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winter_2010_2

# winter_2010_2 - MCV4U CALCULUS AND VECTORS Date TEST...

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Unformatted text preview: MCV4U, CALCULUS AND VECTORS Date January 25, 2010 TEST Chapter 1 LIMITS Teacher: Teodoru Gugoiu Name 1. The function f is deﬁned by the graph represented in the right figure. Find: [KIU 3 marks] a) xﬂtﬂjtx) = L b) lim f(x) :0 .7r——>—4+ 6) lim f(x) DRE. x44 d) lim f(x) == ‘r xm>0 e) limf(x) 10° x46 f) lim f (x) '—'- '2. x—y—z 2. Consider the function defined by the graph at Question #1, {C 4 marks] Analyze the continuity of this function (continuous or discontinuous) and the type of discontinuity (removabte, jump or infinite discontinuity) at the following numbers. Justify your answer (explain why). a)atx=—4 «3'? ‘tb éi\$wm4€huou\$ and if:— UVL ¥v.c>¢) == '1, #‘r Lt“. 4L!) a; 63 RP“? Algmuﬁhbt‘ﬁ) l u-‘J—HJ- , #4) “‘tf b)atx=0 «h 5 t K O {I * , . " q -: ‘ ‘ f . -°- 4:. “5’ 3‘5““ WW t tub om: (vewqulo\€ d‘s‘w“ m” ‘5) MW L36): L‘" 1’3“- — )L—‘Do c)atx=—2 Kim 3r“) = ‘1— ': {“23 )L—>-'l , Q— ‘5 mu‘uhu ouSq‘t Y =-‘—'2. a. UUL £L.,(') '2 0‘9 L tu‘timi‘tﬁ, obscouﬂuu Hg) 3 f tx)_- gm _ [K/U 2 marks} 3. Given lim f (x) =3 and ling g(x) = —6, use the limits properties to find Iim x-—>0 “,0 2g(x) L‘ '5 qﬁat) TESL“) likaf" 1”)" at“) 1 1) (Etta; PK") ”"8113 -: ‘luL... -.-= 7) -; M Q {as 1°35“) hm [2 ﬁnd] '2.. QjUL 0&0 ’(‘Do that = bww :_ BMW ﬂ 1-}: '2. C— 6) "m, #- r+ o Q‘s: -— r.b/l-l- Page1of4 4. Find each limit. [KIU 12 marks} '3 [1]a)1imx *3 8 _. LIZ—3.” :3: —+ x—ﬂ x- 2 "’ "" 2 —- =, n. --—— '9 H41: +2xA8 I K...>__\+ Cgﬂz) cw 1"‘5-4'." ¥fl '"Q'"? E [2]c) lim 4 ‘6 Lu. CfW/ﬁ CH?) C" “+7 Um. 0H2) CRT-H) 4-68) 1542 _. 3 =—' x s pun, wcilﬂﬂmwzm2¥*ﬂ p); 17—52.th- \ _. - (\l‘ tr ‘r [21mm ”‘16 “Um.” ‘6 ,W'H' =\A~M_ (Msth-k xalﬁJE— 4 ﬁ—‘J‘Q w_|+ W 41"? ﬁ.-§\Q X _ {é {’b‘a l .. “TE-Ur l x 7‘5 [21e) 1m x 3 2:2. = 7’ “HUT-3| ‘wab‘ .--\ 3 x 45 . x—3 .9. x—‘a but" "Cc—31“ ?‘ Kim. K =4 .1 ML- \x 3‘ ONE- K»; alt-var “("51 x» 5 '" 5. Analyse the continuity of the foilowing function. Graph the function. ' [A 4 marks] x2—2x+1 , x30 ‘ ﬁx): 1 , 0<x<4 X J; x24 “1‘ " = ° w 1 _ ’_ \ 5‘ L’- wa X4!) =\J‘~\M Li‘ ’2’“) " L5; Um. Kim) ”(Li-tr X‘JJ’Q. K96- K.) Lt" y. (L. x a: |+ :1" L. *‘| Kim Q“) 5L4“ . K a: Um ho :Um — K :1 #va 5 1-)0'1' ,L-w'au'k at r, ‘t 1» §C°U=omﬂlﬂﬁf|51 L—‘I‘bk -~'1- =9“ “*0 .. 939“”? obsmﬂuuﬂ‘ﬂ ~ JUMUWS J #91:: ‘Q‘ ‘5 w“ '0 Page 2 of 4 6. Consider the following position function: s(t) = 2:2 + t3 [A 6 marks] [1] a) Find the average velocity over the time interval [1,2] £\=‘-»i 35‘ "-—*~ 50):: l'i‘lﬁ'b 51:13S1r—5C23" 9.031453} =1“; 1w: 5......1'5‘ e. W“ e i": :. AV =--\’5 “4% ’czm'tt 1—" [3] b) FInd the Instantaneous velocity at the generic moment t: a. Show your work. 3 emu-s 2.0? ’cu ' , v” «ME-NH“ may“ «(MiG-tot stemm- -—'LCQ-Ht) the“) k.” Dew await) -\- tot-eh) - 9.qu . a Dem -t \$09 M '1... 