practice_final_fa05 - Practice Final Exam Math 191 Fall...

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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 , 0 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, 0 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 [ - π/ 2 , π/ 2] . iii. f ( x ) = sec 2 x with domain ( - π/ 2 , π/ 2) . (b) i. Show that for - 1 < x < 1 , d dx sin - 1 x = 1 1 - x 2 . ii. Which of the following functions, both with domain (1 , ) , grows faster?
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