Final answer:
The equation of the tangent plane to the surface at point P(1,2,-1) is 4x + y + 4z + 3 = 0, determined by calculating the partial derivatives and using the gradient vector at P.
Step-by-step explanation:
To find the equation of the tangent plane to the given surface at the point P(1,2,-1), we first need to find the gradient of the surface function at that point. The surface is represented by the equation F(x,y,z) = x²yz³ - 10z - 8. We will calculate the partial derivatives of F with respect to x, y, and z, which are Fx, Fy, and Fz, respectively.
The partial derivative with respect to x is Fx = 2xyz³. At point P(1,2,-1), Fx(P) = 2·1·2·(-1)³ = -4.
The partial derivative with respect to y is Fy = x²z³. At point P(1,2,-1), Fy(P) = 1²·(-1)³ = -1.
The partial derivative with respect to z is Fz = 3x²yz² - 10. At point P(1,2,-1), Fz(P) = 3·1²·2·(-1)² - 10 = 6 - 10 = -4.
The gradient vector at P is thus ∇F(P) = <-4, -1, -4>. The equation of the tangent plane at P is given by the dot product of the gradient and the vector (x-1, y-2, z+1) being equal to zero, which gives us -4(x-1) -1(y-2) -4(z+1) = 0. Simplifying, we obtain the equation of the tangent plane: 4x + y + 4z + 3 = 0.