Question

Find the equation of the tangent plane to z=ey+x+x4+8 at the point (3,0,93). z= ( 1 point) Find the (linear) equation of the tangent plane to the surface z=xe /xy at the point (7,0,7).

Answer 1

To find the equation of the tangent plane to the given surface at a specific point, we need to calculate the partial derivatives of z with respect to x and y. Using the point-slope form of a line, we can then find the equation of the tangent plane.

Step-by-step explanation:

To find the equation of the tangent plane to the surface z=ey+x+x4+8 at the point (3,0,93), we need to find the partial derivatives of z with respect to x and y at that point. The partial derivative of z with respect to x gives the slope of the tangent line in the x direction, and the partial derivative with respect to y gives the slope of the tangent line in the y direction. Let's calculate these derivatives:

dz/dx = 1 + 4x3 = 1 + 4(3)3 = 109
dz/dy = e = e

Now we can use the point-slope form of a line to find the equation of the tangent plane:

z - z0 = (dz/dx)(x - x0) + (dz/dy)(y - y0)
z - 93 = 109(x - 3) + e(y - 0)
z = 109x - 327 + ey