(b) (5%) ∫ dx

(a) (10%) For a certain real number a, the following integral  converges. Determine a and evaluate the integral.

(b) (10%) For what values of the constants b and cwill the following limit exist and be equal to 1? dx .

3. (15%) If dis an arbitrary real number, let Sn(d) =. Determine the following limit: .

4. (10%) Let f(x) be a polynomial of degree m. Show that ∫exf(x)dx = ex(f(x) - f'(x) + f"(x) - ... + (-1)mf(m)(x))

5. (15%) Evaluate the following integral  along the curve C = {(x,y,z) ∈ R³ | x² + y² = 4, z = -3}, oriented counterclockwise as seen by a person standing at the origin and with respect to the right-handed Cartesian coordinates, where F = yi + xz³j – zy³k, r' = dr/ds is the unit tangent vector, and *sis the arc length of C

6. (15%) Evaluate the following integral I = ∫L(2xyz²) dx + (x²z² + z cos(yz)) dy + (2x²yz + y cos(yz)) dzalong the line segment L from P: (0,0,1) to Q: (2,π/4,2)

7. (15%) Compute the area of the hemisphere of radius t>0 and centered at the origin given by S = {(x,y,z) ∈ R³| x² + y² + z² = t², z≥0}

1.Source https://www.weforum.org/agenda/2022/02/future-of-the-metaverse-vr-ar-and-brain-computen/ In January 2020, author and investor Matthew Ball published the most commonly referenced description of the metaverse: "an expansive network of persistent, real-time rendered 3D worlds and simulations that [...] can be experienced synchronously by an effectively unlimited number of users, each with an individual sense of presence." According to his definition, the metaverse is a product or service, with seven core attributes including persistence, synchronicity, and interoperability. The technology strategist Ben Thompson has argued, however, that this description is in fact not far from what the internet already is and does, only "with a 3D layer on top of it". ............. The third definition is compelling in part because it focuses on those who will actually build and use the metaverse: people. Answering questions about how the metaverse might look and feel -rather than pondering about its features could prove a useful viewpoint to predict the wave of socioeconomic change that the metaverse is expected to unleash. After all, the future is built by ruthless pragmatists； not the armchair theorizers who "meander the forest of their own words".

2. Sourcehttps://www.forbes.com/sites/jackkkelly/2021/07/15/the-new-post-pandemic-era-is-all-about-change-and-wide-open-possibilities/7sh=50689c37ee0c  The new post-pandemic era is all about change and wide-open possibilities. Now is the time to act. We are in a unique, fast-changing time period in history. After being locked down for so long, there's a hunger and lust for something new, exciting and adventurous. Jeff Bezos, Elon Musk and Richard Branson are sending rockets to explore space and make Mars habitable for humans. ............ We are in an era of rapid change and it feels like anything is possible. You need to take advantage of it and spend your time on yourself. The pandemic has changed us. Our collective mindset has shifted toward being open and receptive to new opportunities and limitless possibilities.

二、中翻英 Translate the highlighted parts in the following editorial into English (498 characters). 人口減少是危機 也是契機        我國連續兩年「生不如死」,人口負成長將逐年加速。政府雖加碼生育與托育補助,但八成以上民眾並不會因此生育；因為拉拔孩子長大,面臨的經濟和心理壓力,千百倍於此。 已開發國家皆呈現人口減少現象,而經濟困難的非洲、南亞國家,生育仍持續增長。       已開發國家競爭激烈,高房價、工作不穩定,人民注重工作與生活平衡,養育非首選考量。美國、南韓和中國的年輕人,甚至興起「躺平主義」,抗議貧富不均和資本剝削。       少子化並非全然壞事,對大自然與環境反倒有好處。人類減少對食物、水、電力、能源的消耗及製造汙染；氣候暖化及過度消費之害,可獲得改善。未來智能機器人代替人工,過往以勞動力帶動經濟生產的觀念,鼓吹生育反而製造失業,是逆潮流的不智政策,針對年輕世代投資教育方為上策。       瑞典國土是臺灣的十二倍,人口僅一千萬人,GDP產值卻是我國的兩倍多,可見人口不是強國的要項,未來更是如此。臺灣人口逐日減少,更應藉此改善生活環境、平抑房價、實施免學費教育及全面福利照顧。       我國目前老年世代相對較有錢,勿再掠奪資源。美國五成以上老人願拿退休金資助子女,瑞典逐步延遲退休年齡,上下同心越過過渡期,則能化解老齡化社會問題,臺灣可望轉骨迎向青壯活力國家。

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