101-2002&amp;2003-1-F10-January2003

# 101-2002&amp;2003-1-F10-January2003 - ~'I-~...

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Math 101 Calculus I Kuv,·ait University Department of Mathematics Computer Science Final Examination January 12, 2003 Time: 2 hours CalCulators, mobile phones and pagers are not allowed during this exam. Each question is worth 5 points. 1. Find the limit if it exists: (0) lim (X+h)3_X3 h-tO h 1 (b) lim x4 sin3/X x-tO v..0 c 2. Use the definition of the derivative to show that the follmving function IS differentiable at x = ~ : J(x) = { cos(2x - 1) + 4x2 - 2x , 1 2 - 2x if x < 1 - 2 1 if x > 2 \~ Let f (x) = x cos x. Use Rolle's theorem to show that J'I (c) c E (-~,~). 4. Show that the following function is constant on (0, CX») . r3x 1 . F(x) = J - dt x t o for some ( [~EVa)Uate the integral (2 [cos( 1rX) ,_ .,j x3 ] dx J-2 1 +x4 .. 6. Let f(x) = V x . What is the average value of f(x) on [0,4] ? Find a . x2 + 9 . number z that satisfies the conclusion of the mean value theorem for definite integrals. 7. Find the area of the region enclosed by the graph of y = x2 and the lines x ..2 . and y = 0." 8. Find the resulting volume when the region bounded by the graphs of "( . x = (y - 1)2 and x = y + 1 is revolved about the line y = 3.

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101-2002&amp;2003-1-F10-January2003 - ~'I-~...

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