Find the equation of the tangent plane to the surface $3 x y z + x ^ { 3 } + y ^ { 3 } + z ^ { 3 } = 15$ at (1, -3, 2).
A) $- 15 x + 33 y + 3 z = - 108$
B) $- 15 x + 33 y + 3 z = 108$
C) $15 x + 33 y + 3 z = - 108$
D) $- 15 x - 33 y + 3 z = - 108$

Question

Quizplus · Accepted Answer

To find the equation of the tangent plane to the given surface at the point (1, -3, 2), we first need to find the gradient of the surface at that point. The gradient is given by the partial derivatives with respect to each variable. The given surface is $3xyz + x^3 + y^3 + z^3 = 15$.The partial derivatives are:- With respect to $x$: $3yz + 3x^2$- With respect to $y$: $3xz + 3y^2$- With respect to $z$: $3xy + 3z^2$Evaluating these at the point (1, -3, 2) gives:- $3(-3)(2) + 3(1)^2 = -18 + 3 = -15$- $3(1)(2) + 3(-3)^2 = 6 + 27 = 33$- $3(1)(-3) + 3(2)^2 = -9 + 12 = 3$Thus, the gradient vector at (1, -3, 2) is $(-15, 33, 3)$.The equation of the tangent plane at a point $(x_0, y_0, z_0)$ with gradient vector $(A, B, C)$ is given by $A(x - x_0) + B(y - y_0) + C(z - z_0) = 0$. Substituting the values, we get:$-15(x - 1) + 33(y + 3) + 3(z - 2) = 0$Expanding and simplifying, we get:$-15x + 15 + 33y + 99 + 3z - 6 = 0$$-15x + 33y + 3z = -108$Therefore, the correct answer is A) $-15x + 33y + 3z = -108$.

Find the Equation of the Tangent Plane to the Surface 3xyz+x3+y3+z3=153 x y z + x ^ { 3 } + y ^ { 3 } + z ^ { 3 } = 153xyz+x3+y3+z3=15

Find the Equation of the Tangent Plane to the Surface $3 x y z + x ^ { 3 } + y ^ { 3 } + z ^ { 3 } = 15$