Question

1) Show that the follwing limit does not exist

lim_(x,y)-(0,0)(xy+x²)/(x^²+y²)

2) Fin all the second orderpartial derivatives of

f(x,y,z)=x²tan^-1(y+z)

3) Find the equation of thetangent plane to the surface

z=x²-4y² atthe piont (1,2,-15).

4) Fin the linear approximation tothe function

f(x,y,z)=e^-2xy+cos(2yz) at the point (1,-1,2)

5) Fin the normal vector to thesurface

z²=x²-y² at the point (5,3,4)

6) You are standing on a surfacegiven by the equation

z=x^2-2y². Ifyou are standing at the poin (2,1,0), in which directio is thesteepest slope?

7) The temperature in the solarsystem is given by

T(x,y,z)=10⁵/(x²+y²+z²). If acomet travels along the path r(t)=(t, t²-4, 2t), howfast is the temperatur changing when t=2?

8) Fin the critical point of thefunction f(x,y)= e^4x-e^-y. Use the secondderivative test to classifyf them, if possible.

9) Fin the extreme values off(x,y)=4x2+3y2 on the square 0 is less than or equal x,y less thanor equal to 1.