Question

Find an equation for the tangent plane to the graph of f(x, y) = p 8 - 3x 2 - y 2 at the point p(1, 2, f(1, 2))

Answer 1

The equation for the tangent plane to the graph of f(x, y) = 8 - 3x^2 - y^2 at the point P(1, 2, f(1,2)) is z = 7 - 6x - 4y.

Step-by-step explanation:

To find the equation for the tangent plane to the graph of f(x, y) = 8 - 3x^2 - y^2 at the point P(1, 2, f(1,2)), we need to find the partial derivatives of f(x, y) with respect to x and y. The equation of a tangent plane is given by:

z = f(a, b) + (x - a)fx(a, b) + (y - b)fy(a, b)

Substituting the values a = 1, b = 2, f(a, b) = 8 - 3(1)^2 - (2)^2 = 1, fx(a, b) = -6, and fy(a, b) = -4 into the equation, we get:

z = 1 - 6(x - 1) - 4(y - 2)

So, the equation for the tangent plane is z = 7 - 6x - 4y.