To find the equation of the tangent plane at a point on the given surface, we need to follow a few steps. First, let’s rewrite the equation of the surface:
z = 3y² – 2x² + x + 1
Next, we’ll denote the coordinates of the specified point as (x0, y0, z0). It’s important to substitute the values for x0 and y0 into the surface equation to find z0.
Next, we need to calculate the partial derivatives of the function:
∂z/∂x = -4x + 1
∂z/∂y = 6y
Now, we evaluate these derivatives at the point (x0, y0):
∂z/∂x | (x0, y0) = -4(x0) + 1
∂z/∂y | (x0, y0) = 6(y0)
The equation of the tangent plane can be expressed as:
z – z0 = ∂z/∂x | (x0, y0)(x – x0) + ∂z/∂y | (x0, y0)(y – y0)
Substituting the evaluated derivatives and the coordinates of the point into this equation will give you the desired tangent plane equation. Thus, the general form is:
z = z0 + (-4(x0) + 1)(x – x0) + 6(y0)(y – y0)
Make sure to replace (x0, y0, z0) with the specific values to obtain the final equation of the tangent plane.