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國立臺中教育大學
107 學年度學士班日間部轉學招生考試
微積分試題
適用學系：數學教育學系二、三年級 【注意事項：1.請將答案寫在答案卷上，並標示題號 2.本科目得以鉛筆作答】 ㄧ、填充題(20％) 1. 求lim
𝑥𝑥→0tan𝑥𝑥𝑥𝑥=
( ) (5%) 2. 求∫
0∞√𝑥𝑥(𝑥𝑥+1)𝑑𝑑𝑥𝑥=
( ) (5%) 3. 求以圖形y = x3+ x + 1、y = 1，和x = 1為界的區域，繞直線 x = 2 旋轉 所得旋轉體的體積？ ( ) (5%) 4. 正弦曲線與餘弦曲線相交無窮多次，圍出的區域面積都相等，請求出單 一個圍出區域的面積？( ) (5%) 二、計算證明題(80％)1. Find numbers a and b such that
0 2 lim 1 x ax b x → + − = . (10%) 2. Let ( ) 3 4sin1 if 0, 0 if 0. x x f x x x ≠ = = (a) Is f x( ) continuous at x =0? (5%)
(b) Compute f x′( ) for x ≠0 and f ′(0). (5%)
3. Find the average value of ( ) 4 2
3
0 2
9 ,
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4. Determine whether the series is absolutely convergent, conditionally convergent, or divergent. Please state the tests which you use.
(a) 1 sin( 6) 1 n n n n π ∞ = +
∑
(5%) (b) 2 ( 1) ln n n n n = ∞ −∑
(5%)5. Find the area of the region enclosed by y x= 2+3 and y= − +x2 2x+3. (10%)
6. Calculate
∫
02ln x dx. (10%)7. Evaluate the iterated integral 1 1 2 0 x 2sin( ) y dydx
∫ ∫
. (10%)8. Find the local maximum and minimum values and saddle points of
4 4
( , ) 4 8
