Math 1A - Fall 2003 - Dales - Midterm 1

Math 1A - Fall 2003 - Dales - Midterm 1 - Mathematics 111,...

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Unformatted text preview: Mathematics 111, Fall Semester 2'30?- InstruetOr: Garth Dales September 25, mm Midterm Examination 1 L’d'fi‘f' r Your Name: Your SID: . {3-61 : Emfianfl \{ao Sec.,'lDZ. Directions: Do not open your exam until your are instructed to do so. You may not use any external aids during the exam: NU hooks, ND lecture notes. N0 formula sheets, ND cell phones. Answers without explanation will not receive credit. You must show and justifil your work If necessary, use the backs of the pages or the extra pages attached to your exam. and indicate you have done so. When time is called, you must stop working and close your exam. All questions are worth 12 points each {for a total of 50 points]. Good Lack! Score 1. {a} ('4 points) Lot 3'" he a function defined nearl :r = o. Write down the definition {invoking E and {5) of the following: Li florid varoja‘é‘ PG,WW Cg; O L [K‘ml -r— 5 W /]CG<)* 6( LE ‘ {b} {4 points) Prove fiwm the 6—6 dofinitz'on of limit that n. lim£2=4 o a m J, mm a“ 2E. _ x" .13.; 04IX*&l¢g IK’IXZ_/+|4i W -|i- '- I I '9311 a: a {Waldo} (a [HM/E (e) {4 paints) Evaluate the following limits by using the rules of limits [state at each step which rule yen are using): ii. mamEEE—I+IT M 31 + :3; —l’ {W @Aegfezfl H J— I? a x + .__l : M 5+C’FC’ {mafiatfiranfcflfi 3‘7” 1 *0-1-13 / C: I! 2. {a} (3 points} Let f : .5' —1 T he a function. Explain what it means for f to be DflE-tfl-Dflfl. Fat, #05 "2'55" {26 O'f'LE'.fo-"U'm I 3W6, 5 w 5 W W e: a; avg-5136 é wdig.“ T Mo? W. {h} (4 points) Draw the graphs of the following functions on one diagram: y=exp[;1:]1 y: x, y =1oglfx} {Show where the curves cross each of the two axes, and the relationship between the ouwee] 5 i w W ’0 3 .q /o:fi 4 [c] {'5 points} Let 2+I—I2 (————M {X-H fiwm : (x ewe-3) QWhat is the [maximal possible] demajn ef fgCaleulete 111311; “EL £31 fix}: fir}: £1111ij flit-'1 I“; Write down the equations of One vertical asymptflte and one horizentel asymptote to the curve y = f(x). 21!: I Q “I‘M cm W iii/~22 WoEmC-Mm W W 0 n A“ lat-7' x '1' x 2 i ‘3‘ 2 -M (:0 2 M HM} ( MPH/3 :(pegg Obi) X—j?5+ g 195+ - d - e thew) n M_ (1% M, (“5+ r” M“ SJ () xag 2? X713 W3“) W) :: 41.03 :: a-W _ 3-, .L—l {In} em 3%)" x3355: 3“ +" = —-’— : -r ITEM iii-1% 1—] i . 314$?! N) (X): 15W“ iflhfi : _x ( Rm 4') I?“ L‘iIrEH-t-f 5 w _.__"_7 a? Wm '14? d “a w i [any 3. [a] {5 points) Find the number c such that the following function f is continuous at I: _ fl . rim/{em 2:2: 75 M f @kfln-moaw I I Glxngl/befi MW XZc—CXFSJa'fXT-E X _. (3)1 Bessie : 9 — 3o '5 = ‘3‘; *4 [h] (5 points) Let f be a function which is continuous at .13: = oi and let g be 3 function which is continuous at y = f{a}. Prove from the precise definition that the composition 9 o f is continuous at 1: = o. flfgcm: m fiLWéjww/évmcmfiéanflmew i)fltangf. W 3 m we) jg) £35132) 9:.“ 3) 5-343), m fig; new, {:13 WW; W = wow) Tit/em , ego-D = fng 4. {e} (5 points) State without proof the Intermediate Value Theorem fer a funetien f defined on atleeed interval [mt]. M mg M em, W5 01,5] I)[ W a j,mwjfl@ é: 3 5- 70(5) .fléXéb WM 3000 :3 [h] {'5 points} Use the Intermediate 1|nur'edue Theorem to show that there is a number .13: ’1 in the interval [I], 1] such that 5. {a} {4 points) Let f be a function which is difierentiable at m = (I. Write Elm-m: i. the definition 0f f’flx} in terms [If a. limit; 30%): M €Ca+h3 4300 __._--__r__________'_____ h-‘FO h .551 ML: xii-7:1 f" Y‘fl ii. the furmula. for the tangent t0 the curve 3; = f [1:] at the point {:11 f {11]}. 3: {Va} ‘* ffifix *03 or an #Lj.‘ ‘3- W‘l(Pity—K1W 9,3041): ffdafl ><~© L. {b} {3 points) Calculate the derived funetien f’ ef eeflh of the foflewing funetiens f. [In each case, you should use stanciard rules of differentiation.) i. fix} = £31 sin{2:t} ’ .- 3a a ’ Mmesz (a? (Wax) + 6% mam-2x) J = e 1‘ “(3x)” (5m 61):) -t~ @E‘xfl sin m3 (1x) 1: 153‘! .5 ffig'flax + 65X. = 3r {ifix(51'n Rx) + 3L5? (5m 22:) mé'uflz' 2. m. re)=lag(3m2+1) ("fit")! 13,257 {’50: '3 L-rl - (3xitky IR ail/k“) A f t : F : I .. EX T sz‘tl : EX 2!; 3x1+1 iv.f{2:}=ten"1: f ’é) : mitt-1X) ._ I - l l. [-|' X ...
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Math 1A - Fall 2003 - Dales - Midterm 1 - Mathematics 111,...

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