Question

4. (4 pts) Consider the surface z=x2y+y3.(a) Find the normal direction of the tangent plane to the surface through (1,1,2).(b) Find the equation of the tangent plane in (a).(c) Determine the value a so that the vector−→v=−−→i+ 2−→j+a−→k is parallel to the tangent plane in (a).(d) Find the equation of the tangent line to the level curve of the surface through (1,1).

4. (4 pts) Consider the surface z = z2y + y). (a) Find the normal direction of the tangent plane to the surface through (1,1,2). (b) Find the equation of the tangent plane in (a). (c) Determine the value a so that the vector of plane in (a). -1 + 2] +ak is parallel to the tangent (d) Find the equation of the tangent line to the level curve of the surface through (1,1).

Answer 1

the given Z=_x2y + ys = this define finy, z) = x²y +43 - 2 f(x, y, z)=0 gin abore sarfon of is normal to surface the o f = of 2xy x² + 3y2, -1) @ + of 02 = thus, DA(1,1,2) = (2, 173,-1) = (2,4,-1) is the normal veetu to surfou at (11/12) 6 Now of = n = (2,41-1) and ā= (1/12) le the tangent plane gius 8 ñ = á -> (78,2) 2,4-1) =(4,-) (2:41 =) [2x+4y - 2 - 41 6rey plane

Page No. Dato C © Note n is plallal to v = -1 +2jt akan an = ( 21, , -d) = (1, 2,-2) = -1 2 this a = -x = 2 the a = 2 с © Note (2x+4y-z = 6) is place passing through (1,1,2) hea thee Ż = 2 28+ ay-2=4 2x+6y = 6 x+2y=3 leni F reg.