MATH 101 Exam 2, Fall 2005

AMERICAN UNIVERSITY OF BEIRUT Mathematics Department Math 101 - Quiz I Fall 2005-2006 Lecure: Ms. Diana Audi Lecture: Ms. Marwa E] Houri Section 3: Th 08:00 - 09:00 Section 1: T 08:00 - 09:00 Section 4: Th 02:00 - 03:00 Section 2: T 11:00 - 12:00 Section 5: Th 11:00 - 12:00 Section 6: Th 09:30 - 10:30 Time: 60 min Directions: Write your name and ID number and circle your section number. Answer the questions in the allocated spaces, if more space is needed continue on the back. NO CALCULATORS ARE ALLOWED!!! d I- (20 points) Find the derivative = of the followings: 1 2+2 | dy o{s'n)- Cay) dx ) a) y = sin( Cvs (Aw Aa c) sin(ry*) + 2y? =3r+2 Im pret A ffacchebon cos(ryt): (7b 2x77 ys fais” > 290s (xy") + Axa y ‘Cosity) + ey = z | pleats = GF: 3-4 Cos(xq") >, 3-4 cos (45) a Dv y Cos (ryt) 484 II- (15 points) Find the equation of the tangent line to the parametric curve. i 3 z=Ve+t = +t | y = sin(3t +7) +2 at t= 0 IlI- (20 points) Let f(z) = 2° — 327 +2 . Find f’. . Find critical values of f. . Find intervals where the graph of f is increasing or decreasing. . Find local extremes. . Find intervals where the graph of f is concave up or concave down. . Does f have an inflection point? If yes where? . Sketch the graph of f. ca oo ot — L\ ? Das ve oan " x x= asa al mia ok x2 2. Serene! 2. - "Oye X= 6 5 5 } aa - O «£2 X= 4 Blank page x 5 2 O i|-ad IV- a) (6 points) Show that Y1 —S¢+1—azatr= 0 b) (4 points) Use part a) to find _ Wi-bSa-1 lim ———-——— r—0 x Qn £0 V- (15 points) Let f(s) =4 2? +2r O<a<1 3x zr>i where is f differentiable? Explain. 1 l+27 Show that g has an absolute maximum and an absolute minimum on [0, 1] and find phen Deed raberval Wan 5 fas abs min and max. 1s Cowl on ac Ae tex LO on Po, 1]. (14x) ~ S Ovvkical pore x =O ba points X20 and x-7 Dlod= 4 a alssaude mad \ DU = > abseil min. VI- a) (6 points) Let g(x) = A. b) (4 points) Let f be a function such that f(0) = 0 and its derivative 1 f'(z) . Use the Mean Value Theorem to show that 1+? 5sfa) si fis Conk on Loot ac Don (0,1) ewe by VT Mec rs a pot cin [ov") such Hak ¥'Co) = } (iy - Pe - 4) )-O VII- a) (5 points) Let f and g be two continuous functions on fa, 6] and differentiable on (a, b) such that f(a) = g(a) and f(b) = g(b) for some real numbers a and J. Show that there exists a number,c in (a,b) sul n eed fe sh OW at g'(e) ( Me lon O \ ov Coo hae Ror = POTS , fa)7O h lad > Hear Be hb) = plb>- 3! ; vet crn lab? QP’ Potere ire rs ap hath Yb Leer =0 z? $e) - Aloe o Zz? Pere § 6 b) (5 points) Let f and g be two continuous functions on (0, 2] and differentiable on (0,2) such that f(0) = 2, g(0) = 3, f(2) =1, and by RMPe's Be such MA WC) xO dalled. O GOOD LUCK}! 2? 39 (er oo 3 $oe TACO