winter_2009_2

winter_2009_2 - MCV4U, CALCULUS AND VECTORS TEST Chapter 1...

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Unformatted text preview: MCV4U, CALCULUS AND VECTORS TEST Chapter 1 LIMITS Teacher: Teodoru Gugoiu 1. The function f is defined by the graph represented in the right figure. Find: [3] 3) lim f(x) = «L x-->0 b) limi f(x) ii x—>—3 x—>2Jr d) \LL= an?) 5'" 5 “Es K-a mt QJVUL 5%“) m B 75’???) mu} £442.) 1:: 755—? " b 2. The function f is defined by the graph represented in the above figure. Analyse the continuity (continuous. discontinuous, type of discontinuity) of the function f at each number specified below. {3] a)at x=—1 brLb-Ccyfi') z: 1‘) 90“» in") :2, ‘> :1 C-D “a dgihumck. fi—a ——i' ~'—>—i+ \ o“ 9 is no". coU'i‘W-‘fi'e L Ni if 1' mi (1"“"“i:*‘ b) at x :0 -. Sum. fim r: 1'; Li (cu-d ‘; :. X i s chased-i Wow 5" “ KW‘W-m mic ) x~>o " ’ (3)3t 95:4 94‘” {if} 1» .h .i y“; 7.. 1. u; at in, nominations an y .- "am yrs —'2 3. Consider the piecewise defined function [3] x+], x<—1 f(x): x2, —1$x$1 J}, 3: >1 Find: 3) limlflx) Muh- Cy‘} 2:. ——l+'i 70 |_) Um lira!) -.:_(_\37 3‘ a) :. K.)ka HF} x—>- ¥h>w\_ {v3 ‘ b)limf(x) -.::.. 0'1 ‘:-'-O .x—->0 \f‘f:\' ' KN 'f‘fl 7“ ’1': l ‘ ‘(V’A' '5:- .0 c) limf(x) mm. it!) = t \) KM}: j ) ) if) xal {n} r- _ f " i 4. Consider the piecewise defined function. Analyse the continuity of this function. [3] x2 -2, x<—2 Act V: -L i 1 WM ' r. — — 7 2 I - .. _ — 1'; ('2 f(::)— x, ZSxSO my (1:0 = (a) #1. =1, )Mmi‘w) , L n z ) \ Vl+x, x>0 :. ii: ukkuuous at V's-L 2.1.0 “UL- Q’C7l3: —-D:D) MW q {-t > {6‘5 0" 1 ~‘> 0* ' at it: mt muiiwiius aft ‘f'rc Limit?) ("03) 63 0h. (.0) 0°) 0‘ Q n is CBwHVkUUuS‘ on. Page 1 of 4 5. Given iim f (x) = 2 and ling g(x) = —4, use the limits laws to evaiuate: [3} x—> x92 [118)1hn[f(x)+g(x)] s in» the s 9w» fin) —_ 1 “an “L H2 #31 f-r'z, ~ i [11biiianiif(x)/g(x)] a M s. 1}“; -.~ — LL hm Cu Cr“) ,L-V'L .. a = G 8:“? [1]c) lin;[3f(x)—2g(x)] :3 513m 4M5 fl), m an) =35 (a 2 L Q r $1.31 “"1. 6. Find each limit. [6] 2 _ r _ ‘ ’ a) limx 4 =2 Mu». We}: 50m. CAI-13 1—1-7 :"‘L" H72 x+2 ‘0’.) iii-Ff, fag} / i .M .4 W \ ,, w {3) lim x } ___ SUM ._._ wt. .._.__.—- \ ._ .L _ — 'f :1... ‘ I c)lim‘/;2:M\M_W 2' ‘W Z=Uu~ A?) \fir-t'). H" "’4 “4 pm X—‘t Wei PM (Vt/{0 0’7 ' *9” 7. Redefine the function f to remove the discontinuity and to make it continuous at any number. [3] _x3+] i ifs-kt ( 1w)! *1.) W x“ i 9‘": 7271' WWS "" m w on m «r we =5 x—>-\ Y” {-04 . x5+i 1’. ‘A‘fl: - M" 3 H4 '5 ) Y"'i 8. Find the value of the parameter a such that the function f is continuous atany number. [3] f() 2x2—ax, x<l x : a+v‘x~1, x21 hm} it?!) :7 (Lu (1‘ #13:} P; rants... 