MIDTERM#1Sol.

# MIDTERM#1Sol. - Student number Total marks out of 28...

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Unformatted text preview: Student number: , Total marks: out of 28 Question 1. [4 points] Find the derivative of the following function: ._.. 1+ 61 2 7332 _ ln(1+ ex) +1126 ‘ f (33) 3(66") = 2m" + x16,“ (-29 = (2x 43); t1 M cm “ALL, (n ® (luck WWW “\$93.. EH1 ate km mmamwutmwmcmu run :Mt X é.(RM(\\-ek)) :- 6 x be “A“ bat. CD‘NMLLM'L o\x He \ x “3“ H; _ gem—Ha) _ a g 0111 (M’rc‘) ) QM wag“ Mm [may Vuﬁom 0°} ‘ __, 7-__ H “X1 __ 5x CU CK) 61% SM '— RX 2X )5 MUM—ﬂ UMUKKHLUHf) \ ex K 3 X V0510“ m {M = “- a “t- b“ Qx )6 Question 2. [4 points] Over the course of a year, a City receives a maximum of 16 hours of daylight and a minimum of 8. The maximum occurs in July, the minimum in January. Assume that the number of hours of daylight varies sinusoidaﬂy over a period of one year. Find the parameters in the standard cosine description , i.e., m) = A + Bcos(27r(t h (PVT), where t is in months, and t = 0 correspond to the month of January. Draw the graph of the function and identify the four parameters A, B, (I), T in the graph. Give the names of the four parameters. (D T= (speak! Y: \"Z. wst <19 (brow. ,¢=é waits [M‘t‘°r“°‘=*-‘r VIM] [3 iv “me: t: it?“ 1mm 2 Moms 3“ OMAPUML‘ 1-6:?!th : Liinom'g _ 2 Lt-e} :55 gay—- D. + Li wSCLTr) \iM‘Siou [Q 1 gm = \2 i 3 605(EK—(ljciﬂ) '1 (it's) Uwsou (CJ : Sm = R + 2 cos ( ﬂu ) Question 3. [4 points] Consider the function f(a:) = (36:37)? Find the following limits if they exist. If not, indicate why. Report your results in the form of a. little table. If you need a calculator to work out the answer, give at least 4 values for 3: that you tried. (a) lim f(:z:), (b) lim (x), (c) limf(a:), (d) limf(5c). :c—i’f— z—>'T+ m—i? zit—i0 on <“All. Vw‘ckcu is Continuous ml X'—’Q.Yluu1%n. km 20‘) ‘5 “6‘50 {D )HO VQXSlou LL\ 04ml LC} - Some Question 4. [4 points] Use the deﬁnition of the derivative to calculate the derivative of the function f(33) = (:1: + 2)2 — 4. Sim : WM gmm‘n- SM {:5} MM: .__ kw baht)?”— H - [Gulf—Ll ) MAD L‘ _ hm 12*2X‘rwhx *Mz'rlmﬂi "XI-tht M450 ‘1 _ 2x \4 4r kl 1‘ '1‘q _ , _ 3Tb M , (Pro (DAL. kin) = ZX’H‘T <3 Voﬂou GA MQ SWAM— kg gt“) = Z‘A-é Question 5. [8 points] In order to keep the songbirds in the back yard happy, one person puts out 40g of seeds at the end of each week. During the week, the birds ﬁnd and eat 4/ 5 of the available seeds. The DTDS for the amount of seeds in the back yard is 314.1 = 02313 ‘I' 40, where t is measured in weeks and seeds are counted just before a new supply is provided. (a) [1 point] What is the updating function of the DTDS? (b) [1 point] Find the ﬁxed point of the DTDS if there is one. (c) [2 points] Find the general solution formula for the DTDS, i.e., St = (d) [2 points] Graph the updating function and draw the cobwebbing, starting from 80 = 5, for at least 4 steps. (e) [2 points] Is the ﬁxed point stable or not? Check by using the stability criterion and by Checking your cobwebbing. @ m, updmhﬁ harks. “\s [M = le+ no ® at “it (CD (Ll “L [haul peek Salish; [6) =3 or 0.1? 4: he =3 or §=So (a) 3‘ :- 01 so M) f 31 = 01(0)}... Hm) + Me = (Oily-Sci: 5.14.0 Ho S3;- 0.1[®.z)zsc+ 0.1- to Ho) Hie = @zfse e Q1)? to + 0,1.qu m s , t @ . {-I M St = Q31) 30 3. ho[l +0.2 +0.11 +0.13+... +0.2 ) ‘1‘“ Le) 8* .. M‘s ﬂy.) "’ (\$0kath tummy” [PM Loki suns (D ~—D Slope. 6% WAD-k3 L‘lﬁw is [mum ~l Mel l (D Use this space to solve question 5. \chqu (Q ‘ 3* = 30 gig—fat, + 20M g -* (g)1g)1-,. MO \I \mem (a) I. 3* Question 6. [4 points] Consider the graph of a function f below. Find the critical points of f. Determine the intervals where f’ is positive and the intervals where f ’ is negative. gt 3.: llu. Siege. 0% lie, hwgmh 91m. ls ll». wk c& E (2‘4th (FMS OIL Suck M gilxl‘) ‘0 or 8:03) \mAlewA . ® El < 0 EN 0 < x c 2J1 MA 1C, 4X 4 g‘ [QWWXMOLMLH ...
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