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Today's calculation of Integral 788

Source: 2012 Waseda University entrance exam/Science and Technology

February 17, 2012
calculusintegrationfunctionlogarithmsanalytic geometrycalculus computations

Problem Statement

For a function f(x)=ln⁡(1+1−x2)−1−x2−ln⁡x (0<x<1)f(x)=\ln (1+\sqrt{1-x^2})-\sqrt{1-x^2}-\ln x\ (0<x<1), answer the following questions:
(1) Find f′(x)f'(x).
(2) Sketch the graph of y=f(x)y=f(x).
(3) Let PP be a mobile point on the curve y=f(x)y=f(x) and QQ be a point which is on the tangent at PP on the curve y=f(x)y=f(x) and such that PQ=1PQ=1. Note that the xx-coordinate of QQ is les than that of PP. Find the locus of QQ.
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