Z - z0 = fx(x - x0) + fy(y - y0)

Definition

This equation represents the formula for the tangent plane to a surface defined by a function $z = f(x, y)$ at a specific point $(x_0, y_0, z_0)$. It connects the concept of partial derivatives, $f_x$ and $f_y$, which indicate the slope of the surface in the x and y directions, respectively, to provide a linear approximation of the surface around that point.

5 Must Know Facts For Your Next Test

1. In the equation, $f_x$ and $f_y$ represent the partial derivatives of the function $f(x, y)$ evaluated at the point $(x_0, y_0)$, giving information about the slope in each direction.
2. The tangent plane approximation is valid in a small neighborhood around $(x_0, y_0)$ and is particularly useful for estimating values of $z$ without computing $f(x,y)$ directly.
3. The point $(x_0, y_0, z_0)$ corresponds to the original function evaluated at $(x_0, y_0)$, meaning $z_0 = f(x_0, y_0)$.
4. Understanding tangent planes helps in visualizing how surfaces behave and allows for better grasp of concepts like optimization and curvature.
5. The normal vector to the tangent plane can be found using the gradient of $f$, which is $(f_x, f_y, -1)$.

Review Questions

• How does the equation z - z0 = fx(x - x0) + fy(y - y0) relate to finding local approximations of a function?
  • The equation provides a way to find local approximations of a function by creating a tangent plane at a specific point on a surface. By using the slopes given by partial derivatives $f_x$ and $f_y$, it allows us to estimate values of $z$ for points close to $(x_0, y_0)$. This linear approximation simplifies calculations and gives insights into how the function behaves near that point.
• Discuss how partial derivatives are used in this equation to determine the behavior of the surface at point (x0, y0).
  • Partial derivatives $f_x$ and $f_y$ serve as measures of how the function changes as you vary x or y around the point $(x_0, y_0)$. They indicate the steepness of the surface in both directions and are crucial in constructing the tangent plane. This information allows for understanding how slight changes in input variables affect output values, providing insights into local behavior and trends on the surface.
• Evaluate the significance of normal vectors in relation to tangent planes defined by z - z0 = fx(x - x0) + fy(y - y0), especially in multi-variable calculus applications.
  • Normal vectors play an essential role in multi-variable calculus as they provide crucial information about surface orientation. For tangent planes defined by the equation $z - z_0 = f_x(x - x_0) + f_y(y - y_0)$, the normal vector can be determined from the gradient $(f_x, f_y, -1)$. This normal vector is important for applications like optimization problems and surface integrals since it helps define how surfaces interact with different planes and lines in space. Understanding this relationship enhances our ability to analyze and solve complex problems involving surfaces.

Related terms

Tangent Plane: A flat surface that just touches a curve or curved surface at a given point, representing the best linear approximation of the surface near that point.

Partial Derivative: A derivative taken with respect to one variable while keeping other variables constant, used to analyze how a function changes as its inputs change.

Normal Vector: A vector that is perpendicular to a surface at a given point, often used to describe the orientation of the surface in space.