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Unformatted text preview: CON CORDIA UNIVERSITY Department of Mathematics &: Statistics x—+3 \$-—+3 ’ (a) hm [-3900] x——»3 (iii) Find the value of each of the following: Course Number Section(s) Mathematics 209 All except EC Examination Date Pages Final April 2010 3 Instructors Course Examiner M. Amir, L. Dube, E. Duma, H. Greenspan, R. Raphael B. Rhodes, J. Ruddy, C. Santana, U. Tiwari Special Instructions I> Ruled booklets to be used. [> Only approved calculators are allowed. AJARKS J [9] 1. .(i) Find lim W : 6% “5"? : am 535: or; tom; I_’_°° '33 x\$«,o ~‘XZ \(‘P «co (ii) Given lim f(:z:) : —5 and lim g(:c) = 4, ﬁnd 420:) gig «gm :2 o gig [9(w)/2f(as)l= — on (as 614% e3 smenn _ \$2—3x+2 _ , 332—16 (a) \$133133 (:1:~—1) “‘5 (13):;3135 (x_5) Cb) .3. ONE [‘18] 2. (a) If f(:L‘) = 4 - 652310 — 4:53, find f’(a:). ~gO 7K9” (QXL Z (b) If ﬂag) : xzz—Qi’ffrzl ﬁnd f/(x). ‘ @x‘ngLm —- (KEBMLLXQM _. 3X~6X~3 L z " 2. (c) If HIE) : \$31573” ﬁnd Has). CZX gxey—tv —- 065 5% 6X (’4 “3.2 a 1 (ex-73L (d) If y = 2e<r“-5), then 3/ :? 4Xe><~§ 9 d (e) If y = ln[(:z:2 + 322%], then CT: = ? a 2:163 dy x 3% r (f) lfy 2 an + 51 then — =? 4/; dﬂi é: (x+§) 3 (3) Find y’ given my 2 ey — 2. V / I 9 +96%; y'eg “<7 (6912):? 67‘ g MATH 209 [9] 3. [5] 4 [5i 5 l9] 6 [9] 7 Final Examination April 2010 Page 2 of 3 A manufacturer currently sells sunglasses for \$4 a pair. The price p and the demand :r for these glasses are related by :1: = my) = 7, 000 ~ 500]? If the current price is increased, will revenue increase or decrease? Explain Why. R=><P : Emma—5’00);z E’: 7000‘4000p >0 forlbsq tha ﬂaw . A sphere With a radius of 5 centimeters is coated with ice 0.1 centimeters thick. Use differentials to estimate the volume (V) of the ice. [Recall that V = §7rr3. ] l = 3W3 OW‘VON‘ = Zil—lT'ZOlV: 47352. at :{0T 5 1) 3w . Give a function f : R —> R that is continuous at 0 but not differentiable at 0. Explain. gt: 1 ~1- xﬁO‘P X—a 0" LE. . The price p (in dollars) and the demand-3: for a particular steam iron are related by the equation :3 : 1, 000 ~ 20p (A) Express the price p in terms of the demand :22, and ﬁnd the domain of this 2 5‘0“): 0 < < 000 p 20 are“, m“; X\ l. (B) Find the revenue from the sale of x clock radios. What is the domain f R? .. 0. R= PK - 50X ’ K%O (DOWN Osmmo (C) Find the marginal revenue at a production level of 400 steam irons, and Rt”— 50-* : So’qoqo ‘. {new Media/(RUM (D) Find the marginal revenue at a production level of 650 steam irons, and ‘EVW‘WA (Mame Man mam W FW function. interpret the results. interpret the results. W: 50*691‘1-(9 .. Boyle’s law for enclosed gases states that if the volume is kept constant, the pressure P and the temperature T are related by the equation P/T : k Where It is a constant. If thetemperature is increasing at 3 kelvins per hour, What is the rate of change of pressure when the temperature is 250 kelvins and the pressure is 500 pounds per square inch? -ELOiT 7. o di=3> J“: region T”?