Applied Project in Sec.9.3, Calculus by Stewart Chinese version: 上升快，還是下降快? Suppose you throw a ball into the air. Do you think it takes longer to reach its maximum height or to fall back to earth from its maximum height? We will solve the problem in this project, but before getting started, think about that situation and make a guess based on
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 have fascinated mankind since ancient times and have inspired attempts at scientific explanation since the time of Aristotle. In this project we use the ideas of Descartes and Newton to explain the shape, location, and colors of rainbows. Question 1: The figure shows a ray of
Laboratory Project in Sec.3.10, Calculus by Stewart Chinese version: 泰勒級數 The tangent line approximation $L(x)$ is the best first-degree (linear) approximation to $f(x)$ near $x=a$ because $f(x)$ and $L(x)$ have the same rate of change (derivative) at $x=a$. For a better approximation than a linear one, let’s try a second-degree (quadratic) approximation $P(x)$. In other words, we approximate a curve by a parabola instead of a straight
Applied Project in Sec.8.3, Calculus by Stewart Chinese version: Complementary Coffee Cups Suppose you have a choice of two coffee cups of the type shown, one that bends outward and one inward, and you notice that they have the same height and their shapes fit together snugly. You wonder which cup holds more coffee. Of corse you could fill one cup with water and pour it into the other one
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 $y=L_2(x)$