Final answer:
The equation of the tangent plane to the surface y = x² - z² at the point P(3,8,1) is y = 6x - 2z - 8, found using partial derivatives of the function and the point-slope form of a plane.
Step-by-step explanation:
To find the equation of the tangent plane to the surface y = x² - z² at point P(3,8,1), we first need to find the partial derivatives of the given function at point P. The partial derivatives represent the slope of the tangent lines in the direction of the x-axis and z-axis, and they are required to determine the plane's slope.
To find the partial derivatives, first start with:
Then plug in the point P(3,8,1) into the derivatives:
Now, the equation of the tangent plane can be determined using the point P and the derivatives found:
y - y0 = ∂y/∂x (x - x0) + ∂y/∂z (z - z0)
Plugging in our values for P(3,8,1):
y - 8 = 6(x - 3) - 2(z - 1)
This simplifies to the tangent plane equation:
y - 8 = 6x - 18 - 2z + 2
And with further simplification:
y = 6x - 2z - 8