1:) 1120 “"3 0‘ ‘5“ ’L. ..- L” kgfiimx 2’ 3 3‘ i HUT-1 3 r79! ,. ‘ m$;:\ ° .3 is Qnu‘hwaug ad (13:3 \AUtquH‘ A Page 2 of 4 9. Consider the function [5] ~1 f(x)="‘—_ x+1 [1} a) Find the average rate of change over the interval [1,3] ‘ l “"4 > "31:0 am "2-0 \ .-.'-" x,1> rb-\ I“! 2_g‘:—;‘:“""-"*'; :‘A?( "l’ 2*— _) BIT—flb*‘:. and”; Y}, [3] b) Find the instantaneous rate of change at x = 0 in wire 31:) I . . '2. ‘3‘. zLWJL. ) ) = 1"” +E _, Um... yfi- Y-{V "‘ \JW 7") 0 7t” 0 FM: X W gum Cf“) 7‘ 73>” x p; :, \EC-r- 5L [1] c) Find the equation of the tangent line at P(0,—1) W = 2. A. - : K —\ s— or): 9.0—0) 12 .- “‘3 9* 10. An fireworks is launched verticain upward. Its position is described by the position function [4] 12(1) = 201—:3 (where Mr) is the altitude in meters at the time t in seconds). Find: 44'“ 5 ht'=~Q-0*l=~l‘3 *1" 1° ‘3 V2. re 10 DJ “3"” a 60* Tilt-35 1% [1] a) The average velocity over the interval [1,3] \ AD MIA” 3);. \5 E: t .1“ '2, c=—-—- "Kris '5“ M? c '1 4' [3] b) The instantaneous velocity at the moment t = 2 hug—ht?) «Lg-\w-{bw (ago-1’13) \QC‘F‘ Wm... M Hz «'1' ks; JUL 11. Use technology (scientific calculator) to estimate the instantaneous rate of change at a :1 for the function f(x) =tlzx2 4; using 0:001, 0.0001, and0.000001. CL»=‘ ‘5 im "‘ [3] 3 lam-oi ‘) my 4&*“fi 0. L\.m‘- m '“‘ t \ Qoz’) . __ W 3 I o M 0.1x 'b “7.0.009! SO“°‘)1" JMQOD‘ “ 1: \‘\ 3 \A'l'b 00000:; {H ,._.. fi'l. U-oooom)‘ “\jwmw 4‘ .- 03. mm“ o - about :. \Rc': nos) 12. Find the slope of the tangent line to the graph of the curve [3] f00=x+3 x at the generic point P(a, f(a)). 2 K I» \ \ + —" '— 0c"? "H " “"2; x—m. tr 0 w )L— a» " x—~ in—>-— PM 160-— lb?!» —\ guita- okij ‘40-. a x 0‘ M = t— 2:. (11 13. Evaluate the limit [3] 5—1 W.» x... L id)...- x I 3‘ f7“ -.-_ Shunt, \ L: :0 - w ._..- ------ w - n »-> " ‘5 in}; m 0430:} Kim-‘30 P" Mi \rflL " x 1 y~t :19“ [in ——"'""‘ Eu (>64) Lic‘wjret)..,_wmm : 01m, “H”! ‘ Q4“— ‘tfig—j‘ L 7")! Pa"; = \M. " 1 “ l\( 3 ) fl__>‘ itwiA'a.) gun-t \ #- L—P‘It X (“H 1 Fr; 1 P a, yd\ \ltér‘rb‘lfbgy) U5‘q 2 #:5- 5—4 "5 www ' r ’2“ \ Pb bug int-‘— wg. ""‘ "' w" . U”) 1. n . ‘1‘“ L 1 or» l- UL 7 “’5‘ Li . . ‘5‘" K __ “.13.. o n W “gm-n " '2‘ ...
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winter_2009_2 - MCV4U, CALCULUS AND VECTORS TEST Chapter 1...

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