This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 (ex73L
(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 $4 a pair. The price p and the
demand :r for these glasses are related by :1: = my) = 7, 000 ~ 500]? If the current price is increased, will revenue increase or decrease? Explain Why. R=><P : Emma—5’00);z E’: 7000‘4000p >0 forlbsq
tha ﬂaw . A sphere With a radius of 5 centimeters is coated with ice 0.1 centimeters thick. Use differentials to estimate the volume (V) of the ice. [Recall that V = §7rr3. ] l
= 3W3 OW‘VON‘ = Zil—lT'ZOlV: 47352. at :{0T 5 1) 3w . Give a function f : R —> R that is continuous at 0 but not differentiable at 0. Explain. gt: 1 ~1 xﬁO‘P X—a 0" LE. . The price p (in dollars) and the demand3: for a particular steam iron are related by the equation
:3 : 1, 000 ~ 20p (A) Express the price p in terms of the demand :22, and ﬁnd the domain of this 2 5‘0“): 0 < < 000
p 20 are“, m“; X\ l. (B) Find the revenue from the sale of x clock radios. What is the domain f R? ..
0. R= PK  50X ’ K%O (DOWN Osmmo
(C) Find the marginal revenue at a production level of 400 steam irons, and Rt”— 50* : So’qoqo ‘. {new Media/(RUM (D) Find the marginal revenue at a production level of 650 steam irons, and ‘EVW‘WA
(Mame Man mam
W FW function. interpret the results. interpret the results. W: 50*691‘1(9 .. Boyle’s law for enclosed gases states that if the volume is kept constant, the pressure P and the temperature T are related by the equation P/T : k Where It
is a constant. If thetemperature is increasing at 3 kelvins per hour, What is the
rate of change of pressure when the temperature is 250 kelvins and the pressure is 500 pounds per square inch? ELOiT 7. o di=3> J“: region T”? = 6 poms para). (not wharf / a L“ q ~
SGJZ'ﬂbdx: gaglm 300 >< 5x  >(Lx 33 MATH 209 Final Examination April 2010 Page 3 of 3
[12] 8. Compute the following:
—2
(a) /(g;—4)3 dm :2 0"” +C
“2
53:
' 3/2 . V2 3/;
(c) x dm 3 “ 31m=atc+7m +2.92%? ~+c
x~7 “ml W W a 3( Mmlﬂ
5 A 1
(d) /(3:132+52:)d$ 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:. LQEL) 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+3ylxlnl).“"?_ 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 $2yze2y+7. 2X3 1383‘ = iglegg) % :— ﬁezgdxl [8] ’4. Does the line tangent to the graph of f = e'“ at a: = 1 pass through the origin?
Are there any other lines tangent to the graph of f that pass through the origin? Explain. 4‘60: ex .. eat 04. (ﬁlm: AA = WOW—m + elm = eXO (we) +ex° = )Q’Cxwodc (3.
ORIGIN: on a) : (om ) hence. , Q = QXOCO ~xo+g 3, 02 slides?
' [6] 5. A cube With a Side of 12 centime ers IS coated with we 02 centimeter thick, Use differentials to estimate the volume of the ice. ' 8X0 > O
1‘ 2 '2
ﬁzzwbﬁl V: a}: 51V = 3a am : 3‘12(20.a)=wl¢»(‘2¥473 (~xo: o
. .v . . . . Cl bin/Wat;
_ \Z [6] 6. Proﬁt analysrs. The total proﬁt (in dollars) from the sale of :1: charcoal grills is f 35 L Q VL r}
a” Pay) = 202: — 0.02952 — 320, 0 s :1: 3 17000 W
(a) Find the average proﬁt per grill if 40 grills are produced. [3 = 2%0 ‘
i Y <
(b) Find the marginal average proﬁt at a production level of 40 grills: and :2 O
' \l ‘0.
interpret the results. p : m 002 + : ,00260‘ 2 = 0.13 § Oak‘sgg
(c) Use the results from parts (a) and to estimate the average proﬁt per c ,\,\0
grill if 41 grills are produced. a'f\
_.. — .. / \‘
POM): Mach [Dwell = H,2+O.18 2 {[‘9 /(e [9] 7. Suppose a point is moving along the graph of 3:2 312 = 8. When the point is at
(2, 2), its :2: coordinate is increasing at the rate of 0.3 units per second. How fast is the y coordinate changing at that moment?
_ A _ _. ><
xol7<+gctj A0 04;] gdx 60.)?“ OLE: ~§OI3
[xt‘lieotﬂ : "' LLVLNj MWMﬂ‘ MATH 209 Final Examination December 2009 Page 3 of
3 . [12] 8. Compute the following: g L (a) /(4:1;2—7x)$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 Lastt7 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’: 251000032 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" $6226»— . 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+52+Z l (I 2_ ;_’
2m 33: 2 . (X*L)C_2¥+l> ~22L‘M 2x+( (s) lim M = mg 2 _
2 x +95 6 chzxxB) 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 [QXL [EXAM
at“? ._
(b)f(x)=—7—+\/E : IN 8 + ifs);
7 ?> MATH 209 Final Examination April2009 Page 2 of 3 [9] 3. Find (do not simplify): [7] Let=3z4—6:132——7 [13] 5. Let f(:c) =(z2)(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)? + $3
J}; 12265,sz
v g/aﬁtaadzZﬁz
46/2) :3. {6.5%
3 (7 (d) y z: (4 + x2ln$)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“; «(2ZAZ]: 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 72548 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 ...
View
Full Document
 Fall '07
 Chekhov
 Convex function, Concave function, Course Number Section, J. Ruddy

Click to edit the document details