[試題]

[iwantowaylaw (我要成為太鼓達人)]

課程名稱︰微積分甲下暑修
課程性質︰
課程教師︰周青松
開課學院：
開課系所︰
考試日期（年月日）︰2009/9/9
考試時限（分鐘）：110分鐘
是否需發放獎勵金：要
（如未明確表示，則不予發放）

試題 :


A.
(a)
Find the directional derivative of f(x,y,z) = Ax^2 + Bxyz + Cy^2 at P(1,2,1)
in the direction of Ai + Bj + Ck where A,B,C are given constant with
A^2 + B^2 + C^2 ≠ 0

(b)
Find the directional derivative of f(x,y,z)=xe^(y^2-z^2) at P(1,2,-2)in the
direction of the path γ(t)=ti+2cos(t-1)j-2e^(T-1)k.

B.
(a)
Find the rate of change of f(x,y,z)= (x^2)y + zcosx with respect to t along the
curve r(t)= t i + t^2 j + t^3 k

(b)
Find the rate of change of f(x,y,z)= ln(x^2+y^2+z^2) with respect to t along
the curve r(t)= sint i + cost j + e^2t k

C.
(a)
Use double integration to calculate the area of the region enclosed by y=x^2
and x+y=2

(b)
Calculate the volume within the cylinder x^2 + y^2 = b^2 between the planes
y+z=a and z=0 given that a≧b＞0.

D.
(a)
Evaluate the double integral

             3
    ∫∫ (3xy - y) dxdy ,Ω is the region between y = |x| and y = -|x|,
     Ω

    x in [-1,1].

(b)
Evaluate the trible integral

              x
    ∫∫∫ 2ye dxdydz, where T is the solial
      T

   ∕ given by 0 ≦ y ≦ 1 , 0 ≦ x ≦ y
   ﹨ and 0 ≦ z ≦ x + y.


E.
(a)
Find the mass of a solid right circular cylinder of radius r and height h given
that the mass density varies directly with distance from the lower base , that
is , the density function is λ(x,y,z) = kz where k > 0 is a given constant

(b) Find the volume enclosed by the ellipsed
           2    2    2
          x    y    z
          — + — + —   = 1
           2    2    2
          a    b    c

Remark : You may show that the area bounded by the ellipse

           2    2
          x    y
          — + — = 1    is πab first
           2    2
          a    b


參考答案:

A.
(a) ( 2A^2 + B^2 + 2AB + 6BC ) / (A^2+B^2+C^)^(1/2)
(b) -7/5^(1/2)
B.
(a) 4t^3 - (t^3)sint + 3t^2cost
(b) 4e^(4t)/(1+e^(4t))
C.
(a) 9/2
(b) πab^2
D.
(a) 0
(b) 4e-29/3
E.
(a) πk(rh)^2 / 2
(b) 3abcπ/4


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※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.24.223
推 emitsu      :E(a) 課本答案是 πk(rh)^2 / 2
→ emitsu      :E(b)不是課本Example 是習題題目 答案不一樣吧？
謝謝 E(a) 是我打錯 E(b)我根本不會寫XD
推 vchenkoshe  :B(b)怎麼會是常數呢...
→ vchenkoshe  :C(b)好像還缺了3b^3/2π吧
→ vchenkoshe  :C(b)的答案我不記得...但你應該是錯的...
→ vchenkoshe  :B(b)好像是4e^(4t)/(1+e^(4t))
→ iwantowaylaw:C(b)請見課本885~886 這答案沒錯
→ iwantowaylaw:B(b)我真的錯了
※ 編輯: iwantowaylaw    來自: 140.112.24.223       (09/10 00:36)
推 vchenkoshe  :SH T  我看錯題目了...
→ iwantowaylaw:我也是把B(b)分母的1省略 才約分成4...