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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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