微積分

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國立臺中教育大學 108 學年度學士班日間部轉學生招生考試

微積分試題

【本考科得以鉛筆作答】 適用學系：數學教育學系二、三年級 ㄧ、填充題（每題 5%，共 75%） 1. 求lim_𝑥→∞�1 +𝑘 𝑥� 𝓍 = 2. 請問級數∑ 1 𝓃! ∞ 𝓃=0 ，是否收斂？ 3. 求∫ 2₋₁2 𝑥𝑑𝑑 U 4. 求∫ 𝑥3 √𝑥2₊₉ 3 0 𝑑𝑑 U 5. 求右列兩函數圍出區域圖形的面積：𝑓(𝑑) = √𝑑 − 13 、𝑔(𝑑) = 𝑑 − 1？ 6. 求以右列函數圖形為界的區域：𝑦 = 2𝑑 − 𝑑2, 𝑦 = 0，繞直線𝑑 = 4旋轉，所得 旋轉體的體積？ 7. Let 2 2 if 4 ( ) 20 if 4 x x g x cx c x  < =  + ≥ −

 . Find the constant c that makes g continuous on (−∞, ∞). 𝑐= .

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8. Let 𝑦 = 𝑓(𝑢) and 𝑢 = 𝑔(𝑑) , where 𝑓 and g are twice differentiable functions with f(1)= −1 , f ′(1)= −2 , f ′′(1)=3 , g(0)=1 , g′(0)=2 , and

(0) 3 g′′ = . Then 𝑑2𝑦 𝑑𝑥2�_𝑥=0 = ____________. 9. Let 𝑓(𝑑) = ∫ 1 √1+𝑡3 g(𝑥) 0 𝑑𝑑 , where g(𝑑) = ∫ (1 + sin(𝑑2)) 𝑑𝑑 cos 𝑥 0 . Find 𝑓′ �𝜋 2� = ____________. 10. Given that 𝑓(𝑑) = ln 𝑥

𝑥 ¸ the maximum value of 𝑓(𝑑) is ____________.

11. Evaluate lim_𝑡→0� 1

𝑡√1+𝑡− 1

𝑡� = ____________.

12. Find the inverse function f−1( )x of ( ) 4 1 2 3 x f x x − = + . 1 ( ) f− x =____________.

13. Find the area of the region bounded by the curves y=sin2x, y=sin3x,

0≤ ≤x π. ____________.

14. Find the directions in which the directional derivative of 2

( , ) sin

f x y =x + xy at

the point (1, 0) has the value 1. ____________.

15. The area under the curve 𝒴 = sin √𝑑 from 𝑑 = 0 to 𝑑 = 𝜋2 is .

第 3 頁，共 4 頁 二、計算證明題（請列出計算過程或畫圖證明，每題 5%，共 25%） 1. 求以拋物面z＝4 − 𝑑2− 2𝑦2，和𝑑𝑦 −平面為界的立體區域體積。 2. If lim_𝑥→∞�𝑥+𝑎 𝑥−𝑎� 𝑥 = 𝑒，then 𝑎 = ?

3. Find the interval of convergence of the series

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4. Find the maximum positive number _______ such that: If then