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sample midterm

sample midterm - gg NWERSITY 0F wﬁTERLoo MATH 11.7 SYDE...

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Unformatted text preview: gg NWERSITY 0F wﬁTERLoo MATH 11.7 / SYDE £11 MIDTERM EXAM - Fall .2004 Name (Please Prim”: I.D. # : Signahre : Inshucfors: E] B.I'ngalls .. Div.l+ (Comp. Eng.) [3 D. Harmsmex - Div.5 (Comp.Eng.) D G.T2nH - Div. 6 (Elect. Eng.) D K. Lamb - Div. 10 (Soﬁwane Eng.) D “’1‘ StastAa — 67552“ D653? E- Matmials allowed. : Formu‘a. shed: (attached) Doraﬁon : 1.5 hours- [N] i . (a) The functionf(_x) has formulay =x2 for O s x < 1. Sketch the graph of ﬁx) for —4 < x < 4 when f(x) is odd and pen'odic with period 2 (b) Express as Partial fractions ' 1 ' . (x + 1)(x2 — 4) P632 2. 0,5— 8 (c) Two voltage eignals I 17.1%) and 1139:) I are ‘ given by me) = 3 sow) } 11;L£3=2m(+) . (4) State er amPlLEude and angv\ar frequency (Nﬂ’k SI mi’cs)o§ +he two Sanals. (ii) (3an an GXPYCSSIOH 5:0,- {rke signal '0“; given by '1);Uc) = mm + mat?) . (iii) Reduce H19 expression obi-aimed in 125% (b) +0 ‘51 single sinusoid, and hence Slca’ce i-Es am PWcu de and phase (“3in «‘65). [NOTE : WSW/a. x035 m4 '1 ‘ ' ' Page 3 04 8 [l1] 2: . (a) Calwlate the ﬁrst six terms of the following sequence {a,,} and draw a graph of (1,, against 72. What is the behaviour of an as n -> oo? n2+1 n+1 an= (n20) ( Find the least value of N such that when n 2 N, n2+2n >100 (0) Explain H18 precise meaning (i.e.,b3 using a maHnemchal statemen’c) 0; H12 concepl: 0;. con’cino‘tiy 0518 {motion § on an interval I. Wagon-03:8 [10] Sketch the graph the function (‘5) xH(x) -— (x — 1)H(x— 1) + (x — 2)H(x — 2) The parts produced by three machines along a factory aisle (shown in Figure 2.97 as the x axis) Draw the graph of d(x) and ﬁnd the optimal are to be taken to a nearby bench for assembly position of the bench. before they undergo further processing. Each assembly takes one part from each machine. There is a ﬁxed cost per metre for moving any of the parts. Show that if x represents the position of the assembly bench the cost C(x) of moving the parts for each assembled item is given by M cmxdm —4-3—2—1 012345;- Figure2.97 where d(x)=|x+3l+|x-—2|+|x—4| #3225048 [3J4' (a) For which .[Zundcwvs b and Q can we {and a 1Cunabion x such mat (we): 50:) x02) +c(4:) =0 {-0“ all numbers E (b) What ConoLLHOI/ts musk H12 {Undzioms CL de, lo \$365161 “were is 420 be a function X such Hyat- OLGc) XUc) + Mk) :0 1Cox“ all ’watber; 4: page 7 04 8 [4a] 5_ The 7function {.(x): ' never {sakes Hie x—l Valve 2&0; 38% {(0): —1 <0 and. §Cz)= I >0, and 50, by Hue Iyd'ermeoLLal'e Valuz Theorem ) Here skould, be a point c ~0in 0<c<2 whefe ¥(_C):O. Explain {he ”Para&ox ". 'Page3ou‘8 Some trigonometric formulas Sine law: Cosine law: sineace—cosescce—tanecote—l cos 0 sin 0- sin0 tan 0— —, c050 cot-0 — sec’ﬂ—l-i-lan’ 0, 5c‘0—1+cot‘0 - sin‘0+cos’0-l sin(0 :1; go) — sin 0 cos 9 :hcos 0 sin 9: cos(0 i?) - cos 8cos 9:? sin 95in 9: tan 0 :1: tan 9 mw‘h'ﬁ'npmnamp cotﬂcoto¥l °°‘“’ W" W - sin ZB-Zsin 0cos 0 sin30-3sin9-4sin‘0 sin40-8cos'05in0-4cosasin0 cosZB—Zcos'B—l -cos‘9-sin‘9 cos 30—4cos’ 0—3cosB cos40-8c_os‘9-8cos’0+l sinﬂisimp-Zsin 54(9zh9!)COS 5494: 9') 'cosU-l-coscp—Zcos 1,£(9+9)cos “(9-7) mSB—w5?-—2Sin %(9+qa)5in%(9-?) sin(0 d; 4?) sin 0 sin 9) sin(9 J: ?) cps 0 cos gu' I 9 1_cos Va 0 l+cos H)” sm-i-i-( 2 8)! cgsi-i( 2 0 l—cosﬁ sine ._:I_-_(1'-¢°59)"s “‘5‘ sinB '1+cos0 l+cos9 sin Bisin glitzy/“at” cotaicoup- i ulna-:an- cosB+cos¢p a b C sinB-sinq: sin!!! a:_b=+c=—2bccosa ' b=-c=+a‘-2cac°\$? ...
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