The equation for the tangent plane to the surface z=26−x²−y² at the point (2,1,23) is 4x + 2y - z = 15.
The equation for a tangent plane of an explicitly defined surface z=f(x,y) at the point (a,b,c) is:
f_x(x - a) + f_y(y - b) + (z - c) = 0
where f_x and f_y are the partial derivatives of f with respect to x and y, evaluated at the point (a,b).
In your case, the surface is defined as z=26−x²−y² and the point of interest is (2,1,23).
Therefore, we need to find f_x and f_y, evaluate them at (2,1), and plug those values into the equation above.
Here are the steps:
Find f_x and f_y:
f_x = -2x
f_y = -2y
Evaluate f_x and f_y at (2,1):
f_x(2,1) = -4
f_y(2,1) = -2
Plug the values into the equation for the tangent plane:
-4(x - 2) - 2(y - 1) + (z - 23) = 0
Simplifying the equation, we get:
-4x + 8 - 2y + 2 + z - 23 = 0
Rearranging the terms, we get the final equation for the tangent plane:
4x + 2y - z = 15
Therefore, the equation for the tangent plane to the surface z=26−x²−y² at the point (2,1,23) is 4x + 2y - z = 15.