So here's what I was vaguely told:
1. Tangent Plane is always z = L( x,y )
2. Knowing that "think of when they are parallel"
So if a plane is parallel that just means that it has an extra + c or something too?
1. Find the points on the graph of at which the tangent plane is parallel to
Can someone explain this to me/start this for me?
2. Suppose that the plane tangent to the surface at has equations . Estimate
Same as above; I'm not really sure where I'm supposed to start. Do I integrate or something?
1. Use gradient vectors:
Let be the unit vector of <10,-4,1> (I multiplied all of the components by 2) and let be the unit vector of the gradient vector of F(x,y,z)
Set the components of and equal to each other and solve for the values. You get 3 equations and you have 3 variables.
An alternative method would be to solve both equations and ) for z and set them equal to each other. See where that gets you.
2. Just plug in -2.1 for x and 3.1 for y in the equation and solve for z.
Two planes, Ax+ By+ Cz= D (which would be the same as z= (D- Ax- By)/C) and Px+ Qr+ Rz= S (which would be the same as z= (S- Px- Qy)/R) are parallel if and only if their normal vectors, <A, B, C> and <P, Q, R> are parallel- and that means that one is a multiple of the other.
A plane, Ax+ By+ Cz= D is tangent to a surface f(x, y, z)= constant, if and only if the normal vector to the plane, <A, B, C>, is parallel to the normal vector of the surface which is the same as the gradient of the function that lilaziz1 mentioned: