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) -- My Weblog...http://blog.pixnet.net/nk52129 你一定要來參觀喔~~^^ --

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