MATH 201 Exam 2 With Solution, Fall 2009

Math 201 — Fall 2009-10 Calculus and Analytic Geometry III, sections 1~-8, 24-26 Quiz 2, December 2 — Duration: 1 hour GRADES: 1(/15) | 2(/15) | 3(/15) | 4(/i8) | 5(/19) | 6(/20) | TOTAL/i00 | YOUR NAME: < ‘ a\oh ONS YOUR AUB ID#: PLEASE CIRCLE YOUR SECTION: Section ] Section 2 Section 3 Section 4 Lecture MWF 3 Professor Makdisi Recitation F 11 Ms. Nassif Lecture MWF 3 Professor Makdisi Recitation F 2 Ms. Nassif Section 6 Lecture MWF 10 Professor Raji Recitation T 3:30 Ms. Itani Section 5 Lecture MWF 10 Professor Raji Recitation T 11 Professor Raji Section 25 Lecture MWF 2 Professor Tlas Recitation F 12 Dr. Yamani Section 24 Lecture MWF 2 Professor Tlas Recitation F 11 Dr. Yamani Lecture MWF 3 Professor Makdisi Recitation F 4 Ms. Nassif Section 7 Lecture MWF 10 Professor Raji Recitation T 8 Ms. Itani Section 26 Lecture MWF 2 Professor Tlas Recitation F 3 Professor Tlas Lecture MWF 3 Professor Makdisi Recitation F 9 Ms. Nassif Section 8 Lecture MWF 10 Professor Raji Recitation T 2 Ms. Itani INSTRUCTIONS: 1. Write your NAME and AUB ID number, and circle your SECTION above. 2. Solve the problems inside the booklet. Explain your steps precisely and clearly to ensure full credit. Partial solutions will receive partial credit. 3. You may use the back of each page for scratchwork OR for solutions. There are three extra blank sheets at the end, for extra scratchwork or solutions. If you need to continue a solution on another page, INDICATE CLEARLY WHERE THE GRADER SHOULD CONTINUE READING, 4. Closed book and notes. NO CALCULATORS ALLOWED. Turn OFF and put away any cell phones. GOOD LUCK! An overview of the exam problems. Take a minute to look at all the questions, THEN solve each problem on its corresponding page INSIDE the booklet. 1. Let the function f(z) be given by 0, when 0 <a2<7 f(z)=< 2-7, when 1 <2 < 2n and f(z) is periodic with period 27. a) (5 pts) Sketch the graph of f(z) for x € [—2m, 4m]. b) (10 pts) The Fourier series of f(r) is 30,59 @n cosnz+)>,,5 on sinnz. Find ONLY the coefficients 5,. ~ ~ 2. a) (6 pts) Plot the polar graph of the curve C : r = 1+sin@. Also draw the line L:y = 4/9 on your graph. b) (3 pts) Convert the equation of Z to polar coordinates. c) (6 pts) Find the (r, @)-coordinates of the two points of intersection on LN C. 3. Consider the following moving point in space: 1 P(t) = (3t, V6 et, 5¢): a) (5 pts) Find the velocity and the speed of P(t) at the instant t = 0. b) (5 pts) What is the arclength of the curve given by P(t) for 0 <? < In5? Simplify your answer. c) (5 pts) Suppose we have a function f(x,y, z) with the property Vil a,vee,be2) = (e?, V6 e, 5). Find < (PO) at the instant when the point P(t) passes through (3, V6 e, 50). 2 . ry 4. a) {8 pts) Show that lim ) 8 pis) (zy) (0,0) a4 + y? b) (8 pts) Show that lim —2Y Ww P (z,y)+(0,0) 2? + y? does not exist. does exist (hint: the limit is eqnal to 0). 