summer_2009 - MCV4U CALCULUS AND VECTORS Date TEST Chapter...

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Unformatted text preview: MCV4U, CALCULUS AND VECTORS Date Aug 28, 2009 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] 3) lim f(x) T- —~\ .taf)" b) lim f(x) 1 'b raw C) iim f(x) ‘3 H E r—>O d) 1imf(x) =1 7) I»! e) lim f(x) s—‘\ .\‘—>3 Xv) * f) iimlflx) ENE 2. Consider the following function defined by its graph: [G 3 marks] Analyze the discontinuity of this function (continuous or discontinuous) and the type of discontinuity (removable, jump or infinite discontinuity) at the following numbers. Justify your answer (explain why). a) at x = —5 fill C’Ovcklwugu E; r butt M) i (—537: Mm re} (up -7 33" treat 2:. l b) at x = 2 cfigcmfiwi cu b ‘C Am” F Chan?“ WA 2:5) 7' I H I d huts. “ism *? '7! [21.12%ij it“? xv)?” ffigfi c) at x=3 k ‘ ,~~tt‘ft~U‘\kUUOU E. C \"L9‘1--‘“‘f”~“¢*\‘ d‘gm“‘\"m’fl 5) 94% 'thr'“) r '2 ill-jg-JV t5 X-b “a 3. Analyse the limit and the continuity of the signum function 5gn(x) at x = 0. Graph and explain. [K/U 4 marks] —l ifx<0 sgn(x): 0 ifx=0 l5 r'fx>0 Q3“ Sync!) DWE- hmp 33w U)=u\ r90 K-DJQ‘ . k '5. \ Max—v” QM” cc“ 0 R 7% tacit if” cask». ( 03 "-1" ‘3 gamut) ts; augusbil “U005: Qt xu—to "they? ice 0“ ‘3‘”? u . M . 7.: Quit» L3H") ark Y 7—- o ‘09wa EM“ swam) w A 1% \ x—oo" {ad‘ _ PagetofS 4. Find each limit. [KJU, A 8 marks] . x2—9 “3 ma) x+3 2 .71: = H g {1 51b) hm x2 6x+8 fxy‘gtxflz) L iii—1‘ Lt “I” 7- .. L x—>4x2_2x,8 11mm,“ =” “L; "J- M 1;. "a" a“ 5 yrs-v1 96(th gal-f )t-t'L- Hut.» ‘3’: +5 szk‘fi {1.51c) ling3x32+27 :— QJW K D 1:- X a 'x_ wa-B QfQCYVQ *d'5 x" f , ( fi-tfl'tfl __ 2i : m f; w%~5"' Q 1 [21Qiim “H‘WH‘ . \TL-tx fibrx ' he“; atmi‘L x—)02;2+x_ i2_x\ ,r ~- Ww ’- _-»—r-“} 1-H; “t 1‘ \IV‘W“ / \S g -t' .-5( r bk (Hm—cw) We «Ex 92% 1" 1 o -_-__ —-;-”—”' ‘ 2‘*>o Elfin)... (1-x) WU:th PL.» Vt)! *WK .... [H.32- _ 2:3“, r \Yt-t‘rt " "2.. “"’ Q— [2] Jl‘i-rfl:z‘3:] U, 733’. u f X 8 at“ D “AM Kan-J “(5—4 hm. (Ll-flu *Kt -\— .. azu-u] #- % M g Lu.” : \ r2 X">\ Kira»! “‘3‘ “Lt—q Ukw>‘ Ku‘fifiu 't: Unet 3 ~—-- r"— ’ .Lw a) \m, i; Hut—“5% ° Mist. _ Um __ 1 __ L J(mm t FF t-t‘UTTe't" arc—)0 if-Wt—“td'k‘ (\*‘r\:¥) 5. Consider the piecewise defined function below. Find the values of the constants a, and 5; such that the function y = f (x) to be continuous at any number. Show your work. [A 4 marks} 3 —a ,x<1 x+2 f(x): '1. ,x:l xix-t!) ,x>] . t :L...” ...n —-Q_ Dd‘M. Jitfi) z: why a" "" \ \_m.=_1 _,_=> 0.2:: -"\ K-r'n‘ K-‘é‘t" 430 = _____ it‘s —.—.7 3 (L —.=‘) true '55\f\/ X w my —: m» ma E3 about **>\+ Pagezafs 6. Consider the following position function: s(r) = 2‘}: —1 {A 4 marks] {1] a) Find the average velocity over the time interval [5,10] Jet: 59‘; e‘ :sc's) —— iii—€11 e Wu .g T \ML. CNQW'ofi-Q—a Mgr we; $2 r 5003 51m .—.«- Qwu “mu\§cmeh_ AV ._.‘ $33 $2fij “Hi I F)... 