105年 - 105 國立臺灣師範大學_轉學生招生考試試題_數學系二年級：微積分#120213

科目：研究所、轉學考(插大)-微積分 | 年份：105年 | 選擇題數：0 | 申論題數：11

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所屬科目：研究所、轉學考(插大)-微積分

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申論題 (11)

1. Suppose f is continuous on [2,5] and 1 ≤ f'(x) ≤ 4 for all x in (2,5). Show that 3 ≤ f(5)-f(2) ≤ 12.

2. Show that, for any c, the equation x⁵ - 15x + c = 0 has at most one root in the interval [-1,1].

3. (a). Expand f(x) = x/(1-x)² as a power series.

(b). Use part (a) to find the sum of the series .

4. Find the interval on which the curve

is concave upward.

5.(a) Find the limit

Calculate .

(a)

(b)

7. Evaluate ∫∫ᴅ(2y - x)dA, where D is the region bounded by the parabolas y = 2x² and y = 1 + x².

8. Evaluate ∫c(1 + xy²)ds, where C is the upper half of the unit circle x² + y² = 1.