Question

Find the equation of the tangent plane to the surface given by z⁴ = xeʸ cos(z) at the point (4, 0, 0).

Answer 1

The equation of the tangent plane to the surface given by z⁴ = xeʸ cos(z) at the point (4, 0, 0) is 0 = 0.

Step-by-step explanation:

To find the equation of the tangent plane to the surface given by z⁴ = xeʸ cos(z) at the point (4, 0, 0), we can use partial derivatives.

1. Take the partial derivative with respect to x: ∂/∂x (z⁴) = 4z³
2. Take the partial derivative with respect to y: ∂/∂y (z⁴) = 0
3. Take the partial derivative with respect to z: ∂/∂z (z⁴) = 4z³
4. At the point (4, 0, 0), substitute the values into the partial derivatives: ∂/∂x (z⁴) = 4(0)³ = 0, ∂/∂y (z⁴) = 0, ∂/∂z (z⁴) = 4(0)³ = 0
5. The equation of the tangent plane is therefore: 0(x - 4) + 0(y - 0) + 0(z - 0) = 0

Thus, the equation of the tangent plane to the surface is 0 = 0.