5. Consider the function f(z, y, z) = ze Y, a) (6 pts) Find the gradient of f{z, y,z) at Po(1,1,—1). b) (7 pts) Find the equation of the tangent plane tg the surface f(z, y,z) = —e at Po. c) (6 pts) Determine the direction in which f(z, y, z) increases most rapidly when the point (z,y,z) moves away from Py. Your answer should be a unit vector. 6. Given a function f(z, y) satisfying f(1,2) = 4, Vf a »o (3, 4). a) (6 pts) Approximately how much is f(1.03, 1.99)? b) (7 pts} Find a direction @ in which the directional derivative Dg f = 0. Your (1,2) answer t should be a unit vector. c) (7 pts) Let S be the graph of f. In other words, S = {(z,y,z) € R9| z= f(z,y)}. Find the equation of the tangent plane to S at the point Fo(1,2,4) € S. (Be careful.) ii 1. Let the function f(z) be given by 0, when0<a<7 f(z)=¢ 2-7, when 7 <2 < 27 and f(z) is periodic with period 27. a) (5 pts) Sketch the graph of f(x) for 2 € [—27, 4n]. b) (10 pts) The Fourier series of f(x) is 37,55 @n coSNT+) > y51 bn sinnz. Find ONLY the coefficients dy. \ Lt \ ™\ re bys aq Vee seaxdy = Sosa da + \ G4) Snneds X= 9 = ‘ ek Cony other inkerval & 1e) Venatvn Ve on be used (Astead & (0,281 ) an La 2 _ \ — CoS AK\ \ fre _ Gs NX _ — Vari ~ } : 5960-3 2 X2% xen - — os 2n% asa on 7 OE ene ~ ") -\ ~ _ Ces Noe dy 2. a) (6 pts) Plot the polar graph of the curve C : r = 1+sin@. Also draw the line = 4/9 on your graph. Ley [r, RPL owe {, pert c be low| b) (3 pts) Convert the equation of Z to polar coordinates. L. 4 . V/q ARCOAS, con “- or c) (6 pts) Find the (r, @)-coordinates of the two points of intersection on LMC. & ushe c= \4c.0 = “\ an Asin© Ss snD esi Oz Ye pre on is Arob L Ours Va _ i s we Severe pou HAUTE it 8 yey ” om om 3 (.- ; Queue Qa 44 o - = QurNGv-1) =o) is imQoss We Ginke sn € Cir \ L yes Mise te we values LG: Sin Q = % yes = t Se P, (%, »() D O- sw"), 6% . ~4 . @® O- a- sis’) , ct le su = an 4 73, Te sic’ = | iT. pela- Coordimbes Sin Or ve aA AS pessMe 3. Consider the following moving point in space: P(t) = (3t, Vee’, se), a) (5 pts) Find the velocity and the speed of P(t) at the instant t = 0. = — ioe os Tz OP= C34, Veet, be) > > Vv: de = c= (3 Gt, e*) ak t=, at = (3, YG, 1) is Na vetrctly me Kia =4 ‘s Mus Speed, b) (5 pts) What is the a of the curve given by P(t) for 0 <¢ < 1n5? Simplify your answer. \ , \nS The ardeseth x d. Wwlde = Vere Me. jee dt \o5 nS Vb Wn = Vaid) de = \ wee = BlnS tee ~o-% {=> Se => BSe Paes =~ |S m2), c) (5 pts) Suppose we have a function f(x,y, z) with the property , Vila.vee yer) = (27, - —V6e, 5). Find | — F(P)) at the instant when the point P(t) passes through (3, Vee, 46 5f 2), By Yee chain © rhe, qo - ve 47 ? ue \nore, PE) passes Narva P2(%,%ee, her) mw Bebe - eb ne® 2 -laGe ,€& ) (ae eet, 4e*) —_ seas i Seo Ha ow — — § Fines ~ Vel nN \ = (cr, -We,5)- (svic,¢ are GNe ye) io = (AD-6e+5e% = | Ze 2 4. a) (8 pts) Show that lim at q does not exist. (x,y) (0,0) 24 + y? Use. the Ame path sec on pevaloalas ot Ke form ue On, Lek os express Atais veins a oath Pl) ~ (tak), Un Le, Poi —» (2,0). howeve \ean LR) ~ \ee ce) ( ak?) a eB \ = we ob = \ ben Now — . E—-o NC) to lhot {rat This Vimar depends ar A) & Apprackey (3,2) along tac Beret pathe PaCS yields LiRene Values \te £ (PCY) . be Vasrelere , re Limit does Nol cxuk toy Mae. Avo - ya Keck 2 b) (8 pts) Show that — lim vy _ does exist (hiut: the limit is equal to 0). (z,y)(0,0) £7 + y? aan Wwe war to show Mul \ va IS - 8 | eo A On) en, Frc wo Vie Gelar cow dialer Ye Cea re bie Gon Gu») bb @,s) Yasin onl THtete |. e\urent\ <r ad ree aes & Xs errar AG . Dong Second way Re (2,0) eelor by O18 y= (Ay Ay), Ng fake byt (Nx\ (4 | (erer\ | eatey| é (xs) > \ > AS CAD” (asy~ Se we As 9, Mag eter Oo, r%@ x Mird aa oe xe xa Se os com 2, So 444 < 3 (=. \ KS ay\ Cow yl sree yon ke s.) any Lim W\ 29 sls —Tyt, Now Use Mas Songun cls Meorens | (An) 20,7) Cray 5. Consider the function f(z, y,z) = zerY, a) (6 pts) Find the gradient of n ,Y,2) at Po(1,1,-1). — < % VE = come qt {= (2 ane" 1 & ve ) e. ") . Se Te = 2d en, e | \ ae & Se, OV bere ) Ge -e 3, b) (7 pts) Find the equation of the tangent plane to the surface f(x,y, z) = —e at Po. Ve lp. Sub ten. ploe level cf Con Ba: Rl fe) (Carnie 4+ (-3e, ~e,e) meaning Cree) = efy-\) telzaiyz oO La eee nen = |~3e-yse = 5) Ceonek “e@ fachar som ) c) (6 pts) Determine the direction in which f(z, y, z) increases most rapidly when the point (x, y,z) moves away from Po. Your answer should be a unit vector. We Leck on of mosk rapid incheage is nal ole Vp (Se-€,e), an make Xin a onik rector , we dake. = Nis \o, _ (-%e, ~€y e@) ~ (-%e,-e,e ) Q = ee oe a \ Fel,,| Nircrrrrs ee 6. Given a function f(z, y) satisfying f(1,2) = 4, VP l: (3, 4). a) (6 pts) Approximately how much is f(1.03, 1.99)? P= CW, Pe (1.03, 149} Tv: Be = (1y,By) = (9.03) - 0.01) woe Wan LP = Lp) + and, £ (Pe) + VE, ~2AQr = u + (3,4) . (49.03 ,~0,01) = Yro-o% ~oe = [4.05 | cpg ute Valve. dr Erp) Note tus u te fame ay wonhy Fe ECR + Sl A + Sle, OY + Ankee | RAS a Ka You caw alin solve Kats Using Ae Liredppaa\ Lectkve, ley oye b) (7 pts) Find a direction @ in which the directional derivative Dzf ls > 0. Your answer @ should be a unit vector. we oe) a) Won tr 3 Se NO Wak 0 L (as), ca ass note Her (a4) 1 (4-3) o Comeck 4) - u-2) = 2 Pee) ve : hoe («, -3) 4 AN a oni Vecher: We Gin pormalixe Ve owe vers so [62 GED 2 GED es (He, EY) ore sbhe | [ (lu, -3)| Se ee ) | Cle shar cbsbun in B= (-% 1%). ) c) (7 pts) Let S be the graph of f. In other words, S$ = {{z, y,z) € R? | z= f(z,y)}. Find the equation of the tangent plane to S at the point Po(1,2,4) € S. (Be careful.) 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