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Department of Mathematics
15 March 2006
8:30 - 9:20 am
MATH 155 - CALCULUS II
EXAM 2
INSTRUCTIONS
1. DO NOT OPEN THIS BOOKLET UNTILINSTRUCTEDTO DO SO
- Write your name and student number on the front page.
3. For each question, you must SHOW ALL YOUR WORK to receivefull credit.
- No book, paper, or device other than the usual writing instruments, this booklet, and scientific
calculators are allowed. No graphing or programmable calculators are permitted.
- During this examination, speaking to, communicating with, or exposing written papers to the view of other students is forbidden.
- You may use the back of the previous page for rough work or if you run out of space.
- Stop writing when you are instructed to do so. Failure to follow instructions may result in penalties.
Last Name
Given Name(s) Student Number -
Signature
1. Evaluate the followingintegrals:
[4] (^1)
(a) 2 dx 1 (x-l)
[5] (b) 11 00 Inx X2^ dx
lA-:;. (",-)(.
d-u-~~ ;(
~1;tJ^ .lJ^ ~ "Q.¥.;;- 11 --:- - J 1-
I'Ll8-,).~. v-J - J) ~
~.
l (^) - - ~' L( W))JSf);or-'/. Yi- .{;; I.~ t;1&J ( t--7~ -/ 0
-- 1~ (^) S 1 ~^ dL
Y; J
I'
- ~ .3 (?<_I) J 1:-?(~ (^) t
::: j jp;. (^) [ g^ i~^ (.^ t.-^ (^ I(^ J 1 -t 7 t" - '/ J
::: ~ ('t ("" ,l,- t--i'~ J ~
=- fL [
'I~K 1<- r,;" J
t
t-7p .- x: ~) r. I
~ L;
[
_~_l J
t
t -JIP )C )1.. j
fl.
~
( lL-l. t ( (^). .)
c
. L.,..i.- {
/ t ;'):D ... t
~
2. (a) The single compartment model is used to describe the concentration, C(t), of the
solution in a water tank, where ~~ = 3(9 - C), t ;2: O. Solve the differential equationwhen C(O) = 2.
---I - LAltt'- c.):::.. b t- -r ~ +~ f- L(G) -: l ~7 - ( L"'\ =I- ~ ~ ,,"yl-
- 1 L--.'t- c .-t ( 1 ~ 3. t q.-.; C (^) -::: e..--'1 t- -, -1 t CLf:)~ Cf,-1- e- ..,--- ~
[4]
r d~ J ~_e =^ f:>^ JA-
tV (^) [", (J -- C) - ( 1-=- -.1 t cI'Y""
[1] (^) (b) Find lirnt >oo C(t) based on your solution.
crV
~ C Lf) -:: -t-'/o
9-7[D)=
[2]
:::::-
(c) Use your answer from part (a) to find time t* when the concentration ofthe solution
in the tank reaches exactly k of the concentration of the incomingsolution.
CLk¥. ):::- 3 .-::: 1-1- eJ~ -.) tJ{ e ---^ - ~ 1- L I (^) I c:. 1:-~ -;:.. - J IA It
- (^) CJ I lJ S- IJ'l IV ,.., 0.05/
[3] (^) 3. (a) Usethe trapezoidal r,ulewith n = 3 to estimate 12 X2 dx.
i.- .f'>t} 'l -f-
[2]
-;: S [~()-f{X§)<'ft1J~) 2- -r ~sg )~jU} - J ...,- z....
-== -t U(tHL-f{t)-fL.f<f).f~j
. :: ~ [ -f.: I) r:y -Ie «"l0-rf(J1 ~
£".f tt~ f l. j
-=- J [j
- ~ l- &4 J 5" l.^ l'^ 'ill fJ^ 7-;2.~ =1^ --::: 2.^ ~?^ J^ I^ ~^ J I f' .~1'
(b) Howclose is this estimate ~h the exact value of the integral?
f
L ~ II -~ - " L^ :
1.- g i:;_ .- 2.., J 1 ~ -. ~~- ~..1. (^) I - 3---'1'-J I
I (^11) - .)
,...6"^ ~ 3
Gr- ytJV' .: (^) .1--1- fl.1- ...-~~ ~^
- ", ~ ~/. ~ -: tt7 L""".^ {-~ Y:!'"4-1 ...^ ~/' I^ I;; ~/ . ~e (~ Or&;{)
~k~ ~-1A-1--^ h^ y--. [3] 4. (a) The Taylor Polynomial of degree 4 about x = 0 for the function f(x) = e-x is given by X^2 X^3 X^4 p( x) = 1 - x + 2! - 3! + 4! Usethis to approximate the value of ve. -'I-.. ~ ~ rLX)
X ~ -l/L t'dh'
e l;~ ~~~ ft::t.ll-)
{( ( .::::;.. i +{ + 2: +t~ t S1Jlf
~ (. h 4- .g 'fLf L~ (.6~'i72-)
