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微積分

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第 1 頁,共 3 頁

國立臺中教育大學

109 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 一、填充題(每題4%,共 84%) 1. Evaluate lim𝑥𝑥→0|3𝑥𝑥−2|−|3𝑥𝑥+2|𝑥𝑥 .

2. Let ( ) 1 cossin if 0

if 0 x x f x x mx b x −  >  =   +

. Suppose that f is differentiable on ( , )−π π . Find the constants m and b.

3. Let 𝑓𝑓(𝑥𝑥) = (cos 𝑥𝑥)𝑥𝑥. Find 𝑓𝑓(𝑥𝑥).

4. At what points does the curve x3+y3 =6xy have a horizontal tangent?

5. A box with a square base and open top must have a volume of 32,000 cm3. Find the

dimensions of the box that minimize the amount of material used. 6. Find the linearization of ( ) 2 2 1 9

1 x f x dt t + = − +

at x =1.

7. Suppose that 𝑓𝑓 is continuous, 𝑓𝑓(0) = 0 , 𝑓𝑓(2) = 1 , 𝑓𝑓′(𝑥𝑥) > 0 ∀𝑥𝑥 ∈ ℝ , and

∫ 𝑓𝑓(𝑥𝑥)𝑑𝑑𝑥𝑥 =02 13. Find the value of the integral ∫ 𝑓𝑓1 −1(𝑦𝑦) 𝑑𝑑𝑦𝑦

0 . 8. Let 4x22y2 = . Then 9 2 2 d y dx = . 9. Calculate 2

(

2

)

3 0 x x +1 dx=

. (背面尚有試題)

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第 2 頁,共 3 頁

10. Let f x( )=x x5+ +1. Then

( )

f−1 ′(1)= .

11. Evaluate

0ln3 ex 1+e dxx = .

12. Find lim𝑥𝑥→0(1 + sin 𝑥𝑥)1/𝑥𝑥= .

13. Evaluate ∫ 𝑥𝑥2√4−𝑥𝑥𝑑𝑑𝑥𝑥 2 √2 1 = . 14. Evaluate ∫ 𝑑𝑑𝑥𝑥𝑥𝑥3 +∞ 1 = . 15. 求lim𝑥𝑥→02 tan 𝑥𝑥𝑥𝑥 = . 16. 求lim𝑥𝑥→0𝑒𝑒𝑥𝑥𝑥𝑥 = . 17. 請問級數∑∞ 𝑛𝑛 ln 𝑛𝑛1 𝑛𝑛=2 ,是否收斂? . 18. 請問級數∑∞ 𝑒𝑒𝑛𝑛2𝑛𝑛𝑛𝑛 𝑛𝑛=1 ,是否收斂? . 19. 請求出一個以 0 為中心的冪級數(power series)代表𝑔𝑔(𝑥𝑥) = 𝑡𝑡𝑡𝑡𝑡𝑡−1𝑥𝑥? (請 至少寫出前三項),而其收斂區間為何? . 20. 求∫ 𝑥𝑥𝑒𝑒0 𝑥𝑥𝑑𝑑𝑥𝑥 −∞ = .

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第 3 頁,共 3 頁 21. 求∫ 𝑥𝑥12𝑑𝑑𝑥𝑥 1 −1 = . 二、計算證明題(每題8%,共 16%) 1. Let 2 2 4 if ( , ) (0,0), ( , ) 0 if ( , ) (0,0). xy x y f x y x x y y  ≠ = = +    

(a) Show that f is not continuous at(0,0). (b) Find f x yx( , ).

2. Evaluate the integral∫ ∫ 𝑒𝑒1 𝑚𝑚𝑚𝑚𝑥𝑥�𝑥𝑥2,𝑦𝑦2

0 𝑑𝑑𝑦𝑦𝑑𝑑𝑥𝑥

1

0 , where max{𝑥𝑥2, 𝑦𝑦2} means the larger

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