Question

Consider x=h(y,z) as a parametrized surface in the natural way. Write the equation of the tangent plane to the surface at the point (−1,−3,5) given that ∂h/∂y(−3,5)=−2 and ∂h/∂z(−3,5)=5.

Answer 1

The equation of the tangent plane to the surface at the point (-1,-3,5) with the given partial derivatives is x - 2y + 5z = 12.

Step-by-step explanation:

To find the equation of the tangent plane to a parametrized surface x=h(y,z) at a specific point, we can use the gradients of h with respect to y and z. The point given is (−1,−3,5), and we have the partial derivatives ∂h/∂y(−3,5) = −2 and ∂h/∂z(−3,5) = 5. Using this, the tangent plane equation at the point (x_0, y_0, z_0), where x_0 is the output of the surface function h at (y_0, z_0), is given by:

∂h/∂y(y_0,z_0)(y - y_0) + ∂h/∂z(y_0,z_0)(z - z_0) = x - x_0

Plug in the known values to get the equation of the tangent plane:

(−2)(y - (−3)) + (5)(z - 5) = x - (−1)

Which simplifies to:

−2y - 5z + x = −3 − 10 − 1

The final equation of the tangent plane is:

x - 2y + 5z = 12