9— Um-tkf— 11 "r “ﬂux-D} . \r : Lt am: '5'»: — M 04 L36 Lq—th—MU -Ir 04') Bat-thsa't (a +030 tea " w ":- ‘L L2 1:. Hi} 1- LCQ £02 t. cat 91') (a) rec-Q [1] c) Use the formula you get at part b) to find the velocity at l= —2s . .. m ,__ \r' : tot—65% “'3 “1"“ == Ltbz) *AL-Z) —; —- 2 +0 I:— ' \r t—Z) :Ltuws [1] d) Find the moments when the particle is at rest. \r‘mo ecu-tau. at“) =10 :. t t -11; ,3 RVTJ'tQ'E'ECL-z ﬁ> (215—6 0'? QE‘F-E-tg A. “AMEN-a “b -:...0 ‘3 0V ‘3’ l 2x 7. Find the equation of the tangent line to the graph of y= f (x) H x 3(- U'th) -- £10 at the point P(1,l). Show your work. \J U \ \ [A4 marks] “1.: ‘\M._ VJ»- __._..—-— =— -- \A">o \A. \i’k': bx—Qo Q'JC‘A' L a. (t-H/O , \ .. i. \u -— — _. 2:: Li». W 2. z) “3 - = L2,. ('1‘! 45 "D “’7 o \n P U)” Attic-'4' ‘0 I L. . _. \JU-K— 9K :. '=- “i E + ()— 't M40 ﬁlm“) W ‘ _ bus. at fClai“, '-“-'— \W’“ mulch“) \N") o 8. Determine the instantaneous rate of change In the volume of a spherical balloon (as it is inflated) with respect to it radius at the point in time when the radius reaches 20cm. [A 4 marks] 47: 3 H t: V =—r — '" 3 limit his LCQQwIH'IoW’C‘Dﬂgl Ike: Uh- ig: 3’ W” Alt—>0 ‘) :‘ﬂ: 02\$ (1°31- : \Qooh _ we. M ,7; “-20 M '3 (—- II M Lei new? _ “L“; U») \9_Q=\Coo\l ‘1‘“ “L. M new \A '5 '5 Lily: SUM— QLM“) "(m 5 Ut-‘aa \A 1 3 3 .102) th at UM“ (1‘ “k _ Lg Kim—- 03’) Lt'Lo 3 \ _ \A —? a W Page 3 of 4 L‘- 9. Use technology (a scientiﬁc calculator) to estimate the slope of the tangent line to the curve y = '1de H at the point P(0,l) by using 11] : 0.l. h2 =0.01, gnd I23 = 0.00] . Show your work. [A 4 marks] _ _, n. —| MaW°)=‘W~__—lﬂ 30.002319; 0-\ O-\ "i w (Eh- m N i(o+o.o\\-— L° -.—. Mun-Jo -‘ [email protected]>’>.a°2‘2”> 2 — o.o\ (Lot '3 1PM) “imam *‘ ~O.coovp’b33 o.oo\ 0-60! .-—-- -— We 10. Consider the piecewise deﬁned function below. Find the values of the constants a, and I) such that the function y 2 f (x) to be continuous at any number. Show your work. [A 4 marks] x , x50 ﬁx): ax2+b , 0<x<l x3 , x2! 1:"? H: 3h=o NC 13‘ Cl) geek ® L: Sum *:° Lakiw. Lmt1i53=qib 0L:\ the 0" l‘r‘ﬂ' ‘1;me tail—W3 =5 ll: Um. 17"” =\ X->D"' ﬁ..:;\*‘ . Mo) = 0 Sim =.\"a=\ Ltztriba) [:2- LZR:R—Ci~)\-ﬁ‘ b: 0 6) Guide "-'- \ ® at ““5 vow-URN” V4" 1'. 5:— ‘ie muhuuoua» q=\ was. ‘050 . . v‘1+x -v'l —x 11. Compute the limit 1im-——-—. P HO Jig—m [Ti S3marks] 1: EM \ii—u —\li——x \iw «WI—7” \lm—tx «hr-x O . _____________..____. KWDO W ._m th “La-.1 "6 L-wﬁ Mao—mix) h-tx ’t 1"" ':—~\1W~4 ' \$-‘>o [if—em “Vi—*3 \Si—tx ‘UIT'S‘ :UM M1“ . {1—H til-”K 1"“.- W 3:. poo 9" Wt vx ”tint. Wm M “(ii—ﬂwth *'"’° QM; 1 tF-‘ﬁ 9" \l Page4of4 ...
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