Final answer:
The equation of the tangent plane to the surface z = 7x^5 + 3y^5 + 6xy at the point (2, -1, 209) can be found using partial derivatives. The final equation of the tangent plane is obtained by substituting the values of the partial derivatives at the given point.
Step-by-step explanation:
The equation of the tangent plane to the surface z = 7x^5 + 3y^5 + 6xy at the point (2, -1, 209) can be found using partial derivatives. Let's start by finding the partial derivative with respect to x (denoted as ∂z/∂x):
∂z/∂x = 35x^4 + 6y
Next, let's find the partial derivative with respect to y (denoted as ∂z/∂y):
∂z/∂y = 15y^4 + 6x
Now, substitute the point (2, -1, 209) into these partial derivatives:
∂z/∂x = 35(2)^4 + 6(-1) = 2726
∂z/∂y = 15(-1)^4 + 6(2) = 12
Using these values, we can write the equation of the tangent plane as:
209 = 2726(x - 2) + 12(y + 1)
Simplifying this equation will give you the final answer for z.