Problem 4 (25 points). State the mean value theorem.

I. Translation (每小題10分，共20分) 1. 科學家們早就知道蜜蜂透過搖晃身體來跳舞。這是他們分享關於蜂巢周圍花朵的方向和距離的訊息的方式。花會產生一種叫做花蜜的甜味液體和一種叫做花粉的黃色粉末。它們都是蜜蜂的食物。蜜蜂甚至可以用它們的舞蹈動作互相告訴對方某些花有多美味。現在一項新的研究顯示，蜜蜂在出生時並不具備這些舞蹈技能。年輕的蜜蜂透過觀察年長的蜜蜂來學習它們。像其他一些昆蟲一樣，蜜蜂可以透過相互模仿來學習新的行為。這就是所謂的「社會學習」。

2. 以下文字是一本書的簡介，請翻譯為中文。書名請譯為中文，第一次提及時請附帶原文。作者姓名以原文呈現即可。       The Invisible Kingdom: Reimagining Chronic Illness, by Meghan O’Rourke: Beginning in the late 1990s, Meghan O’Rourke was tormented by mysterious symptoms that would consume her life for years to follow. She describes her wrenching experience searching for a diagnosis in The Invisible Kingdom, a 2022 National Book Award finalist. O’Rourke’s reported memoir is an indictment of the U.S. health care system and its approach—or lack thereof—to identifying and treating chronic illnesses, which take a grave toll on millions of Americans. Moving between her own medical journey, the history of illness in the U.S., and the crisis faced by millions currently suffering from long COVID, O’Rourke writes with an empathetic hand to argue why and how we need to change our systems to better support patients. The book is a bold and brave exploration into a much-overlooked topic, one that she punctuates with candor and urgency.

II. Essay Question (20分)       Cultural diversity refers to the variety of human cultures, languages, beliefs, practices, and traditions that exist and coexist within a society or among different societies, and the recognition and appreciation of these differences. For you, as an English teacher in Taiwan, what role does cultural diversity play in the classroom? How can you create a learning environment that values and respects the diversity of your students' backgrounds and experiences? Please provide specific examples of strategies that can be applied to promote cultural understanding and inclusivity in the classroom.

III. Question Design (每小題10分，共20分) The following passage is a news article.(1) Summarize the passage in around 250 words.(2) Based on your summary, design five cloze test questions suitable for high school twelfth graders.Provide answers to the questions.       Changes are coming to TikTok accounts held by young people.        Soon every account held by a user under the age of 18 will have a 60-minute daily screen time limit, TikTok said. However, users can choose to keep watching. The changes arrive during a period in which there are growing concerns among government leaders in different countries about the video-sharing app. TikTok’s security and ability to alter its algorithm to push certain posts have been questioned.       Social media algorithms work by identifying topics and content of interest to a user. The update, announced on March 1, 2023, also mirrors gaming rules for minors in China, where TikTok’s parent company ByteDance was formerly based. It has relocated to Singapore.       In 2021, Chinese authorities issued new rules limiting the amount of time that minors could play online games. Young people in China may play only an hour a day and only on Fridays, weekends and public holidays. The rules are part of an effort to curb addiction to Internet activities.        In the United States, families have struggled with limiting the amount of time their children spend on the Chinese-owned app. About two-thirds of American teens use TikTok, says the Pew Research Center. It’s an organization that informs the public about trends and attitudes.        Cormac Keenan is head of trust and safety at TikTok. He said in a blog post on March 1 that when the 60-minute limit is reached, minors will be prompted to enter a passcode. They will be asked to make an “active decision” to keep watching. For accounts on which the user is under the age of 13, a parent or guardian will have to set or enter an existing passcode to allow more time. That will give 30 minutes of extra viewing time after the initial 60-minute limit is reached.        TikTok said it came up with the 60-minute mark by consulting academic research. The company also said it checked with experts from the Digital Wellness Lab at Boston Children’s Hospital in Massachusetts.        There have long been concerns about what minors are exposed to on social media and the possible harm it might do. A report released late last year suggested that TikTok’s algorithms are promoting videos harmful to vulnerable teens. Instagram, which is owned by Facebook’s parent company, Meta, has faced similar accusations.         Algorithms on social media are a way of sorting posts. They predict what users will want to see based on their interests. Then the user is sent more of the same, as a way to keep the person on the site. However, social media critics say it’s not that straightforward. The same algorithms that promote content about a sports team, hobby or dance craze can send users down a hole of harmful content, they say.       TikTok also said on March 1 that it will begin asking teens to set their own daily screen time limit. That would be for users who opt out of the 60-minute setting. The company will send weekly inbox notifications to teen accounts with a screen time recap.       The app has already been using some safety features for teen accounts. Accounts start out as set to private for those between the ages of 13 and 15. Additionally, direct messaging is available only to those accounts with a user who is 16 or older.       TikTok announced a number of changes for all users. Those include the ability to set customized screen time limits for each day of the week and allowing users to set a schedule to mute notifications.       The company is launching a sleep reminder to help people plan when they want to be offline at night. For the sleep feature, users will be able to set an end time. When the time arrives, a pop-up will remind the user that it’s time to log off.       Concerns about the app go beyond its use by minors. Around the world, there is growing concern about the app’s security. In the United States, leaders warn that ByteDance could give user data to the Chinese government or push misinformation. The European Parliament, the European Commission and the Council of the European Union have banned TikTok from being installed on official devices.       That follows similar actions taken by the U.S. government, Congress and more than half of the 50 U.S. states. Canada has also banned TikTok from government devices.       Republicans in the U.S. House of Representatives are pushing a bill to give President Joe Biden the ability to ban the app nationwide. A possible ban has faced opposition from some organizations that argue it would violate rights protected by law. The legislation passed a committee in Congress on March 1. The bill must still be voted on in the full House and Senate.

1.A one-story storage building is to have a volume of 250,000 cubic feet. The roof costs $32 per square foot, the walls cost $10 per square foot, and the floor costs $8 per square foot. Find the dimensions that minimize the cost of the building.

(1)Rewrite the above double integral by exchanging the order of integration.

(2)Evaluate the double integral derived in (1).

3.Find the least squares approximation h(x) = ax + ax2 for f(x) = e-x, 0 ≤ x ≤ 1. [Hint: Solve a by minimizing the integral of the squared difference between f(x) and h(x).]

(1)Set up and solve a linked pair of differential equations to determine respectively the amount of money in the savings account and stock fund t years from now.

(2)How long does it take for the savings account to be exhausted? How much money is in the stock fund at this time point?

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