The formula to calculate the equation of the tangent plane to a surface at a given point is:
\[ z = f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \]
Where:
The equation of the tangent plane is a mathematical formula that represents a plane which just touches a surface at a given point, without cutting into it. This plane is parallel to the surface at the point of tangency. In calculus, it is used to approximate the surface near that point, and its equation can be derived from the gradient of the function that defines the surface.
Let's assume the following values:
Step 1: Calculate the z-coordinate using the formula:
\[ z = 4 + 2(2 - 1) + 3(3 - 2) \]
Step 2: Simplify the expression:
\[ z = 4 + 2 \cdot 1 + 3 \cdot 1 = 4 + 2 + 3 = 9 \]
The z-coordinate of the point on the tangent plane is 9.