[試題] 102上 朱樺 微積分甲 期中考

[honsan (honsan)]

課程名稱︰微積分甲
課程性質︰
課程教師︰朱樺
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2013/11/2
考試時限（分鐘）：150分鐘(兩個半小時)
是否需發放獎勵金：是
（如未明確表示，則不予發放）

試題 :

(1) (20%) Find the limits.

(a) lim x^(2/3)*(sqrt(x+2)-2*sqrt(x+1)+sqrt(x)) as x to inf
(b) lim [x+x^2*ln(1-2/x)] as x to inf
(c) lim (tanx)^cosx as x to (pi/2)-

                     {|x|^x if x!=0
(2) (15%) Let f(x) = {
                     {  1   if x=0

(a) Determine whether f(x) is continuous at 0.
(b) Determine whether f(x) is differentiable at 0.

(3) (15%)
(a) Show that the function f(x)=x^3+x+1 is strictly increasing on R.
(b) If g(x) is the inverse function to the function of f(x) of part(a).
    Find g'(5) and g''(5).

(4) (10%) Prove the identity
          / x-1 \
    arcsin|-----| = 2*arctan(sqrt(x)) - pi/2.
          \ x+1 /

(5) (10%) The minute hand on a watch is 8mm long and the hour hand is 4mm long.
    How fast is the distance between the tips of the hands changing at
    two o'clock.

(6) (15%) Two vertical poles PQ and ST are secured by a rope PRS as shown
    in the picture.

         S
        /|
    P  / |
    |\/  |
    Q R  T

    Given that PQ = 1 m, ST = 3 m, QT = 2 m,
    find the position of R such that
    (a) the length of the rope PRS is maximized.
    (b) the angle θ= PRS is maximized.

(7) (25%) Let f(x) = (x-1)^(5/3)(x^2-1)^(-1/3).
    (a) What is the domain of f(x).
    (b) Find all the asymptotes of f(x).
    (c) Determine the intervals of increase or decrease.
    (d) Determine the intervals of conccavity and the inflection points.
    (e) Find the local maximum and minimum values.
    (f) Sketch the graph of f(x).
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◆ From: 140.112.242.15
※ 編輯: honsan          來自: 140.112.242.15       (11/03 19:52)
推 winston1907 :奇怪怎麼記得有一題是純微分題? 而且印象中沒第4題
→ winston1907 :不好意思再順便請問一下 第一題的c我記得是cosh?
推 th1279sky   :這是數學系的考題？樓上講的是03班吧。
推 winston1907 :原來如此，感謝解惑~ 不過好多題目是重複的XD
→ winston1907 :不過 感覺數學系的考法應該不是這樣才對?
推 yuseke      :朱樺這學期有兩個班  統一教學跟非統一教學
推 yinyang102  :這是13班的，工管/地質/生工/生機