- = 6 poms para). (not wharf / a L“ q ~ SGJZ'ﬂbdx: gaglm 300 >< 5x - >(Lx 33 MATH 209 Final Examination April 2010 Page 3 of 3 [12] 8. Compute the following: —2 (a) /(g;—4)-3 dm :2 0"” +C “2 53: ' 3/2 . V2 3/; (c) x dm 3 “ 31m=atc+7m +2.92%? ~+c x~7 “ml W W a 3( Mmlﬂ 5 A 1 (d) /(3:132+52:)d\$ 2 323 *5 £+C— Xstgk’l—i—C, 5 2 2 3:2 __ l 3 (e) /4+\$3dcr 3 KWUHX 1+0, 2 12 l (Katya ~ (5 (f) +1) mdx : l- \__+C:. LQE-L) t’C lg 2g [6] 9. Evaluate the following integrals [accurate to 2 decimals]. 5 9; A I - w 3 t , 5_ lZS' (a) Aug 4W (— Well ‘ —,;~20= (igzzléazlw J. J? 3: 3 (b) / ehz h dh : 2 2, 3— 9 ’- iCQ “eql '3 (3503 ~59.6)=433.z [9] 10. Find the area bounded by f(a:) : 52:2 — :1: and g(;r) = 2x for ~2 S :E g 3. o 3 _ Z )(\:o x223 ‘ [\$2 (XL—sxwxl +1 éCXEBXXalxlzlsé- [9] 11. Consider the function f(:1:) 2 x4 — 2:123. Show its intercepts, Where it is increasing, decreasing, concave up and concave down. Sketch its graph. x \e‘ : axial: XLCW‘é} l3“: (2 X142 y: \g f 4(1)?- “l Lil!“ \ _ v 0 A 3,; v X~mecep+3 'HJc ©-—- [Maura ﬂoat/K3092) CONCORDIA UNIVERSITY Department of Mathematics 85 Statistics Course Number Section(s) Mathematics 209 All except EC Examination Date Pages Final December 2009 3 Instructors Course Examiner M. Amir, V. Enolski, H. Greenspan, T. Koulis, R. Raphael R. ~Mearns, R. Raphael, B. Rhodes, F. Romanelli, J. Ruddy, Special Instructions Ruled booklets to be used. Only approved calculators are allowed. D D MARKS rv —32:3 + 53c? — r 1. Find lim 5‘\$2 25600 [11] ~5 and lim 9(1) 2 4, ﬁnd x~+6 '(ii) Given lim ﬁx) x—sﬁ (a) hm l—3g(\$)l=‘ll (b) Hm V9(\$)=2 3:-—>6 :3—r6 (c) g [me/2m = — 0% (iii) Find the value of each of the following: x2~164quz , 1132—393—l—2 2—44 ecu/x = 2 (SW) 1. mil—Bl (5Z3 — f(a+h)—f(a) h 7 ﬁnd the derivative if 2. Using the deﬁnition of the derivative fling) ﬁr) = 3 ~ :23. ” gym a ~(xm°’— 3+)? : (‘3x1+3ylx-lnl).“"?_ 3 X7. lo [21] 3. (a) If ﬁx) 2 26~5z15 — 3:34, ﬁnd f’(:r). : (b) If : ﬁnd M X?- 3 x2 k ~3 x k1 F? +23 L. A v 2‘ r \w‘90 ~30x”~t1x3, Wﬁﬂ+2xf§ ,— CKZFZJ Exiting.) “Mug—3F 1)“ (75%K+3) 4—3 KL~7>7C7/X/3 (7 ekKlt‘b)?” €51Y C‘ .— 22~9 \$3+2\$+5 9 .z'—3 Tln.(:z:)+3 If ﬁx) = ﬁnd f’ (2:) . (C) (d C‘ ) lfy = 65—2-}, then 3/ z? a MATH 209 Final Examination December] 2009 Page 2 of 3. (Continued) l§><1 d (e) Ify : 111(5233 —5)3, then 3% 2 ? 3 (W3ﬂ§> ‘7/é 1 dy _ —l/ «l _ __L - (f) Hy: 6\$_7,thenE£—? («[6])(X—7) 6 _. 6 [Y . . ' a ; 2x a (g) Find y’given \$2yze2y+7. 2X3 1383‘ = iglegg) % :— ﬁezgdxl [8] ’4. Does the line tangent to the graph of f = e'“ at a: = 1 pass through the origin? Are there any other lines tangent to the graph of f that pass through the origin? Explain. 