Calculus

Applied Project in Sec.3.1, Calculus by Stewart Chinese version: 建造較佳的雲霄飛車 Suppose you are asked to design the first ascent and drop for a new roller coaster. By studying the photographs of your favorite coasters, you decide to make the slope of the ascent $0.8$ and the slope of the drops $-1.6$. You decide to connect these two straight stretches $y=L_1(x)$ and $y=L_2(x)$

Applied Project 1 in Sec.4.7, Calculus by Stewart 英文版請見 The shape of a can 給定一個體積為 $v$ 的圓柱體罐頭，我們想找到一個高 $h$ 和半徑 $r$ ，將製成罐頭的金屬量減到最小。 如果我們將製程中金

Laboratory Project in Sec.3.10, Calculus by Stewart 英文版請見 Taylor polynomials 在近似函數 $f(x)$ 在 $x=a$ 的值時，切線逼近函數(tangent line approximation) $L(x)$ 是線性逼近(linear approximation)中

Applied Project in Sec.3.4, Calculus by Stewart 英文版請見 Where should a pilot start descent? 上圖是一台飛行器的降落軌跡近似圖，它必須滿足以下條件： 巡航高度為$h$，而飛行器在與著陸地點(同時也是圖

Section 11.1, exercise 79: Find the limit of $$ \{\sqrt{2}, \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}, \cdots\} $$ Remark: 這個數列也可以用以下表示方式 $$ a_1 = \sqrt{2}, \quad a_{n+1} = \sqrt{2 a_n}, \quad n\in \mathbb{N}, \quad n>1. $$ Method 1 我們首先證明它有極限 $a_n<2$ (我們使用數學歸納法) 已知

這裡我們要介紹 $r^2=\cos(4\theta)$ 這個方程式的圖形怎麼畫. 首先, 因為 $r^2 \ge 0$, 所以整個方程式只有在 $\cos 4\theta \ge 0$ 的地方有定義. 我們先找離 $\theta=0$ 最近的一段定義域, 也就是 $$ -\frac{\pi}{8}\le \theta

Applied Project 2 in Sec.4.7, Calculus by Stewart Chinese version: 飛機和鳥：能量最小化 Small birds like finches alternate between flapping their wings and keeping them folded while gliding. In this project we analyze this phenomenon and try to determine how frequently a bird should flap its wings. Some of the principles are the same as for fixed-wing aircraft and so we begin by

Applied Project 1 in Sec.4.7, Calculus by Stewart Chinese version: 罐頭的形狀 Given a cylindrical can with fixed volume $v$, we want to find the height $h$ and radius $r$ that minimize the cost of metal to make the can. If we disregard any waste metal in the manufacturing process, then the problem is to minimize the surface area of the cylinder. It can be found that

在微積分課程裡我們有學到積分的’中點法', ‘梯形法’以及’辛普森法’. 這裡我們簡

這幾天有個朋友問我一個關於不定積分的問題, 看起來很詭異. 我們一起來看看. 我們以分部積分(integration by part)來求 $\tan$ 的不定積分 $\int