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Sec.4.1 - The Calculus of ranbows

Applied Project in Sec.4.1, Calculus by Stewart Chinese version: 彩虹的微積分學 Rainbows are created when raindrops scatter sunlight. They …
Last updated on Jan 6, 2021 2 min read Calculus

Sec.4.7 - 飛機和鳥：能量最小化

Applied Project 2 in Sec.4.7, Calculus by Stewart 英文版請見 Planes and birds: Minimizing energy 像雀科這樣的小鳥滑翔時，會在拍打翅膀和保持翅膀折疊之間進行交替。在這裡，我們觀察這種現 …
Last updated on Jan 6, 2021 2 min read Calculus

Sec.3.10 - Taylor polynomials

Laboratory Project in Sec.3.10, Calculus by Stewart Chinese version: 泰勒級數 The tangent line approximation $L(x)$ is the best …
Last updated on Jan 6, 2021 2 min read Calculus

Sec.8.3 - Complementary Coffee Cups

Applied Project in Sec.8.3, Calculus by Stewart Chinese version: Complementary Coffee Cups Suppose you have a choice of two coffee cups …
Last updated on Jan 6, 2021 1 min read Calculus

Sec.3.1 - Building a better roller coaster

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 …
Last updated on Jan 6, 2021 1 min read Calculus

Sec.4.7 - 罐頭的形狀

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 …
Last updated on Jan 6, 2021 2 min read Calculus

Sec.3.10 - 泰勒級數

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 …
Last updated on Jan 6, 2021 3 min read Calculus

Sec.3.4 - 飛行員應該何時開始下降高度?

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 …
Last updated on Jan 6, 2021 2 min read Calculus

Limit of a sequence - Example

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 …
Last updated on Mar 6, 2020 1 min read calculus

Polar curve - Example

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 $$ …
Last updated on Mar 3, 2020 1 min read calculus