Sec.10.3 - 極座標曲線家族

Laboratory Project in Sec.10.3, Calculus by Stewart English version: Families of Polar Curves 在這個研究中,你將發現極座標曲線家族有趣又漂亮的形狀。同時,當常數改變時,你也會觀察到曲線形狀的變化。 問題 1: (a) 探討

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.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

Sec.3.1 - 建造較佳的雲霄飛車

Applied Project in Sec.3.1, Calculus by Stewart 英文版請見 Building a better roller coaster 假設你被要求設計一架新的雲霄飛車的第一個上升與下降。在研究你最喜歡的雲霄飛車的圖片後,你決定新的雲霄飛車先

Sec.6.5 - Where to Sit at the Movies

Applied Project in Sec.6.5, Calculus by Stewart Chinese version 電影院要坐哪? A movie theater has a screen that is positioned $3$m off the floor and is $10$m high. The first row of seats is placed $3$m from the screen and the rows are set $1$m apart. The floor of the seating area is inclined at an angle of $\alpha=20^{\circ}$ above the horizontal and the distance

Sec.6.5 - 電影院要坐哪?

Applied Project in Sec.6.5, Calculus by Stewart English version Where to Sit at the Movies? 有家電影院的螢幕高10米且離地3公尺,第一排的位子距離螢幕3公尺且每排間隔1公尺,座位區的地板以與水平20度夾

Sec.11.11 - Radiation From The Stars

Applied Project in Sec.11.11, Calculus by Stewart Chinese version 恆星的輻射 Any object emits radiation when heated. A blackbody is a system that absorbs all the radiation that falls on it. For instance, a matte black surface or a large cavity with a small hole in its wall (like a blastfurnace) is a blackbody and emits blackbody radiation. Even the radiation from the sun is close to