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summer_2010_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: [KIU 2 marks] a) 1m} f(x) = —l b) liril+f(x) =1 9... c) iim f(x) ONE x—>4 d) lim f(x) = -’- x—)0 2. Find the numbers x where the function given graphically below is not continuous. Explain what kind (type) of discontinuity is there. [C 4 marks} -P is vice“? Cou‘dmuoue qt e)\x=‘+ -‘sz‘Qﬂ)7L—’£CLH=L W”) Li- Lye‘wft ov Y'E‘Wm‘f “ML ens. mud-Kw i '\* ID 3. Given lingl f(x) =—1 and iin; g(x) =3 . use the limits properties to ﬁnd him. [KIU 3 marks] x—>2 g(x) + 2 Q.) 9. 41C!) "‘ ‘350 We it—v), tacit) -’c L Q, but. 47—90 -- 0AM. %cx> 1-4)?— 1—9'1- ll _ xﬁ‘ti—lﬁ“ T35”) ® 35—31%va + 3212‘ - 0.5» [(3.00 +9.1 x49— __ Lt—i) —-‘b . kiVuLE-Q‘ 59-093 -%i_§5_%£p © "- ,5 _\_ 9.: G) __ it") . ‘— Rim. (3'00 4? LVN 9" — ”fix—(b 1,ch .. -— _ ) (300 ﬁ—VL had“ 5‘ .1 km. =—l -__-_ -1 ® K—M. 334*) "i“ 9* F..._—a Page 1 of4 C) 4. The function y = f (x) is given by its graph in the ﬁgure below. [KIU 4 marks] [1] 8) Find the rate of change in the 3/ variable over the interval [0,2] . X\'—"-0' ) ‘ﬂ‘zo [3] b) Find the slope of the tangent line at . 7 . . . the point P(2,4). 1 j "_ ' ‘ f f . : A (4?) 4) 5) B (3 )3) , ' .. 5 is 739...!31...1;1 1;. 1. "ii ”*1. "1h— " = r M h M __n 3- (-4) ~— M e. - (545) . V“; £05893 5. Find each limit. [Km 11 marks] '5 [1Ia)linéx3+ll = 95'“ ____ 8».ng r5 [1]b)xij1,1_13«/x+3 75245 Ii) x+ 0.1+‘ . QjMX+1=\$ ”5‘3- 1—». K14 Kﬁhw+mnm= [2.5]e) lim ‘ _ \l—X +"l' [2.5m lim ”‘3” ﬁ *‘ﬁ— x—)—2x2_4 _ 2n QJwL, Gaff) LXm-ZX 414-9 zbma iA-(g'; x.—>-2_ (gt/(2)002.) 0.. b ‘ =-%i~ 7" ;‘2..:"c‘l- ‘....":-- up "9—1-5 ﬂ pals WAC“, X —9. -‘).. *7. 4: .._—— ”'1 fig; C j 1. i 3 tr “3‘ \G -* "’ g .. 9JVML. x_\g =. E #L‘. on. Quin-L. x1]- =‘3 sis-w; X—>-‘L X —H— Page20f4 = "“5 6. Analyse the continuity of the following function. Graph the function. [A 4 marks] x2+2x+1 , st At x =5.“- x: — , < < . '- g m % x:4x 4 \_==- til-quf—w 2. '1. £3.92 muti‘muoue elven. (“\$3,533 a K. =“W‘ﬁ‘m -.=. M (33%,) oh. Uh“) yuan-t 3-05 7— ﬁn ""”’ w :. J; at: mumuwg A ”A? L.— = Um. Lx1+zxm3 =. l ’3 - x—m" L_—.f-iL - J} 'v: “at @ﬁuvou{> at he 41(0):! ' J; i: muhmoovs uC’t' ““3 mumhez. 1,7ch x :0. .0 7. Consider the following position function: 5(1) = t4 + 4! [A 5 marks] [1} a) Find the average velocity over the time interval [0,1] 4“: 5 s, .-. 0* actual} =0 tr— '3 v.1 =\$£\)=\“'rttm =5 AN:— LEQ‘ 1‘ 5—0 =-—-- 5‘ .0. The. A-V oven. Loaf"; i‘a ‘3 “/5. \—-‘o [3] b) Find the instantaneous velocity at the generic moment t = a. Show your work. 5 (an :er + tre- 9’ CM“): (My? +‘+Lu+\1) b - '+ u- - Mm UH‘O‘H Him-m _ (arrows - that ”c . -= (my ('1.er +H— \A—>c \A :- TVL \V aft "k \n W. L" Mot-Hi — ”3 =‘ijWLCQ-th’ “*M‘“ Q‘ttﬁk u—gLi-q-t-“r ‘>° ‘ ~10 (“WE-=91 ‘1':— Uwc )‘(QMHOL + huh“? U-—>o )‘C kg)“ 8. Find the equation of the tangent line to the graph of y = f(x) = 3 at the point P(I,3) . Show your work. WV: QM»; £CH‘0“¥C0 21:: Q [A4marks] “—90 \N = £2: Huh, '5 -— '5 s Um. 39:211.... w“... a 9, “‘3 k n3... *5:- e Lx —~O 1-. “civic. 9§”%Ec’(+ln/B m3 .._;_ ’(BX Jre't") M» o ['1— Lem—Q \~ 4’ TM, eczuﬁm‘ou. 94V “Hua. EU?“ ﬂue. in“, “C” “at P039 \‘3: \'L-->o Gamay my. ——G x-t‘il ._._____\ (5 Page 3 of4 x2—4 |x+2| 9. Analyse the continuity of the function y = f(x) = 9—i— “; - I‘ X “9... "lﬂ'Fi X+94 X 9. :> > o. _ * Xv‘t' =L—X37'NL 5L . Graph an xpiain. {A 5 marks] puma.) Ab X = *9» L—‘=-—Li\M- [ll—"'x’): L!- ﬁ-‘s'r‘l.’ x—a—aft J; (-2.) DRE :9 {l ‘1': Mat Codi mucus: aJL X=""1- LAM”? O‘szuﬁ RUNS.) 10. 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] x2” , xS‘2 r: is caiﬁn‘VNWB Mm, (—053.— 25 9 (—2323 oh. f(x)= |x| , —2<x<2 L :1, on) \fbx+1 , x22 ﬁt -- ”'1- PrJL' ‘tf = ’1.— L_=Um_ Lx1+av=4t°v L= Um... \x\ “HA-:9" x-—>—-'L' art-‘91," 'L H-n—j=em)ﬂ=‘ﬁ°~ 59w) srfb-H _ . \ -_-_— _9_‘ =9... ‘ -.-.-_ \ k_bu. \Xl \ \ 9‘th W W rte—1* +- rL-a’i— Hi- 5; is cod-Numb “It x2—L . {1 \‘a cow-tavvoub at 7‘ 5 L H" L\--’C°K.. =04 '1. -—=- Lb—H °h‘ I It L‘s-H g—‘L z Ob ‘o :— 2“. 'L- g, k» T‘s Coti‘ihuoub at: 0013 “hum bf; q:—Q_ unJ» 5: '5h).- 11.Computethelimit lim “6‘x‘3. (Sex +\$ [TIPS4marks] 3 JH-3 x +27 \Sc—x 1'27 ______________,..._...— (<3 Aces—t3) K '5‘”)? yep-x - 93 \ 5mm. "" " __....—— \ x—e—b X?) "C 9’} ‘61: "= f is!» Page 4 of 4 ...
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