1. 證明 f(x) = x² - 2x + 1, -2 ≤ x ≤ 3，滿足均值定理之假設，並求定理結論指定之c值。

4. 求出兩曲線 r = 1 - cosθ 與 r = sinθ 之交點，並以直角座標表示 = ___________(可能不只一點)。

5. 求冪級數之收斂區間 = __________。

6. 求 f(x) 之泰勒級數展開式，f(x) = e-2x(x-1) 在 (x+1) 之冪次 =__________。

7. 求 f(x,y,z) = x²y + yz + z²x 之方向導數，其在點 (1,0,1) 上，朝向點 (1,3,0) =__________。

2. 求不定積分 ∫ ( x² (2x - 1) ) dx。

3. 一矩形盒其三面在座標面(xy, yz, zx 面)上，且其一頂點在第一八分區(the first octant)之拋物面 z = 4 - x² - y² 上，求此矩形盒之最大體積。

4. 求(x+y) dxdy，Ω是平行四邊形由 x - y = 0, x - y = π, x + 2y = 0, x + 2y = π，所圍成。

5. 畫出由曲線 y = x, y = 2 - x², 0 ≤ x ≤ 1. 圍成之區域，求出此區域之形心與此區域各別繞兩座標軸所產生之體積。

V. Short Essay Writing 20%本大題請於答案卷上作答Directions: Read carefully the below passage about the issue of long-term care condition in Taiwan.Then write a well-organized short essay of approximately 200 words in English as your response to this passage. (Writing competency 40%, argumentation 30%, organization 30%)TAIPEI (Taiwan News) -- Taiwan is likely going to become a "hyper-aged" society within a mere eight years, according to the National Development Council (NDC), reported CNA.The NDC estimates that people over the age of 65 will make up over 20 percent of Taiwan's population in only eight year's time, thus making it a "hyper-aged" society.In fact, the NDC estimates that Taiwan will reach that milestone even faster than Japan (11 years), the U.S. (14 years), France (29 years), and the UK (51 years). Meanwhile, it is on a similar track as South Korea (8 years) and Singapore (7 years).

1. Find .