1,5“ 0.br\m,b LEJ\6§\ .1) m at F, "Ca-WM“ to—Ia g D‘qmtb [3] b) Find the instantaneous velocity at t = 5 . Show your work. -— \l —-L+ hm“: M \Ar-‘m; k 1m '61" e..ch “WMUAK f ML w seat-ff"; L‘— ‘N—N‘J \Il (9-\l%4M+‘E3 rpm 1M szu '99 m we f 7. Find the equation of the tangent line to the graph of f (x) = 3x —2 at the point P(2,5) . Show your work. 3;. k _n x __ '2. _/6 \ [A4 marks] mg: QM (1+ 3 43(1) : {hm— ’b_(9.4m. T“ 1' M“) o in M”) Q A h an“ — "2'1; QM-r'bkq’a—flrrh 7/ “(Fl- V50 8. Consider the function: f(x) = 42:2 +22:3 . [A 5 marks] [3] a) Use the alternate formula in = Iim M x to find the slope of the tangent line to the graph of the curve at 1—H! — a the generic point P(a, f (a)). .- ’Z. a, _._ 7" '5 UK: hwy bx utlx .. C 3n +2013 *mfll 2mm. x, K = “BB-nil.» Fm +351; than. xvi...“ X'J‘“ —_u, —5 thin. (w) _t L QJW UAQizikv—m?) 54». X'- Kata. 7; cc. '2. a 7’. -=- —?;Q_iuL. CY—kq] +2, but, a: FQcL—t- Go} K—‘Ja- .o. v“, -= .» G a. vb Gear." [2] b) Find the point(s) where the tangent line is horizontal. W\.-.—_o x=os 23:0 “=> RUDD") __ 7—: .. —ni Goa-rem fade) lfigflBFk2=w\ i> B(\s D QKCq"3”° :. 'flm taug‘m't- tab \AoYiFNHQ/ “At Our-*0 (“I-q: 940,0} on. Poms-4) Pae30f5 \rb 9. Analyse the continuity of the function. Graph the function. {A4marks} XIX—ll xi] f(x)= H ’ A Us 0 ,x=l 7k x :7 \ 0) X "all -w K )4 4—l 3 >X (but. #90: —~\ fi*>t'" ‘ . 5 l1 t =>t$yflx3 \‘9 5°3m“l‘“”°“ 5min q g “A: X r’ I‘ . flwi‘ TVQVD. l5 0.— AUWP 3:353qu WINE) Cd: {1.4 U) =0 10. An oil tank is being drained for cleaning. After 1 minutes there are V litres of oil left in the tank, where m} = 35(25—32, a 5:5 25. [A 5 marks] [1] a. Determine what V(0) and V(25) represent. °. We): "be Cats—.037— = 2x818 tum. Re the w.in Vonnegut an. :. \J(r;5):35t_25v2531 :43 t5 the. that Wtan at at); Hum 95 WAqu [1] b. Determine the average rate of chan e ofvolume during the first 15 minutes. JCLT—O '3 V1:- =. 7 L JEFF?» '3 V2 =~\l(.ts) :35 (25-453 =‘3Quo Di "‘— w ’ "LL’h-Kl '" 0 ‘ U ‘.’. The. QUEVOQQ Wi‘l— Bl ctran 9:\ math ckqu l 4L ‘5 WW“ ‘5 “Vi-9.5 RiNeS/uuuute. Team—9.5 in; Cl" [3] 0. Determine the rate of change of volume at the time I = 15 minutes. *2. n.— —— \e \I t t -V @5"L‘5fl6& “LC: huh CS ‘4') ({5}: 008 MW “‘50 \A klij M '1. (to—tum W —_: 233 Slim» ._ 23c; but M \M-éa l“ \A—>o :. The... \ua’tquiome—DU5WQR (A Chat}? N M“? vmiezd Meg/ mu wowed +0 Jam. 06c Arts-ts minutes» is» '- Whale Page 4 of 5 11. Use technology (a scientific calculator) to estimate the slope of the tangent line to the curve y = xix hf; at the point P(4, J3) by using 11 = 0.000l. Show your work. [A 2 marks] Mg [H‘fO‘OQQ‘-:\IL\-*o¢;. “'th fr; MW O-Q‘OOI NIH-moot «AI—LEW ._\jq, 0.0001 '5'— -..-:. (3.155163? :3- “M- 31599; oi bested? ad” ? ( LUV-6'3 3%» qfi‘roflwet“ QQNQK 31% o . 155' 12. Find aand 1) such that ling] “ax”? _2 =1. [TIPS 4 marks] x» x Q‘Wv its) =i (stuck; gm» 0 fixmmb " 2— :0 =1) 1‘. K")D ~>o “we (Em/twat)“, cum 4- \imaat +2—— K *9 0 X i090 W7 ii 19? ‘F’ q a W Q Li, 7 ...
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This note was uploaded on 04/11/2011 for the course MATH 1400 taught by Professor Grether during the Fall '08 term at North Texas.

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summer_2009 - MCV4U CALCULUS AND VECTORS Date TEST Chapter...

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