[微積] 一些微積分相關問題
1. f(x)=lnx/x on (0,∞)
(a) Find critical points, the intervals where f(x)increases , decrease
(b) Find the extreme values
(c) Determine the concavity of the graph and find the points of inflection
2. Given x+1=0 and x+y^2=0
(a) Find the area of the bounded region determined by the above two curves
(b) Locate the centroid of the bounded region
(c) Find the volume of the solid generated by revolving the bounded region about the y-axis
3.
(a) Let f(x)= (ax+b)/(cx+d) with x≠ -d/c Show that f is one-to-one if ad-bc≠0
4.
∫ cscx dx = ∫ sinx/(1-cos^2x) dx
Show that ∫ cscx dx= ln√(1-cosx)/(1+cosx) +C
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