Question

(9) Let f(x,y) = 1+ 4- y2. Evaluate f(3,1), find and sketch the domain of f. (10) A thin metal plate, located in the xy-plane, has temperature T(x,y) at the point (x,y). The level curves of T are called isothermals because at all points on such a curve the temperature is the same. Sketch some isothermals if the temperature function is given by 100 T(x, y) = 1 + x2 + 2y2 (11) Show that lim (z2 + y²) In(z2 + y²) exist. (x,y)+(0,0)

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Answer 1

4 =lim 7-907 Info? 172 com ( incr)] lim ro Ya? Cor 2) r- ot dr d - 2772 % 2 (27) lim rot tim [-22 -2/3 8 of von CM2+ y2) In CM2 +42) = 0, (ny) (0,0) CS Scanned with CamScanner

22 lim (12ty2) in Cu²+ y²) (ny)(0,0) im (2+2) in /v1+ y) = (0? +o) in 10t0%) (ny) 10o) (O)(0) undefined First rewnite the given function using polar Coordinates do Twoso, ya msis O, Y>0 Also n²+ y2 = 82 70+ ai(14)-51010) be rewritten as followed The variable The given limit can (M2+ y2) 10 (42+42) - lim (82) in (m2) (wy) -9 1010) (0) Ino?). In lo) undefined By direction substitution, we undefined Value. TO use the have to change One L'hospital rule, first q the indeterminate forms & lim CM2 + y ) in (^2+y?) - lim (Y2) 10 (2) CHH14)-7 (010) rot lim In(72) we the indeterminate form a new we get apply l' hospital's ruler- (n2 + y2) in (x2 + y2) CS Scanned With) o (nelanner rot tim () 11/0?)

2 with this is equation ellipse 100 1 and 9?- b² 50 - - a 100 -1 c and bi $ Y TOUM)=C In the case the level cuores present function the 100 -k + 12+242 took CI+ 12 +242) 14 12+242 = coo 42+242 - 100 -lock L 100 The level curvex y describe a family a k K ellipes 8 K-1 k=2 k=3 2 k=4 6 to 4 6 o 2 4 6 CS Scanned with Cascanner

9) Giren fclusy) 13,1) I+s4-42 it 4-12 it 3 = 2.732 The not function of is defined when 4-y2 zo (boot square root of negative is real). -) 42 42 y2 24 42-2250 =) (4+2)(y-2) 50 =) -25 Y=2 the donain a f is {Cry): NER, -224227 100 10) Given that Tchy) 14 12+242 Ita? + 242 100 M? + 2y? 100-1 72 + 242 100 100 - x2 y² El 100 50-1 CS Scan ed With Cash Slanner