EXAM 2, MATH 132 WEDNESDAY, OCTOBER 23, 2002 This examination has 20 multiple choice questions, and two essay questions. Please check it over and if you find it to be incomplete, notify the proctor. Do all your supporting calculations in this booklet. In case of a doubtful mark on your answer card, we can then check here. When you mark your card, use a soft lead pencil (#2). Erase fully any answers you want to change. Problems 1 through 20 are worth one point apiece. Each of the two essay questions is on a separate sheet. PUT YOUR NAME ON EACH SHEET. Show all your work and indicate clearly your answer to the problem. Partial credit will be given for partially completed solutions. Each of these problems is worth 5 points. There is a total of 30 points for the whole examination. You may use a Mathematics Department approved scientific calculator. You may use a 3 x 5 note card (both sides). (1) Find «fl;;!) (B) cos b (C) sin b sin 1 (D) cosb COB 1 (E) sin ( eb) sin 1 (F) cos{ eb) COB 1 (G) In(eb) (H) coo ( e6) (1) In b (J) lnb 1 tdt t1 =) c= e; b => 1,.( b 0 1.(':' 0 b
2. EXAM 2, MATH 132 WEDNESDAY, OCTOBER 23,2002 (2) Find (A) :t! 1\" (B) 'K,I:8J.::;::\ (!/ (E)... el 11" (F) 11" (G) =l 21\" (H) (I) 'K (J) 1I"2e+l..., te1rt dt e I 1['("" '"'C.." 1t't" JT..". e c c T(t"" 21\" 11'J... 1f e 1("'C () I ttt: trc 1ft ':;7="" te e 0 fe
EXAM 2, MATH 132 WEDNESDAY. OcrOBER 23, 2002 '3 " (3) Find the approximation by the Trapezoidal Rule with n 1 21 dx 1 x 2 to the integral (A) (8) (C) (D) (E) (F) (I) (J) 9 16 7 12 29 48 Q 8 II 48 2 3" 11 :i) 35 48 3 i /JX: { J.. 'f ( 1 + :J.. 1 z t /),X 2;. r + t) '1 I?
. EXAM 2, MATH 132 WEDNESDAY, OCTOBER 23,2002 4) A radar gun was used to record the speed v (in fils) of a runner during the first 2 seconds of a race: Find the Simpson's Rule estimate of the distance in meters the runner covered during those 2 seconds. I.z J 116 sv t (0 + ;)0 0 I... 16
EXAM 2, MATH 132 WEDNESDAY, OCTOBER 23,2002 $ (5) Determine whether the integral :1 J3 (x 1)3 dx is convergent or divergent. If it is convergent, then find its value. (A) (B) (C) t (D) k (E) * l @2,V (H) (I) (J) divergent..,... L 3 1 1 dx I, (X' "() J fyi. tcp..... tjt:i> I 3 c I, I :r.t;;jif d ('t f) '\.
6 EXAM 2, MATH 132 WEDNESDAY, OCTOBER 23, 2002 (6) Evaluate the integral J:lnxdx. (A}2ln 2 (B) 2ln2 (C) 2 (D) 2 (E) ln2+ 2 (F) ln2 2,ln!2 ([2ln2 (1) 2ln2 + 2 (J) The integral is improper and divergent. dx x 2. x
EXAM 2, MATH 132 1 dx is convergeot (i) ii) ill) t dxisdivergeot 1!, dx is divergent IX t:;>, @ (A) i) and li). (B) i) and ill). (C) i) and iv). (D) i) and v). JJ.!l1 42!D (G) ii) and v). (H) ill) and iv). (I) lii) and v). (J) iv) and v). iv) v) [;Tdx=2 U/ [. 0 rgii II " if ' 3 4 a"j cpt Xi' /) ii) I"'; ) t..<»j v.t r a e.i /f 11) J't') [ x dk t::p :;J. X IA.. I
8 EXAM 2, MATH 132 WEDNESDAY, OCTOBER 23,2002 (8) Find the area of the region enclosed by the curves y = X2 and y = x. /' It::: 0 ( xl. )( )... t 3 0 I i 1 3
EXAM 2, MATH 132 WEDNESDAY, OCTOBER 23, 2002 9 (1. d T + /0) (::t :;;>0 /2 )
10 EXAM 2, MATH 132 WEDNESDAY, OCTOBER 23, 2002 (10) Find the volume of the solid whose base is the circular disk centered at the origin with radius 1, and whose crosssections perpendicular to the xaxis are squares. c rosj if :::) ( J 3 ;). == t{k 0 I Q g l 5
