101-2008&2009-3-M10-July2009

101-2008&2009-3-M10-July2009 - Kuwait University...

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KuwaitUniversity Math101 Date: July18,2009 Dept.ofMath.&Comp.Sci. FirstExam Duration: 90minutes Calculators,cellularphonesandallothermobilecommunicationequipments arenotallowed Answer the followingquestions: 1. Use the definition of the limit toshowthat lim x →- 2 (1 - 4 x )=9 (3pts.) 2. Evaluate the following limits,if theyexist: (a) lim x 1 x +8 - 3 x 2 +8 x - 9 (3pts.) (b) lim x →∞ 1 x 2 sin( x - 1) (3pts.) 3. Findtheverticaland horizontal asymptotes, if any,forthegraph of f ( x )= x 3 x 2 +1 x 2 - 6 x +8 . (4pts.) 4. Findthe x -coordinates of thepoints at which the function f is discontinuous, where f ( x )= | x - 2 | ( x - 3) x 2 - 4 , if x< 3 , tan 2 ( x - 3) x - 3 , if x> 3 . Classifythe types of discontinuityof f as removable, jump, orinfinite. (4pts.) 5. (a) State the IntermediateValue Theorem. (1pts.) (b) Let g ( x )= x 2 + f ( x ) , where f is a function with continuous derivative f prime (i.e., f prime is continuous), such that f prime (0)=2 and f prime (1)= - 3 . Show that g prime ( x )=0 has a realsolution.
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