[試題] 98上 陳其誠 微積分甲 第一次考試

[nccuntu (宜)]

課程名稱︰微積分甲
課程性質︰必修
課程教師︰陳其誠
開課學院：理學院,管理學院
開課系所︰生工,生機,地質,工業管理組
考試日期（年月日）︰2009.10.23
考試時限（分鐘）：100分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）

試題 :

◎ x^2 = x的平方, _0_ = 下標

Write down your answers on the answer sheet. You should include all the neces-
sary calcultions and reasoning. This exam contains seven proplems written in
one page.

(1) Calculate the derivatives of the following functions(5 point each)
    a.f(x)=x^10+23x+8  b.f(x)=(x^2+x+1)^-1   c.f(x)=(1+sinx)[(1+cosx)^-1]
    d.f(x)=g(sinx), with g'(t)=(1-t^2)^1/2   e.f(x)=(cosx*cosx+1)^1/9

(2) Calculate the following limits(5 points each)
    a.lim_x→0_sin9x/x     b.lim_x→+∞_(x^2-4)/(x^4+1)^1/2

(3) (10 points) Find the tangent x_0_=-0.7, Calculate x_1_

(4) (10 points) Find the tanget line to the curve x^7+7xy-7y^7=1 at (1,1).

(5) Let f(x)=x(x-10)^3
    a.Determine the intervals where f is increasing or decreasing(6 points).
    b.Determine the intervals wherer f is concave up or concave down(6 points).
    c.Find all points of inflection(3 points.)
    d.Sketch the graph y=f(x) (7 points).
    e.Does f(x) have a maximum or minimum for x in(+∞,∞)? Why(3 points)?

(6) (10 points)You are in a boat that is 10km off the shore. Suppose you are
   heading for a place located 15km down the coast and 5km inland and you plan
   to take your motor bike to the place after you get off the boat. The gasoline
   consumption is 0.2 L/km for the boat, 0.1 L/km for the motor bike.Use your
   computation to show that to save the most amount of the gasoline, you will
   need to get off the boat 5km down the cost.

(7) a.(5 points) Suppose f"(x) exists and is positive for every x in R, and
      suppose we know that f(0)=0, f'(0)=3. Can we conclude that f(x)>3x for
      x>0? Why?
    b.(5 points) Give an example of a degree 4 polynomial that has no point
      of inflection on R


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