91Ó°ÊÓ

In calculus, a partial derivative is a way to take a derivative of a function with multiple variables while holding all but one variable constant. This is particularly important when dealing with surfaces that are a function of two variables, like \( z = f(x, y) \).
For the surface given by the equation \( z = e^{-(x^2 + y^2)} \), the partial derivatives help identify how \( z \) changes with respect to changes in \( x \) and \( y \).
This is crucial for establishing the tangent plane, as the tangent plane represents the linear approximation of the surface at a specific point.

• To calculate the partial derivative with respect to \( x \), treat \( y \) as a constant and differentiate \( z \) concerning \( x \).
• Similarly, to find the partial derivative with respect to \( y \), consider \( x \) as constant and differentiate accordingly.

Understanding partial derivatives provides essential insight into the behavior of the surface in various directions, which in turn informs the shape and position of the tangent plane.

The surface equation represents a function defining values for a surface in three-dimensional space. In our example, the surface is represented by \( z = e^{-(x^2 + y^2)} \). Such equations allow us to understand how variables interact in forming a surface.
The tangent plane is a flat surface that touches the surface equation at exactly one point, without intersecting it. It serves as an approximation of the surface at a specific point, often facilitating easier calculations.
In order to find the equation of the tangent plane, we first need to understand the form of the equation. For a surface given as \( z = f(x, y) \), the tangent plane equation at a point \((x_0, y_0, z_0)\) is expressed as:\[ z = z_0 + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \] where \( f_x \) and \( f_y \) are the partial derivatives previously calculated.

Evaluating derivatives at a specific point allows us to determine the rate of change of a surface precisely at that location. This forms a critical step in calculating the tangent plane.
In our task, we evaluated the partial derivatives of the surface \( z = e^{-(x^2 + y^2)} \) at the point \( (0, 0, 1) \). When we substituted these coordinates into our partial derivatives, additional terms containing \( x \) or \( y \) zero out, simplifying our equation further.
At this point, the evaluated partial derivatives \( f_x(0, 0) = 0 \) and \( f_y(0, 0) = 0 \) reflect no changes, indicating that any small movement from \( (0, 0) \) directions won't change the surface height at this point, given the surface shape around \( (0, 0, 1) \).
Upon substitution into the tangent plane equation, we derive \( z = 1 \), indicating a horizontal plane at \( z = 1 \).