Final answer:
The equation of the tangent plane to the given surface at the specified point is 4x - 4y + 3z = 55.
Step-by-step explanation:
To find the equation of the tangent plane to the surface at the specified point, we need to find the partial derivatives of the surface equation with respect to x and y. Let's start by finding these partial derivatives:
Partial derivative with respect to x: dz/dx = 4(x-1)/3(y^3)^2
Partial derivative with respect to y: dz/dy = -4(x-1)^2/3(y^3)^3
Now, we can plug in the x and y values of the specified point (2, -1, 17) into these partial derivatives to find the slope of the tangent plane at that point:
dz/dx = 4(2-1)/3((-1)^3)^2 = 4/3
dz/dy = -4(2-1)^2/3((-1)^3)^3 = -4/3
The equation of the tangent plane is given by: z - z0 = (dz/dx)(x - x0) + (dz/dy)(y - y0), where z0, x0, and y0 are the coordinates of the specified point. Plugging in the values, we get:
z - 17 = (4/3)(x - 2) + (-4/3)(y + 1)
Simplifying further, we get the equation of the tangent plane as:
3z - 51 = 4x - 8 - 4y - 4
or
4x - 4y + 3z = 55