4‘60: ex .. eat 04. (ﬁlm: AA = WOW—m + elm = eXO (we) +ex° = )Q’Cxwodc (3. ORIGIN: on a) : (om ) hence. , Q = QXOCO ~xo+g 3, 02 slides? ' [6] 5. A cube With a Side of 12 centime ers IS coated with we 02 centimeter thick, Use differentials to estimate the volume of the ice. ' 8X0 > O 1‘ 2 '2 ﬁzzwbﬁl V: a}: 51V = 3a am : 3‘12-(2-0.a)=wl¢»(‘2¥473 (~xo: o . .v . . . . Cl bin/Wat; _ \Z [6] 6. Proﬁt analysrs. The total proﬁt (in dollars) from the sale of :1: charcoal grills is f 35 L Q VL r} a” Pay) = 202: — 0.02952 — 320, 0 s :1: 3 17000 W (a) Find the average proﬁt per grill if 40 grills are produced. [3 = 2%0 ‘ i Y < (b) Find the marginal average proﬁt at a production level of 40 grills: and :2 O ' \l ‘0. interpret the results. p : m 002 + : ,00260‘ 2 = 0.13 § Oak‘sgg (c) Use the results from parts (a) and to estimate the average proﬁt per c ,\,\0 grill if 41 grills are produced. a'f\ _.. — .. / \‘ POM): Mach [Dwell = H,2+O.18 2 {[‘9 /(e [9] 7. Suppose a point is moving along the graph of 3:2 312 = 8. When the point is at (2, 2), its :2: coordinate is increasing at the rate of 0.3 units per second. How fast is the y coordinate changing at that moment? _ A _ _. >< xol7<+gctj A0 04;]- gdx 60.)?“ OLE: ~§OI3 [xt‘lieotﬂ : "' LLVLNj MWMﬂ‘ MATH 209 Final Examination December 2009 Page 3 of 3 . [12] 8. Compute the following: g L (a) /(4:1;2—7x)\$dx :. /% 344332 , g M (c) [<2m3+7>18x2dm = L WWW) M; 3 16 3/ /L d 3’ d :: gt], ; u, 1., c; 7— , Vt () W I Last-t7 3:]; (Fr?) C =w~7 (e) [84% f ,1, e” 92%, f 3 (f) /<x——3)-4dac = CH) analogy“ 4C [7] 9. Evaluate the following integrals [accurate to 2 decimals]. .(a)/12€x2 ivdm : éexll f—z 8‘32: (6%665 2, 7)::251g5 (b) /4(t‘2+3)dti 2. 2 gatlz: : O (9 [4] 10. A note will pay \$25,000 at maturity 10 years from now. How much should you be willing to pay for the note now if money is worth 2%; compounded continuously? 22 000 = P 90'”? ‘ ’O P = 25000 9:9“ 2’: 251000-032 r 2 20,930 [5] 11. Find the area bounded by the graphs of y = 3:2 - 32:; y = 0, —2 S :c g 2. S ngX 59X; éigg} O OCL~3 {4 19‘ "33:0 Xi: 0 c 1’923 Wide, , . z. 2- l £405“)sz + t £0 (\$300424 = 12 [6] 12. Find the interval(s) on which the graph of ﬁx) : 3:3 ~ 6::2 + 9:1: + l is concave upward, the interyaMs) on which the graph of f is concave doxxrnward, and the inﬁec ion oint sl. L /\ £4“--._ ::;+ Uﬁhﬁﬁwp+.le a. Camugakfe/dbcyu,= ~]~Qﬁ(2_[)cmp 22:50C. CONCORDIA UNIVERSITY . Department of Mathematics & Statistics Course Number Section(s) Mathematics 209 . All Examination Date Pages Final April 2009 3 I Instructors Course Examiner S.T. Ali, F. Balogh, L. Dube, E- Duma, E. Cohen H. Greenspan, A. Kokotov, R. Mealrns"y M. Padamadan, JIPark, C. Santana MM”— Special Instructions ' ‘ 1> Ruled booklets to be usd. ﬂgaw not? Calcu’bﬁ‘o" \$62-26»— .- e- ‘5 a "W ' 99mmch 13m New 209 W wad OM e'mr- MARKS 1. (a) Find the following limits 2- ﬂ 3' 2 +2~Z~ ( l 5 5‘ (3) 11m M —— _.___ ,ﬂ 3H2 \$2+3\$+2 22+5-2+Z l (I 2_ -;_’ 2m 33: 2 . (X*L)C_2¥+l> ~22L‘M 2x+( (s) lim M = mg 2 _ 2 x +95 6 chzxx-B) W9. ms — 2 Where is the function. r: -coutinuous? £9 Gym p. ./- Y: 3 > x2 —1 [5] 2. Find the derivative f’(3:) of the functions f (:5): > (Do not simplify) (a) f(x)= 3m4—433+\$— 2 19' 2 [Q-XL [EXAM at“? ._ (b)f(x)=—7—+\/E : IN 8 + ifs); 7 ?