Tangent plane, Equation of the tangent plane, Partial differentiation, Normal vector of the plane.
If we remember, the geometric interpretation of the derivative in case of single variable functions was the slope of the tangent.
The equation of the tangent for function at point is:
The tangent of a single variable function is a line, and the tangent of a two-variable function is a plane.
The number of coordinates is increased by 1, so it is not x and y, but x, y and z.
The equation of the plane tangent to function at point is:
Well, this is the equation of the tangent plane.
Let's see an example.
Here is this function, for instance:
and we are looking for the tangent plane at point .
Here comes the equation of the tangent plane,
and we have to calculate these.
Well, this is the equation of the tangent plane:
If we expand the parentheses, and get all terms on one side,
then we can see the normal vector of the plane.
And here is the normal vector:
The first two coordinates are the derivatives with respect to x and y,
and the third coordinate is negative one.
What should parameter be, so that the tangent at point
to function would also pass through point ?
A plane passes through a point if the equation holds when substituting the point’s coordinates into the equation of the plane.
Here is :
Now, let's see the vector.
The vector in the formula must be of unit length.
Since now is not a unit long,
we turn this into a unit vector.
We divide the vector by its own length:
The equation of the plane that is tangent to the surface given by at point is:
The normal vector of the tangent plane is . This is easy to see, if we move z to the right side of the equation of the tangent plane.