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Unformatted text preview: MAT 1322A W2009 Wed., March 18th 8:30—9:50 Prof. Desjardins MIDTERM TEST 2 Max=20 Student Number: OTime: 80 min. 0 Only basic scientiﬁc calculators are permitted (nongraphing, non—programmable, no
integration or differentiation capabilities). Notes or books are not permitted. 0 Work all problems in the space provided. Use the backs of the pages for rough work if
necessary. Do not use any other paper. 0 Write only in non—erasable ink (ballpoint or pen), not in pencil. Cross out, if necessary,
but do not erase or overwrite. o The problems require complete and clearly presented solutions and carry part marks if
there is substantial correct work towards the solution. U)“
1. (3 points) A casserole is taken out of the refrigerator" which has a constant temperature
of 4 °C, and left on the counter. The temperature in the kitchen is constant at 20°C. One
hour later, the temperature of the casserole is 10 °C. If the temperature of the casserole follows N ewton’s Law of Heating, what is its temperature after being out of the refrigerator
for two hours '? ﬁx 771) @221 Wee 7275 W! a» 54:“ t [,qu 4,; Aw ) ﬂi/MC’ICJ 14M) 0&7. jif: A (if—r26") ~éd’l’ ll 2. (3 points) Biologists place a colony of 2000 bees 011 an island in the Saint Lawrence.
They estimate that the carrying capacity of the island is 20 000 bees. Also, the relative
growth rate in an unconstrained environment is estimated to be k = 0.04 per month.
Assuming that the population follows the Logistic Model, (i) write the differential equation that the population P(t) will satisfy, where t is mea— sured in months and K (ii) given that the solution of the Logistic equation is P(t) = m , ﬁnd the number
e— of bees on the island after 6 months. a2) 3. (4 points) Determine if the series converge or diverge. Explain your reasoning. / °° 1 °° 1
1 __
(a) ”22:2 Mb 71)2 (J) ; x/n2 + ‘2n 1/» ﬂ; 1 A, m w ML
£1: ' , J
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£314 “/5 I / (14‘ IV” £1 : “N IvI‘nLkln 21 [5,»)1 11—) >° b ‘ 4"” ‘
a ‘5 A
: Zm / #2 (/91 A
bags 2 ”(add : va " / ' ' m :3
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to am i“? A I! " ’é‘é ’50
2 if}  (’Z'rww¢ a‘ l.” 4. (4 points) Determine the Iadius and interval of convelgence of 2 m .
n ”=0 Cé'n‘l‘r‘i 4/ ‘5 0 Jimq >
A 4w —~——~ :1»; 22.29; n '
A—Aw ,,:/ n5"; 9"“(‘4'h2’ 44100 /)¢" 2"“ 5/
x“
J!“ {rtH)
 flat; )1! n&( = 13—1“,—
Q n+2 0?
L: W /71( ( I :9 /%/< ’2 :9 f V
A; 2:2
m 1% Man 4W m ~.Z < z < .2
Maw. a 1“ “‘H n a“)
_ ' Z = 1 ~12  Z “' Canaan r
If %— 2 / 4:0 SPCA—H) vital gum“) mo '4“ (44);)
92 . K 2°— . .\ 3
if 91': ,2, Z )L t l ‘3 1 Z, .L. dtucnru
"=03“('\+) “=0 9‘“ (AH) ”W” AH (karma.30 W) 5. (4 points) (a) ’1'
1 + 13
(ii) Then deduce the Madam 111 series of /I (i) Find the Maclain 111 series of and give. its radius of convergence. 1 + 1 dv and give its 1adius of (onvergence.
I: 0.1 1 (111) Express / 1 + 3 (1'1 as a sel 1es and estimate its value with err‘01 less than 10 6.
