Let $S$ be a surface in 3D described by the equation $3 x+\sin (y)+z^{2}=0$.

What is the equation of the plane tangent to $S$ at $(-3, \pi, 3)$ ?
Choose 1 answer:
(A) $3(x-3)-(y+\pi)+6(z+3)=0$
(B) $3(x+3)-(y-\pi)+6(z-3)=0$
(C) $2(z-6)=0$
() $-3(x+3)-\pi(y-1)+3(z-6)=0$

Question

Let $S$ be a surface in 3D described by the equation $3 x+\sin (y)+z^{2}=0$.

What is the equation of the plane tangent to $S$ at $(-3, \pi, 3)$ ?
Choose 1 answer:
(A) $3(x-3)-(y+\pi)+6(z+3)=0$
(B) $3(x+3)-(y-\pi)+6(z-3)=0$
(C) $2(z-6)=0$
() $-3(x+3)-\pi(y-1)+3(z-6)=0$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the equation of the tangent plane to the surface $S$ at a given point, we need to calculate the gradient of the surface at that point. ‖ ‖ step 2 ⋮ The gradient of the surface $S$ described by $f(x, y, z) = 3x + \sin(y) + z^2$ is $ abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} ight)$. ‖ ‖ step 3 ⋮ From the asksia-ll calculation, we have the partial derivatives at the point $(-3, \pi, 3)$: $\frac{\partial f}{\partial x} = 3$, $\frac{\partial f}{\partial y} = \cos(y)$, and $\frac{\partial f}{\partial z} = 2z$. ‖ ‖ step 4 ⋮ Evaluating the partial derivatives at $(-3, \pi, 3)$, we get $ abla f(-3, \pi, 3) = (3, 0, 6)$. ‖ ‖ step 5 ⋮ The equation of the tangent plane is given by $ abla f \cdot (x - x_0, y - y_0, z - z_0) = 0$, where $(x_0, y_0, z_0)$ is the point of tangency. ‖ ‖ step 6 ⋮ Substituting the gradient and the point $(-3, \pi, 3)$ into the equation of the tangent plane, we get $3(x + 3) + 0(y - \pi) + 6(z - 3) = 0$. ‖ ‖ step 7 ⋮ Simplifying the equation, we obtain $3(x + 3) + 6(z - 3) = 0$. ‖ ‖ step 8 ⋮ The final equation of the tangent plane is $3x + 9 + 6z - 18 = 0$, which simplifies to $3x + 6z - 9 = 0$. ‖ ∻[question 1] Answer∻ ***B*** ∻Key Concept∻ ⚹Gradient and Tangent Plane⚹ ∻Explanation∻ ⚹The equation of a tangent plane to a surface at a given point can be found using the gradient of the surface function at that point and the point of tangency.⚹◊What is the formula for the equation of a plane tangent to a surface at a specific point in 3D space?⍭ Generate me a similar question◊

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