Final answer:
To find the equation of the plane tangent to a surface at a given point, we need to determine the slope of the surface at that point and use it in the equation of the plane tangent. The equation of the plane tangent is z - z0 = f(x0, y0)(x - x0) + f(y0, y0)(y - y0).
Step-by-step explanation:
To find the equation of the plane tangent to a surface at a given point, we need to determine the slope of the surface at that point. The slope of a surface is given by its derivative. Let's say the surface is represented by the equation z = f(x, y), and the point is (x0, y0, z0).
To find the equation of the plane tangent to the surface at (x0, y0, z0), we can use the equation: z - z0 = fx(x0, y0)(x - x0) + fy(x0, y0)(y - y0), where fx and fy are the partial derivatives of f with respect to x and y, respectively.
For example, if we have the surface equation z = x2 + y2 and the point (1, 2, 5), the partial derivatives are fx = 2x and fy = 2y. Substituting these values into the equation, we get: z - 5 = 2(1)(x - 1) + 2(2)(y - 2).