課程名稱︰微積分上
課程性質︰必修
課程教師︰黃維信
開課學院：工學院
開課系所︰工程科學及海洋工程學系
考試日期（年月日）︰2008.01.15
考試時限（分鐘）：120分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. The curve x^(2/3) + y^(2/3) = a^(2/3) is called an astroid. Sketch the
astroid and show that the curve can be parametrized by x(θ)=a(cosθ)^3 ,
y(θ)=a(sinθ)^3 , 0≦θ≦2π. Find the length of the astroid, the volume
and surface area of the solid generated by revolving the astroid about
the x-axis. (25%)
2. Let P be a polynomial of degree n.(10%)
(a) Can P have an inverse if n is even? Support your answer.
(b) Can P have an inverse if n is odd? If so, give an example. Then give
an example of a polynomial of odd degree that does not have an inverse.
3. Calculate. (35%)
sinx-cosx -1
(a) ∫———————dx (b)∫sec xdx
sinx+cosx+1
x+1
(c) ∫π^x dx (d)∫———————dx
x^3+x^2-6x
(e) ∫cos√xdx (f) d/dx [arctan(coshx)]
(g) d/dx [(sinx)^cosx]. What is the domain?
4. Find a parametric form for the ellipse ( x^2 / a^2 ) + ( y^2 / b^2 ) = 1 ,
and the tangent lines at the intersections with the line y=x. (10%)
5. Sketch the polar curve r=1+cosθ , and find the area , and the coordinates
of the centroid of the region enclosed by the curve. (15%)
1
6. Show that d/dx[arc(sinhx)] = ——————— , x real. (10%)
√(x^2+1)
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