Final answer:
To find the equation of the tangent plane to the given surface at the specified point, we need to find the partial derivatives of the surface equation with respect to x and y. Then, we can substitute the values from the specified point into the partial derivatives to get the slope of the tangent plane. Finally, we can write the equation of the tangent plane using the point-slope form.
Step-by-step explanation:
To find an equation of the tangent plane to the given surface at the specified point, we need to find the partial derivatives of the surface equation with respect to x and y. Then, the tangent plane can be represented by the equation:
z - z0 = fx(x0, y0)(x - x0) + fy(x0, y0)(y - y0)
where fx(x0, y0) and fy(x0, y0) are the partial derivatives evaluated at the specified point (x0, y0).
In this case, the partial derivatives are:
fx(x, y) = 16xy2
fy(x, y) = 16x2y - 7
Substituting the values from the specified point (1, 3, -4):
fx(1, 3) = 16(1)(32) = 144
fy(1, 3) = 16(12)(3) - 7 = 41
Therefore, the equation of the tangent plane is:
z - (-4) = 144(x - 1) + 41(y - 3)
This can be simplified to:
z = 144x + 41y - 425