Final answer:
The equation of the tangent plane to the surface z = 5x²y - 2xy at the point P(-2, -1, 16) is found by first calculating the partial derivatives with respect to x and y, then evaluating at the point to get the slopes, and finally using the point-slope form to write the equation.
Step-by-step explanation:
To write the equation of the tangent plane to the given surface z = 5x²y - 2xy at the point P(-2, -1, 16), we first need to find the partial derivatives of the equation with respect to x and y, which will give us the slopes of the tangent plane in the x and y directions, respectively.
The partial derivative with respect to x is dz/dx = 10xy - 2y, and with respect to y is dz/dy = 5x² - 2x. We then evaluate these derivatives at the point P(-2, -1) to get the slopes of the tangent plane at P. Substituting x = -2 and y = -1 yields dz/dx = 20 and dz/dy = -12.
With the slopes, we can write the equation of the tangent plane using the point-slope form, which is z - z0 = (dz/dx)(x - x0) + (dz/dy)(y - y0). Substituting the point P and the calculated slopes, we get the final equation of the tangent plane as z - 16 = 20(x + 2) - 12(y + 1).