00:01
In this question, we're asked to find the tangent plane to the given surface, an equation of the tangent plane to the given surface at the point 2 -2 -2.
00:11
Let's call the left -hand side of this equation by f capital of x, y, z.
00:22
Then the general formula for an equation of the tangent plane is the gradient of f at x ,0, y, naught, and z -not, dot product, x minus x knot, y minus y0, and z -m minus z -not, equal zero.
00:45
In our case, we are given the coordinates of x ,0, y -not, and z -not, so we can rewrite this equation as the gradient of f at the point 2 -2.
01:01
Dot product, x minus 2, y minus 2 and z - minus 2 equal to 0.
01:12
Now let's calculate the gradient of the function f.
01:20
First we'll differentiate the function f capital with respect to x to get 7y plus z.
01:30
The second component is a derivative of f with respect to y, which equals to 7x plus 4 z.
01:41
And the last component is a derivative of f with respect to z, which is going to be 4y plus 4 .x.
01:51
Now let's calculate its value at the point 2 to 2...