MATH 201 Final, Fall 2011

Math 201 - Fall 2011-12 Calcul us and Analytic Geometry III, all sections Final Exam, January 21 - Duration: 2 hours 15 minutes GRADES­- 2 (/15) 3 (/15) 4 (/11) 5 (/14) 6 (/15) 7 (/15) Problem 1 (/15) Parte. Part b Part c Total GRAND TOTAL, YOUR NAME, YOUR AUB ID#, PLEASE CIRCLE YOUR SECTION, Section 1 MWF 3, Kobeissi Recitation F 11 Section 5 MWF 10, Abi-Khuzam Recitation T 11 Section 9 MWF 11 , Brock Recitation T 12:30 Section 13 MWF 2, Nahlu, Recitation T h 11 Section 17 MW'F 8, ?vfakdisi Recitation F 2 Section 21 MWF 1, Raji Recitation M 8 Section 24 MWF 10, Egeileb Recitation F 11 Section 2 ?v!V.lF 3, Kobeissi Recitation F 5 Section 6 MWF 10, Abi-Khuzam Recitation T 3:30 Section 10 MWF 11 , Brock Recitation T 2 Section 14 MWF 2, Nahlu, Recitation Th 3:30 Section 18 M\VF 8, Makdisi Recitation Th 8 Section 22 MWF I, Raj; Recitation M 9 Section 25 MWF 10, Egeileb Recitation F 2 Section 3 NrVVF 3, Kobeissi Recitation F 4 SectiOD 7 MWF 10, Abi-Khuzam ReCitation T 5 Section 11 MWF 11, Brock Recitation T 11 Section 15 MWF 2, Nahlu, Recitation Th 8 Section 19 MWF 8, Makdisi Recitation Th 2 Section 23 1v[\VF 1, Raj i Recitation M 4 Section 26 MWF 10, Egeileh Recitation F 3 Section 4 MvVF 3, Kobeissi Recitation F 10 Section 8 MWF 10, Abi-Khuzam Recitation T 2 Section 12 MWF 11 , Brock Recitation T 3:30 Section 16 MWF 2, Nahlu, Recitation Th 5 Section 20 MWF 8, Makdisi Recitation Th 3:30 1. (5 pts each part,15 pts total) (a) Use the Sandwich theorem to find the following limit 1+1+ ... +1 lim} 2 n n __ co 31n;n (b) (UNRELATED) Find, with justification, all values of p for which the following series is convergent 00 ( 1 1 )' L --sin(-) n~l,;n ,;n continued ... (c) (UNRELATED) Compute the nth partial sum Sft of the following series, and use it to find, according to the values of c, the sum of the series 00 c''' - c' c> O. L: ('" + 3)(",+1 + 3)' """,,0 2. (5 pts each part, 15 pts total) (0) Consider the function xy' I (x,y) = 3 ., + " if (x,y)" (0,0) sm x y Prove that 1(0,0) may be defined in sucb a way that 1 becomes continuous at (0,0). (b) If in part (a) 1(0,0) has been defined correctly, prove that 1 is not differentiable at (0,0). For this part, you may use without proof that 1,(0,0) = 1,(0,0) = o. continued ... (c) (UNRELATED) By about how much will h(x, y) = In(x' + y' + Z2) change if the point P(x, y, z) moves from Po(l, 1, -1) a distance of ds = 0.1 unit in the direction of the vector 3i + 6j - 2k? 3. (5 pts each part , 15 pts total) (a) Sketch the region or integration and evaluate the double integral l IS /.'/. I 0 yerydxdy. (b) Evaluate the double integral '"' r' dydx /.o J~I/f>y6+1· continued ... (c) Evaluate the integral 1 00 - ' a e J: dx . 4. (11 pts total: 5 pts for (a), 3 pts for (b), 3 pts for (c)) (0) Find the volume of the solid bounded below by the surface z = 1, and above by the surface x2 + y2 + Z2 = 4. (b) If the density is o(x, y, z) = Z , set up but do not evaluate, a triple integral with av = dxdydz , giving the mass of the solid in part (a). continued" . (c) Set up but do not evaluate the integral in part (b) in spherical coor­ dinates. 5. (14 pts total: 5 pts for (a), 5 pts for (b), 4 pts for (c)) (a) Let f(x) = t:!3' Find the Taylor series expansion of f about a = 1,(Le., centered at a = 1), and use it to find ! (n)(l). (b) (UNRELATED) Find the maximum and minimum values Dfthe func­ tioo l ex, y) = x2 + y' + x - y on the curve x' + y' = 2. continued ... (c) (UNRELATED) Use the transformation x = au, Y = bv,z = cw to find the volume of the region R = {(x,y,z): ~ + ~ + ~ ~ 1) . Here a,b,c are positive constants. 6. (5 pts each part, total 15 pts) Consider the region D = {(x,y): x' + y' ~ 4, Y ~ -I}, and the vector field F(x ,y) = -yi + xj in D. (a) Using appropriate parametri2.tions of the boundary, evaluate the liDe integra.! Ie F · dr where C is the positlvely oriented boundary of D. (b) Use Green's Theorem to evaluate the integral in part (0). continued ... (c) Let G(x,y) ~ =,7,,1+ "~,,j be another vector field defined away from (0 1 0). Evaluate Ie G · dr on the same curve C as before. 7. (5 pts each part, 15 pts total) Consider the vector field F(x ,y,z)~( Y, ,)i+(l X, , +e' cosy)i+(e'siny)k, l+xy +x y and Jet S be any curve in the first octant, starting at A (~, 1f , 1) and ending at B(~, pn3) (a) Is F a conservative field in the first octant {(x,y,z) : x,Y, z > OJ ? Prove your answer. (b) Evaluate the line integral Is F · dr. continued ... (c) (UNRELATED) Suppose I (x ,y) satisfies lu + Iw = 0 in a domain D with positively oriented boundary C. Prove the following identity: fa !'v I .nds = J101"i1 II'dA.