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101-2002&2003-3-M10-July2003

101-2002&2003-3-M10-July2003 - Kuwait University...

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Kuwait University Math 101 Date: July 12, 2003 Dept. of Math. & Compo Sei. First Exam Duration: 75 minutes Calculators, mobile phones, pagers and all other mobile communication equipment are not allowed 1. E\',duate the followinglimits, if exist.: Ixl (a) !~1 sin 2x (b) lim cos(x - sin x). :r~n- 2. Find the vertical aild horizontal asymptotes, if any, of x3 - X I(x) = x2 _ X - 2' 3. Let 2sin(x - 1) if, x> 1, x2 - 1 I(x) = < A+Bsin "2:r if, x = I, xj -1 if, 2A+-- x<l. x-I Find the values of the constants A and B, so that I is cOrltinuous at x = 1. 4. Show that the graphs of !(~J) = x" + 3x3 + 1 and g(x) = BX'I - x3 + 2x - 1 illterl'Cct.. X2/~ 5. Show that the graph of I(x) = -- has a cusp. x-I 6. Find an equation for the tangent line to the graph of I(x) = (x + 1)2 cas x at x = o. 2 x-I ~ 7. Let y = u - 5u + 1 and u = --, find -. x+l dx (3 pts.) (3 pts.) (3 pts.) (4 pc.s.) (:3 pts.) (:J pes.) (3 pts.) (3 pts.)
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Kuwait University Dept. of Math. & Compo Sci. Math 101 Date: First Exam July 12, 2003 Answers Key , I .• Ixl ±2;l; 0J lxl 1. (a.) lim -'-2- =! lim -:--?' = ±~ =? lim -'-2- = DN E. :z:-O± sm x 2:z:-0± S1I1 _x :z:-O Sill :c (b) lim cos(x - sin x) = cos! lim (x - sin x)) = cos(1f - 0) -r=Tl %-~ %-~ ~ 2. x2 - X - 2 = 0 =* (x - 2)(x + 1) = 0 =* x = 2 or x = -1. . () . x(x - 1)(x + 1) [] 11m f x = bo:. (. 0)( ) = -3 =* No V.A. at x = -1. :z:_-I :z:--1 )',,~ ~ x + 1 !im f(x) = ±co =*1 x = 21is V.A :r-.2± xJ (1 _ 1 ) [im I(x) = lim 2( I;:'I '2' = ±co :;, No H.A. :z:_±oo :r-±oo x 1- x - ;?I 3. f(I)=A+Bsin~=A+B,
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