0 1 How many terms we needed? 1 I _ ” " /;¢/< ( {m £=U
‘ M 1M W 7:: “ “EC" ’t A
m _L. = z c—w .141 ”a, e:
[+71 ntd
“:19 . 1 ‘3 :— Acl .3 [£3]
Zé ’3 2 Z (4)0‘IJA ,M W”(% l)! l<l ) IZ/ 6. (2 points) Find the Taylor series of e“ centred at :1? = 2. Bonus (2 points) What are the conditions that the terms, bn , of a convergent alternating @ 00 series 2 (—1)”"1b,l must satisfy? If the nth partial sum, 3n , is used to approximate the
11:0
sum. 3, of the series7 what is the error? Draw a diagram to justify your claim. MAT 1322A W2009. Wed., March 18th 8:30~9:50 Prof. Desjardins MIDTERM TESTX‘o/Z Max=20 Student Number: oTime: 80 min. 0 Only basic scientiﬁc calculators are permitted (nongraphing, non—programmable, no
integration or differentiation capabilities). Notes or books are not permitted. ' 0 Work all problems in the space provided. Use the backs of the pages for rough work if
necessary. Do not use any other paper. 0 Write only in nonerasable ink (ballpoint or pen), not in pencil. Cross out, if necessary,
but do not erase or overwrite. o The problems require complete and clearly presented solutions and carry part marks if
there is substantial correct work towards the solution. 1. (3 points) A casserole is taken out of the refrigerator, which has a constant temperature
of 4°C, and left on the counter. The temperature in the kitchen is constant at 22°C. One
hour later, the temperature of the casserole is 10°C. If the temperature of the casserole
follows Newton’s Law of Heating, what is its temperature after being out of the refrigerator
for three hours? p0 77,5) : J2+ 4e 2. (3 points) Biologists place a colony of 2000 bees on an island in the Saint Lawrence.
They estimate that the carrying capacity of the island is 18000 bees. Also, the relative
growth rate in an unconstrained environment is estimated to be It = 0.05 per month.
Assuming that the population follows the Logistic Model,
(i) write the differential equation that the population P(t) will satisfy, where t is mea
sured in months and K (ii) given that the solution of the Logistic equation is P(t) = m , ﬁnd the number
6— , of bees on the island after 6 months. 3. (4 points) Determine if the series converge or diverge. Explain your reasoning. DO 1
(a) Z n(1n ”)3 ”=2 0“) {17 z ./ 3
)Z/er) I (b) 7121 [)0 1
Z \/ n2 + 3n 90 n
. . . . .1:
4. (4 poults) Deternnne the radms and Interval of convergence of E ”:0 3”(n + 1) '
«4; /52: / z I»; 1!. / M2 2 [it]
"*0” 0,. 9139» 3 mm 3
W [29! < l .3) 2‘: 3
3
40 W A ’3 < I Q 3
e» n 00 ‘
A " (I)‘
Fiﬁ 1: '3 Z 1 ’ Z— 07’1“"
I n20 37M!) “:0 VH4 i“
up 9°
'\ P I a, r
x ,Q, ‘_ m (at
( 1": 3/ Z ‘ _ A met 76"
“=0 3"(9114) "‘0 5. (4 points) 1' @
(i) Find the Maclaurin series of +14 and give its radius of convergence. (ii) Then deduce the Maclaurin series of / 1?
l + £4 d1? and give its radius of convergence. 0.1 Ll» . . . . (111) Express / 1 + 4 (11: as a series and estimate Its value With error less than 10’s.
0 .L‘ How many terms are needed '.7 1”, LU] 6. (2 points) Find the Taylor series of 6‘” centred at .17 = 3. (/) Bonus (2 points) What are the conditions that the terms, I)” , of a convergent alternating ’30
series 2 (—1)"'lbn must satisfy? If the nth partial sum, 3", is used to approximate the
n20 sum, 8, of the series, what is the error '? Draw a diagram to justify your Claim. (523mg as ,ﬂ) ...
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 Winter '10
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