hw4s - MAT 1330, Fall 2011 Assignment 4 Due November 9 at...

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Unformatted text preview: MAT 1330, Fall 2011 Assignment 4 Due November 9 at the beginning of class. Late assignments will not be accepted; nor will unstapled assignments. Instructor (circle one): Robert Smith? Jason Levy Olga Vassilieva Catalin Rada DGD (circle one): 1 2 3 4 Student Name Student Number By signing below, you declare that this work was your own and that you have not copied from any other individual or other source. Signature QUESTION 1. Find the derivatives of the following functions. Do not simplify your answers. (a) f(:c) = COS(3£IJ) sin(7:z-3 + 2x) evfis/nls x)»5in (“3+5le +wolsx)‘ COOHXMW)‘ f@¢=°(a4xa+a) wmekimw) 3glw: e5$+1 CQSXHV‘ ‘ ‘ 5X . ( yx c0152 XL ax (e +15 foz) 6‘6 I 9'($)— _Le”+/ (dMfl=é#mHmw> CDS‘IX)+3 ’ (x5) . ww)=ki_ an (éu%0053mx)i-BLQMXQ'BXZ)‘—j (d) w(a:) = x/sc+3ln$+4 I 3.1. umw=htz§mza'('+ x) _J QUESTION 2. Find the global minimum and the global maximum of f = a: — lnm on the interval [0.1, 2]. QC‘mch 3‘: 506:0 l” J23“) . . X z i POM/(Z) w ed- {:M §(©.l)= 0.1— 614 (0‘1) V :2, [Aware mflééfi km :34!) = l~€mw = 1 SL2) = 4* @“(étl =/,30600 <~Ualu40§&,a/€HL ecwlpoiwé J08». Global maximum at :1: = . Global minimum at m z . QUESTION 3. Consider the following DTDS: 1.5%2 {173+ 0.5. $t+1 = (a) The DTDS has three equilibrium points. Give the equilibrium points in increasing order. (b) Find the derivative of the updating function of the DTDS. LSX “Wm 57X): "5'0?></xa+a5)—/,¢*x*vax 3X5+/.5‘x-3x5 .. M .: Zxélmxfl"1 ‘ ()O’zfloté')”2 In W \ (c) For each of the equilibrium points in (a) find f’ and decide Whether the equilibrium point is stable (circle the correct answer). f/(mi) = E unstable (.5. n r ,y 4 HQ) 2 . O-AIS‘MJ ' ~r‘ stable ——§—_/‘ - ~ , f’($§) z 3'35 ~ (0.5 unstable QUESTION 4. The number of individuals (in thousands) of a certain species satisfies the DTDS: 4115;: 1 + (13,; The population is harvested with the rate It 2 0. Answer the following questions: (n+1: ~hmt, t==0,1,2,... (a) The equilibrium points of this DTDS are (Hint: one of the equilibrium points will depend on h): 3—h and [+h f[X):X L :L‘+l L_hx:x I+X /+x ((+x)(A+i): Li l/X . - =0 r " HT M X W £‘ Lz—h-i . Pa’b ~A_]]:O X- hat! ‘l = h“ - H’h *fl‘ ¥_5~h LN“) Z‘i:h_ (b) Give the largest interval for h such that both equilibrium points in (a) are non-negative (i.e. biologically meaningful). he (0) What is the harvest at the positive equilibrium? (Hint: the answer Will depend on h) (5%) R(h) = A Hh (d) The maximal harvest at the positive equilibrium is R'(h)=o ~3hehz>k~ swat HA ’(Sh‘l’lz—Y‘I_ 5~2h+5h~zl¢~3h 2 & Hh (HM; ~ CHM; : \W‘3 :0 or‘ b4+2_lrw'3=0 Rto)«-—R(3>)=O 32): 4.493226 [4 : ‘3” : 1 I 6‘ , ~I A =”2_-Zj;_ , g , RU): lull )“ ‘2 a. 3 eex'l/Lwled me ln>,o_ Rma,lc = and occurs for the harvesting rate h = (e) For the values of h as in (b) let :I;* be the positive equilibrium point. Then: $* is stable when h is I: 0‘ 3 ) $* is unstable when h is h evu-l I ‘ Z Ssh)°~~i‘\=\@’fl;h) ~i\\ 1 Min z) [Ho'zAma—qh‘q or \l~ah+h2\<‘7’ [(h~l)a|<‘f G,—\)2 <4 hall <2 ~02 (Pr—K <R ~\< h <5 mm Thu/9‘65“ 6: [0‘9 “EM” X: C6 $£ablé ‘ Ohm/wig, (+65 ammo/biz? bee/(7 x42: 0mg egg/:3 ...
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This note was uploaded on 12/19/2011 for the course MAT 1330 taught by Professor Dumitriscu during the Fall '08 term at University of Ottawa.

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hw4s - MAT 1330, Fall 2011 Assignment 4 Due November 9 at...

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