Final answer:
To find the equation of the tangent plane to the given surface at the specified point, we can use the formula z - z₁ = fₓ(x - x₁) + fᵧ(y - y₁), where (x₁, y₁, z₁) is the specified point and fₓ and fᵧ are the partial derivatives of the surface equation with respect to x and y. Substitute the values into the formula and simplify to get the equation of the tangent plane as 10x - 40y + z = -39.
Step-by-step explanation:
To find the equation of the tangent plane to the surface, we can use the formula:
z - z₁ = fₓ(x - x₁) + fᵧ(y - y₁)
where (x₁, y₁, z₁) is the specified point and fₓ and fᵧ are the partial derivatives of the surface equation with respect to x and y.
For the given surface, z = 5(x - 1)² + 5(y - 3)² - 4, and the specified point is (2, -1, 29). We can start by finding fₓ and fᵧ:
fₓ = 10(x - 1)
fᵧ = 10(y - 3)
Now substitute the values into the formula to get the equation of the tangent plane:
z - 29 = 10(2 - 1)(x - 2) + 10(-1 - 3)(y - (-1))
Simplifying further:
z - 29 = 10(x - 2) - 40(y + 1)
Finally, rearrange the equation to get the standard form:
10x - 40y + z = -39