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Unformatted text preview: Practice Final Exam, Math 191, Fall 2005 No calculators. Show your working. Clearly mark each answer. 1. (a) Calculate the length of the curve given by y = 2 x- x 2 , x 2 . (b) Sketch the region between y = sech x (- < x < ) and y = 0 . Calculate the volume of the solid swept out by rotating this region about the x-axis. ( You may quote the fact that d dx tanh x = sech 2 x . ) (c) Find the area of the surface swept out when the curve y = cosh x, x 2 , is rotated around the x-axis. 2. (a) Calculate the degree 1 Taylor polynomial of f ( x ) = x 2- 3 + integraldisplay x 2 2 sec( t- 2) dt at x = 2 . (b) Sketch the region enclosed by the curves x- y 2 = 0 and x + 2 y 2 = 3 and find its area. 3. (a) Which of the following functions have inverses? i. f ( x ) = ( x- 1) 3 with domain (- , ) . ii. f ( x ) = 1 2 + sin x with domain [- / 2 , / 2] . iii. f ( x ) = sec 2 x with domain (- / 2 , / 2) ....
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- Fall '07