EXAM 2, MATH 132 WEDNF3DAY. OCTOBER 23, 2002 u (u) AnMDPhD lays a human liver along a ruler with one end at 0 and the other end at the 6 inch mark. Using an advanced CAT scan technique, she is able to determine that a slice made at x inches, perpendicular to the ruler, has a cross sectional area of A(x) = lx(6 x) in2, for each x along 0 $: x $: 6. Find the volume (in cubic inches) of this liver. (&:) JL (B) 12. 25 t ; G (C) 12.5 D (D) 12.75 I / X) (E) 13 G,CI(It A(l'=, X ( 6 (F).l3.25 (G) 13.5 (H) 13.75 (I) 14 V )" (J) 14.25 { J 0 tp "" [ 3X x x' f 6 "J..::?) t. 12
..11 EXAM 2, MATH 132 WEDNESDAY, OCTOBER 23, 2002 (12) Consider the solid obtained by rotating about the yaxis the region bounded by the curves y = X3, Y = 8, x = 0 Find A(y), the area of the cross section of the solid made by the plane perpendicular to the yaxis at y, when 0 y 8. '8:l J (A) 1rY (B) 1ryl I"") 1 1rY3,.. 1rY" (F) 8y..;; I 1 '/1 (G) 8yl (H) 8y! 2 (I) 8yi (J) 8y2 X;u "'" I/IJ 1
EXAM 2, MATH 132 WEDNESDAY, OCTOBER 23, 2002 13 (13) Find the length of the curve y = x + over 1$x$2.::...f 1+ '\.. 1... I' r. I 1 :r.j ol..s q 'l. (.ff J 1 J I + J s r x I r 1... Q1: (I) '/) (("I 2) )b..ett" So 5
14 EXAM 2, MATH 132 WEDNESDAY, OCTOBER 23, 2002 (14) Find the integral which gives the length of the curve (A),"3 d () jo y3 + 5tidt (D) 103 v'l + 5t2 dt (E} loa va+3'i2 dt (F) J: Vi+3t2 dt (G) J:v'3 dt 3 (H) 10.yI dt (1) [03";3 + 2t2 dt (J) J:.yIl+W dt x = t f, y = ti, 0 t.$ 3 J,x d't 1,... f1, L '"t t1. t.;l 1/",? l....to
EXAM 2, MATH 132 WEDNESDAY, OCTOBER 23,2002 15 c:')td 6 1.3.3
16 EXAM 2, MATH 132 WEDNESDAY, OCTOBER 23,2002 c== (2.) <1 1
EXAM 2, MATH 132 WEDNESDAY, OCTOBER 23, 2002 if (17) An aquarium 6 ft long, 1 ft wide and 4 ft high is full of water (which weighs 62.5 Ibs/ft3). Find the work (in ftibs) neede9 to pump half of the water out of the aquarium over the top edge. / =0 (A) 749 ///'" 5 <r22_d { If '" /; " (E) 751 (F) 751.5 (G) 752 (8) 752.5 (1) 153 (J) 753.5 ( " '). (;.f( () 6 'd (D) 750.5,.. d8 'L,. 1. 0... Itj Ih
13 EXAM 2, MATH 132 WEDNESDAY, OCTOBER 23, 2002 (18) A force of 10 lb is required to hold a spring stretched 5 inches beyond its natural length. Find the amount of work done (in inchpounds) to stretch it from its natural length to 6 inches beyond its natural length. (A) 32 (B) 33 (C) 34 D35 ) 36 (F 37 (G) 38 (H) 39 (I) 40 {J} 41 6., X:l.i /0 J'O x J, "': ': V'{" X:::6 de., w f() (, 16
EXAM 2, MATH 132 WEDNESDAY, OCTOBER 23,2002 (19) Suppose the waiting time for a customer's call to be answered by a company follows an exponentially decreasing density. If the average waiting time is 3 minutes, find the probability that a customer waits more than 6 minutes. l' (A) ei (B) ei (C) el (D) ef (E)et (F) e! (G) ej 1W.J (J!l!::; (J) e2 v (;/ :J<f 6) e f 6 e _t:/, dre {, c e "1"
20 EXAM 2, MATH 132 WEDNESDAY, OCTOBER 23, 2002 (20) If c if 0.$: t.$: 25, f(t) l 0 otherwise. is a probability density function, find the value of the constant c. (A) 1 (B) (C) 1 3 (D)! (E) (F) ft G.l.. H) i\ 30 (J) is I.J jrrow c;t;.'( t tp:j tc ci> bt'lf 2,..) c.l> d1... :: +((/ 0 c tit? CT 5C so
t) A ;.. a.it; :;)d, )I) f.., a... Q, (ol, J ).( _cl,#3 1 ""3 )
1T b) 0 1T \,.. 'I,.., At" 0 7f :;: j " ( f)