Final answer:
To find the equation of the tangent plane to the surface z = ln(x - 8y) at the point (9, 1, 0), calculate the partial derivatives, evaluate them at the given point, and apply the formula for the equation of a tangent plane. The final equation is z = x - 9 - 8y + 8.
Step-by-step explanation:
To find an equation of the tangent plane to the surface z = ln(x - 8y) at the point (9, 1, 0), we start by calculating the partial derivatives of the function with respect to x and y. For a function z = f(x, y), the tangent plane at a point (x_0, y_0, z_0) is given by:
z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)
where f_x and f_y are the partial derivatives of f with respect to x and y, respectively.
The partial derivatives are:
f_x = 1 / (x - 8y)
f_y = -8 / (x - 8y)
Evaluating these at the point (9, 1), we get:
f_x(9, 1) = 1
f_y(9, 1) = -8
Therefore, the equation of the tangent plane at (9, 1, 0) is:
z - 0 = 1 * (x - 9) + (-8) * (y - 1)
Simplifying, the final equation of the tangent plane is:
z = x - 9 - 8y + 8
The complete qestion is: Find an equation of the tangent plane to the surface z = ln(x - 8y) at the point (9, 1, 0). is: