Unformatted text preview: 1910 - CALCULUS FOR ENGINEERS, F ALL 2011, PRELIM II 7:30-9pm, October 27, 2011. Answer the following 5 questions. Show all work. Closed book, no calculators; 1-sided (8.5 x 11) cheat sheet is allowed - individually and uniquely hand-written (without collaboration) - will be collected along with the exams - will not be returned. Academic integrity on the part of each student is presumed. Violations will be dealt with swiftly and justly. 1.(10 pts each) Find derivatives for the following functions: a) x b) xe (x > 0) c) x arctan(ex ) 2.(10 pts each) a) lim Compute the following limits: ln (1 - x) x0 arcsin x b) lim +
x0 3 x3 x 0 t3 dt c) lim (1/x)x x0+ et - 1 d) lim xe-1/x +
x0 3.(15 pts each) a) Compute the following integrals:
/3 2 tan x sec x dx b)
0 1 3 x3 dx c) x2 - 1 2 1 dx dx 2x - x2 4.(10 + 5 + 5 + 10 pts)
a) Show that the following function is one-to-one, and has an inverse: f (x) = earctan x cosh x
b) What is the domain of c) What is the range of d) Compute f -1 ? f -1 ? (f -1 ) (1). 5.(20 pts) Calculate the volume of the solid of revolution formed by rotating the region in the x-y plane between the curves y = e2x and y = ex , 0 x 1, about the y-axis. 1 ...
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