Equation of a tangent plane

[Calpalned]

Homework Statement

What is the difference between the two given equations below? When would you use one or the either?

Homework Equations

$z - z_1 = \frac{∂z}{∂x}(x_1, y_1, z_1)(x - x_1) + \frac{∂z}{∂y}(x_1, y_1, z_1)(y - y_1)$
$\frac{∂f}{∂x}(x_0, y_0, z_0)(x - x_0) + \frac{∂f}{∂y}(x_0, y_0, z_0)(y - y_0) + \frac{∂f}{∂z}(x_0, y_0, z_0)(z - z_o)$

The Attempt at a Solution

I am confused because it seems like there are several equations for the tangent plane.

 

[Mark44]

Homework Statement

What is the difference between the two given equations below? When would you use one or the either?

Homework Equations

$z - z_1 = \frac{∂z}{∂x}(x_1, y_1, z_1)(x - x_1) + \frac{∂z}{∂y}(x_1, y_1, z_1)(y - y_1)$
$\frac{∂f}{∂x}(x_0, y_0, z_0)(x - x_0) + \frac{∂f}{∂y}(x_0, y_0, z_0)(y - y_0) + \frac{∂f}{∂z}(x_0, y_0, z_0)(z - z_o)$

The two equations above are working with different forms for the equation of the surface on which you're asked to find a tangent plane.

In the first equation, you are given z as a function of x and y (i.e., z = g(x, y)).
In the second equation, the surface is given like this: f(x, y, z) = C.

The Attempt at a Solution

I am confused because it seems like there are several equations for the tangent plane.

 

[Calpalned]

The two equations above are working with different forms for the equation of the surface on which you're asked to find a tangent plane.

In the first equation, you are given z as a function of x and y (i.e., z = g(x, y)).
In the second equation, the surface is given like this: f(x, y, z) = C.

thank you

 

[HallsofIvy]

You understand, I hope, that the second one is not an equation! I presume that was a typo and it was supposed to be equal to 0 or some other constant.

Given that, the first is generally used where you are given z as a function of x and y: z= z(x,y), with $z_1= z(x_1, y_1)$.

The second would be used when you given some function of x, y, and z equal to a constant: $$F(x, y, z)= C$$.

That second form is more general since, if you have z= f(x, y), you can write it as F(x, y, z)= z- f(x, y)= 0 but if you are given F(x, y z)= C, you may not be able to solve for z. For example, [tex]F(x, y, z)= x^2+ y^2+ z^2= R^2[/itex] is the equation of a sphere. If you wanted to solve for z, you have to do two cases- [itex]z= \sqrt{R^2- x^2- y^2}['itex] and $z= -\sqrt{R^2- x^2- y^2}$.