102年 - 102 國立臺灣師範大學_學士班轉學生招生考試試題_數學系二年級：微積分#122181

科目：研究所、轉學考(插大)-微積分 | 年份：102年 | 選擇題數：0 | 申論題數：12

試卷資訊

所屬科目：研究所、轉學考(插大)-微積分

選擇題 (0)

申論題 (12)

(a)Write out the ε-δ definition of the limit f(x) = 2. (5 points)

(b)Suppose that f is a continuous function on the interval (a, b), 0 ∈ (a, b) and f(0)> 0. Prove that there exists a δ> 0 such that f(x) > 0 for all x ∈ (−δ, δ). (10 points)

2.Let p, q ＞ 1 with= 1. Prove that ≥ for all x, y ≥ 0. (10 points)

(a)dx (7 points)

(b)3^xsin³x dx (7 points)

(c)cos y² dzdydx (8 points)

(d)e^{-(x²+y²+z²)}dxdydz (8 points)

4.Find the volume of the solid formed by revolving the region bounded by the graphs y = x³ + x + 1, y = 1, and x = 1 about the line x = 3. (10 points)

5.Find the area of the common region bounded by the following polar curves: r = -6cosθ (circle) and r = 2 - 2cosθ (cardioid). (10 points)

6.Use the integral test to show that the series converges if p> 1. (10 points)

(a)Write out the definition of a function f(x, y) to be differentiable at a point (x₀, y₀). (5 points)

(b)Prove that if the function f(x, y) is differentiable at the point (x₀, y₀), then it is continuous at (x₀, y₀). (10 points)