> MATH 209 Final Examination April-2009 Page 2 of 3 [9] 3. Find (do not simplify): [7] Let=3z4—6:132——7 [13] 5. Let f(:c) =(z-2)(xﬁ—4z_8) 4 Find ' dx () 821: a y: I. L 932—4 ((3%) (b) .y:ln(3\$2+5) 3% [:1 6 X A ' . axles (c) y = (2x? +1)3(4x + 6? 3(WX53M}; (Xz’ew‘zy: 3\$CH2¢% Xl’gthBZgiX 8w)? + \$3 J}; 12265,sz v g/aﬁtaadz-Zﬁz 46/2) :3. {6.5% 3 (7 (d) y z: (4 + x2ln\$)3 ( \ » (a) Find the slope of the tangent line to the Curve when a: =" 2 (b) Find the equation of the tangent ﬁne to the curve when m : 2 g a 72002.) + n 400 = xgaef—Hg \ KM: 3x5 lzx W02) 2 6X ~(z (a) the critical and inﬂection points of f(a:) pﬁ' XV: D X: the intervals where is increasing and where it is decreasing I » kW. {2 t. X = L (c). the intervals on which f(:13) is concave up and on which it is concave down ((1) use the above to sketch the graph ' . A student center sells 1600 cups of coffee per day at the price of \$2.40 per cup; A market survey shows that for every \$0.05 reduction in price per cup, 50 more cups of coffee will be sold. How much should the student center charge for a cup of coffee in order to maximize revenue? \G MATH 209 Final Examination April 2009 ‘ Page 3 of 3 [7] 7. Find the absoluteextrerna of the function f(x) : 2:3 432:2 +92: — 6 on the interval H5}- t’ti: 0.. ax‘a i2y+9 = 0 xi tax-+3 =0 @‘Bléwj :0 X13(\>(2,33 a=—u ~“ ( tb”s = ~22 )£0)2‘2‘ 72(3):”.63 ow ong [3] 8. If interest is compounde continuous y and the interest rate is 6.4%, will it take for money invested to double? 6 2 M D O . 05kt» : 29,: A: [e ' V 0.06% [10] 9. Find the equation(s) of the tangent line(s)to the graph of y2 — :ry — 6 '2 0 at the oint s with :c = 1. Z [10] 10. Compute these antiderivativ : (a) /(3\$5—2\$3~7)dm.é 6 (b)/2:e—7\$ d3: "‘ “0% 8W! 2 {132 _‘ Ell ‘ (C) \$~3d\$ “ (1-:\+C£Q+ l1 5/; [10] 11. Evaluate the integrals: + X (a)/bl(\$3,_ 4)dx =(%H-%y>(otz .. (rs/2.7 1de awwei [10] 12. Find the area bounded by the graphsof f(:r) = 3:2 — l and y ': :5 e 2 over the interval #235531- ‘ . ﬂew“); xﬁxwi > 0 7m Maw l 7; + : Xé—-¥_.2“ \ 4 \ g (X X (>0{X “g )21“; «(2-ZAZ]: CON CORDIA UNIVERSITY Department of Mathematics & Statistics Course Number Section(s) Mathematics 209 All Examination Date Pages Final April 2008 4 Instructors Course Examiner A. Atoyan, L. Chekhov, E. Duma, P. Gauthier, J. Ruddy Hughes, T. Mancini, R. Masri, R. Meams, M. Padamadan, J. Park, J. Ruddy W Special Instructions ‘ ' ’ 1> Ruled booklets to be used. {> Only approved calculators are allowed. W MARKS {10] 1. Given f(:z:) = -7 and g(z) == 3, ﬁnd I") :c——) (a) [2f(a:)+g(x)l (b) Given 11(1) = W ' Find (c) lathe) ' I “0 Eh“) (iii) Find (8) Hm 7254-8 2—»00 52— 5 [12] 2. Find the derivative of each of the following (do not simplify): . (a) y = 5:1:4 + 73:3 =— 102:2 + 47 (b) y : (53:3 + 3332)7 (412:2 + 9x)6 ...
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• Fall '07
• Chekhov
• Convex function, Concave function, Course Number Section, J. Ruddy

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