Question

DUE DATE: 23 MARCH 2020 1 1. Let f(x,y) = (x, y) + (0,0) 0. (x, y) = (0,0) evaluate lim(x,y)=(4,3) [5] 2r + 8y 2. Show that lim does not exist. [10] (*.w)-(2,-1) 2.ry + 2 3. Find the first and second partial derivatives of f(x,y) = tan-'(x + 2y). [16] 4. If z is implicitly defined as a function of x and y by I?+y2 + 2 = 1, show az Əz that +y=z [14] ar ду 5. If 6 = 3ray and y = x2? - 2y, evaluate grad (grado) (grad) 12 6. (a) Find the directional derivative of the function f(x, y, z) = sin(xyz at the point P = (7, 1, 1) in the direction of OĂ where A is the vector (0, 1, -1). (b) Let U be a unit vector whose direction is opposite to that of (grad f)(P).

Answer 1

e Solution fin,y)={ 5 nye 40² + y² -- lim 5n2y2 (ny) (4,3) 4 ² + y2 (24) & (0,0) (MY)= (0,0) 5:43 4.42 +3² 5X16X9 64 + 9 720 73 2n+ 8 9 (ny) — 12, 7-22) 2n4t2 lim 2x2 + 8x600) EX EX(-2) + 2 4 - 4 =2+2 O (indeterminate form) Somore investigation is needed We use Linear paths method. In this method we approach to the limit point along any path. If two paths lead to different nalues then limit does not enist Let y=ma where MER as y/ then a 271 + 8 mm (1+4m) x lim žm 891, mu +2. 4mn2t1 (1+4m) (m) 4 m.lt - (1 + 4m) 4mx Ametl} em is lim n n em am 2m.

which gives -(1+4 m) 2+ 2m -1 - 4m 2- 2m Here limit depends on parameter m different limit for different value of m. Hence/ limit does not exist. if we get dim form then we say (nyes (a,b) limit does not enist. l N.B. O where 170 sol 3. are y finy) = tant (ntry) Its first partial derivatines wirts and of and af respectively ay af a (tant (at 2y)) ifpid. wert. & then y treated as constant. an - n 1 = It exteyge .2 2 it (n+2y) 2 of an also af a (tant (atzy) an ay if p.d. wert.y then T + enteyze a treated as constant af an It (utryje Take the partial derivatives of the partial derivatines jhen this is called second partial derivating. . :) = on it exteyze aef -2 (ntry) {1+ exteyjeze required How an2

C 2 he also afl af dy t(atry, e 21-2) (x+2y) {1+ (1+243234 af - 4 (ntry) ay2 which is required {1+ (n+ 2yje e Solna If z is implicitly defined as a function of u andy by x² + y² + z ² 1 ₂2 = 1-1²_y 2 = 51-x² - y2 (1-12_ye) ke taking partial derivative wortese of equo 2 (1-x²-yegole (-44) 2, az - az an VT-xe-ye also taking partial derivative writy of eq" 2z = + (1-22 yeghe (44) Re .y V1-2²-42 Now əz 212 22 az on Ay ay -22 (-y). ty Ji-neye J1-12_y2 vintage - Joxyz -n²-42 √1-21²-ye 1-21²-ye-| DT-21² - y2 1-212_y2 Ji-n2-yz √1-2²-ye = di-xe-ye Ji-xe_yz 2 - 2 -IN Hence heh he e ke n Z- N- S.E.D.