practice_final_fa05 - Practice Final Exam Math 191 Fall...

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Practice Final Exam, Math 191, Fall 2005No calculators. Show your working. Clearly mark each answer.1.(a) Calculate the length of the curve given byy=2x-x2,0x2.(b) Sketch the region betweeny=sechx(-∞< x <) andy= 0. Calculate the volumeof the solid swept out by rotating this region about thex-axis. (You may quote the fact thatddxtanhx=sech2x.)(c) Find the area of the surface swept out when the curvey= coshx,0x2, is rotatedaround thex-axis.2.(a) Calculate the degree 1 Taylor polynomial off(x) =x2-3 +integraldisplayx22sec(t-2)dtatx=2.(b) Sketch the region enclosed by the curvesx-y2= 0andx+ 2y2= 3and find its area.3.(a) Which of the following functions have inverses?i.f(x) = (x-1)3with domain(-∞,).ii.f(x) =12 + sinxwith domain[-π/2, π/2].iii.f(x) = sec2xwith domain(-π/2, π/2).(b)i. Show that for-1< x <1,ddxsin-1x=11-x2.ii. Which of the following functions, both with domain(1,), grows faster?

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