一空心圓球 (如一顆壁球)、一實心圓柱(如一根鋼條)、與一空心圓柱(如一根鉛管)同

Sec.5.2 - Area functions

Discovery Project in Sec.5.2, Calculus by Stewart Chinese version 面積函數 Question 1(a): Draw the line $y = 2t+1$ and use geometry to find the area under this line, above the t-axis, and between the vertical lines $t=1$ and $t = 3$. Answer: let $f(x)=y=2t+1$, then $f(1)=3$ and $f(3)=7$. So, Area$= \frac{1}{2}(3+7)(3-1) = 10$ Question 1(b): If $x>1$, let $A(x)$ be the area of the region that lies

Sec.5.2 - 面積函數

Discovery Project in Sec.5.2, Calculus by Stewart 英文版請見 Area functions Question 1(a): 畫出 $y = 2t+1$ 且用幾何方法找出在此線下方、$t$軸上方、與$t=1$和$t=3$兩條垂直線所圍出的面積. Answer: 令 $f(x)=y=2t+1$, 則


前情提要: 數值積分初探 連續函數可以用多項式來逼近它, 因此直覺來講, 既然我們已經找到一個離給定函數"很近"的多項式了, 何不


前情提要: 數值積分初探 前情提要: 以內插多項式來做數值積分 我們想要找到一些插值點使得以內插多項式來做數值積分會最準. 這就是高斯積分. Goal: 任意給定

Sec.13.4 - Kepler’s Laws

Applied Project in Sec.13.4, Calculus by Stewart Chinese version 克卜勒定律 Johannes Kepler stated the following three laws of planetary motion on the basis of massive amounts of data on the positions of the planets at various times. Kepler’s Laws A planet revolves around the sun in an elliptical orbit with the sun at one focus. The line joining the