課程名稱︰微積分甲下
課程性質︰系內必帶
課程教師︰陳宏
開課學院：工學院
開課系所︰土木工程學系
考試日期（年月日）︰2010/4/13
考試時限（分鐘）：50min
是否需發放獎勵金：是，謝謝
（如未明確表示，則不予發放）
試題 :
Make sure to give sufficient reason in each problem or you will NOT get any
credit for your answer.
1.(25 points) Let f(x)=(x^3-xy^2)/(x^2+y^2).
(a) (10 points) Define f(0,0) such that f(x,y) is continuous at (0,0).
(b) (10 points) Find fx(x,y) and fx(0,0).
(c) (5 points) Is fx(x,y) continuous at (0,0)?
Solution. Refer to Q2 of 89 midterm.
http://www.math.ntu.edu.tw/~mathcal/download/exam/cala892_II_mid.pdf
2.(10 points) Let u=f(r,θ,z) where x=rcosθ and y=rsinθ. When u is
differentiable, write ∂u/∂x and ∂u/∂y in terms of ∂u/∂r, ∂u/∂θ,r,
and θ.
Solution. Refer to Q3 of 89 midterm.
http://www.math.ntu.edu.tw/~mathcal/download/exam/cala892_II_mid.pdf
3.(20 points) For the curve given by r(t)=(t^(-1),2lnt,2t). Find
(a) (7 points) the unit tangent vector T.
(b) (8 points) the unit normal vector N.
(c) (5 points) the curvature (i.e. It is defined as |dT/ds|)
Solution. Refer to Q2 of 92 midterm.
http://www.math.ntu.edu.tw/~mathcal/download/exam/92midA.pdf
4.(15 points) Find the arc length s(t) of the curve r(u) from u=0 to u=t,
t>0 where
1-u^2 2u
r(u) = -------i + -------j
1+u^2 1+u^2
Solution. Refer to Q1(i) of 94 midterm.
http://www.math.ntu.edu.tw/~mathcal/download/exam/942cala1_mid.pdf
5.(20 points) Let
z
F(x,y,z)=x^2+2z+∫√(t^2+1-y^2)dt
y
Find the plane tangent to the surface {(x,y,z)∈R^3|F(x,y,z)=2} at (2,-1,-1).
Solution. Refer to Q6(a) of 95 midterm.
http://www.math.ntu.edu.tw/~mathcal/download/exam/952A1Mid_So1.pdf
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※ 編輯: KenYang0531 來自: 118.168.68.158 (07/10 18:15)