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101-2001&2002-1-M10-October2001

101-2001&2002-1-M10-October2001 - -t_~_J Kuwait...

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\-t;:- . ."._~_J Kuwait University Math 101 Date: October 25, 2001 Dept of Ivlaths. & Camp. Sci. First Exam. Duration: 75 minutes 1. (9 points) Find the following limits, if they exist .5 c .'] 1 . <f::t. .- 4 (a) lim _x_-_ .;f- j)I)Il-~ :z .- b' x-d ~ _ 1 j-7 b (b) lim I x - 2\ sin ( 'if ) x--2 X - 2 ( ) 1. 1- cos(3x) c In1---- x--o sin 2 (x) Calculators and Mobile Phones are not allowed 2. (4 points) Find the vertical and horizontal asymptotes, if any of f (x) = 2x + j x2 + 1. x+2 3. (4 points) State the Intermediate Value Theorem. Then use it to show that the graphs of f and g intersect, where f (x) = x3 + 7 x2 - 5 and g (x) = 5 - x - 2x2 . 4. (4 points) \i\Trite the definition of the derivative of f at 1. Then use it to find l' (1) if it exists , where f (x) = vi x2 - 2x + 1. Justify your answer. 5. (4 points) Find the points on the graph of f at which the tangent line is perpendicular to the line 7y + x = 1, where f (x) = x3 + x2 + 6x - 5 .
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x-I -- = X2/3 + XI/3 + 1 ==? 3 when x ==? 1 ~-1 For (b), we have , \ , -}, v.u .L.L -Ix - 2/ S; )x - 21 sin C = 2) S; Ix - 21 the Sandwich theorem will then imply Ix - 2/ sin (_71"_) ==? 0 when x==?2 x-2 For (c), we have L ; + Cn];( I-fGA;;.x .9 2. Problem 2. We have 2x+~ x+2 2x + Ixl!I + I/x2 x(I + 2/x) 2 + /1 + l/x2 so when x > 0, we have f(x) = / which implies 1+2 x lim f(x) = 3 that is y = 3 is B.A. X->OO . 2-/1+I/x2 and if x < 0, we have f(x) = / which implies 1 + 2 x lim f(x) = 1 that is y = 1 is B.A. X->OO
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