Problem 180 - Calculus Volume 3

Question

Find the equation for the tangent plane to the surface at the indicated point. \(h(x, y)=\ln \sqrt{x^{2}+y^{2}}, P(3,4)\)

Step-by-Step Solution

Answer

Tangent plane: \( z = \ln 5 + \frac{3}{25}(x - 3) + \frac{4}{25}(y - 4) \).

Step 1: Understand the Problem

We need to find the equation of the tangent plane to the surface described by the function \( h(x, y) = \ln \sqrt{x^2 + y^2} \) at the point \( P(3, 4) \). A tangent plane at a point involves finding the partial derivatives of the function with respect to \( x \) and \( y \) and calculating them at the given point.

Step 2: Calculate Partial Derivatives

First, we find the partial derivative of \( h(x, y) \) with respect to \( x \):\( h_x(x, y) = \frac{\partial}{\partial x} \left(\ln \sqrt{x^2 + y^2}\right) = \frac{x}{x^2 + y^2} \).Next, we find the partial derivative with respect to \( y \):\( h_y(x, y) = \frac{\partial}{\partial y} \left(\ln \sqrt{x^2 + y^2}\right) = \frac{y}{x^2 + y^2} \).

Step 3: Evaluate Partial Derivatives at the Point

We evaluate \( h_x(3, 4) \) and \( h_y(3, 4) \):\( h_x(3, 4) = \frac{3}{3^2 + 4^2} = \frac{3}{25} \).\( h_y(3, 4) = \frac{4}{3^2 + 4^2} = \frac{4}{25} \).

Step 4: Find the Tangent Plane Equation

The equation of the tangent plane at a point \( (x_0, y_0, z_0) \) on a surface \( z = f(x, y) \) is given by:\( z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \).We have \( x_0 = 3 \), \( y_0 = 4 \), and we find \( z_0 = h(3, 4) = \ln \sqrt{3^2 + 4^2} = \ln \sqrt{25} = \ln 5 \).Substituting the values:\( z - \ln 5 = \frac{3}{25}(x - 3) + \frac{4}{25}(y - 4) \).

Step 5: Write the Final Equation

Rearrange the equation obtained in Step 4:\( z = \ln 5 + \frac{3}{25}(x - 3) + \frac{4}{25}(y - 4) \).This is the equation of the tangent plane.

Key Concepts

Partial Derivatives

Partial derivatives are essential in understanding how a function changes with respect to its variables. They are derivatives concerning each variable while keeping the other variables constant. In our problem, we are dealing with the function \[ h(x, y) = \ln \sqrt{x^2 + y^2} \] and need to find its partial derivatives with respect to both \( x \) and \( y \).This process involves taking the derivative of the function as if the other variable doesn't change.

• The partial derivative with respect to \( x \), denoted \( h_x(x, y) \), gives us information about how the function changes as \( x \) changes.
• Similarly, the partial derivative with respect to \( y \), denoted \( h_y(x, y) \), explains how the function varies with \( y \).

Using these derivatives, we can analyze the slope and behavior of the function in any direction in the \( xy \)-plane. They form the building blocks for understanding tangent planes on surfaces.

Surface

When solving problems related to surfaces, both visualizing and understanding the geometric implications is crucial.In mathematics, a surface like \( h(x, y) = \ln \sqrt{x^2 + y^2} \) represents a 3D surface over a plane. Think of it as a sheet that stretches in space, depending on \( x \) and \( y \) values. This invoking function tells us the height at any given point \( (x, y) \).Surfaces can be complex, but it boils down to how the \( z \)-values vary with \( x \) and \( y \).This specific surface logs the distance from any point \( (x, y) \) to the origin in terms of \( z \)-value. Identifying points like \( P(3,4) \) on this surface is essential for tasks like finding tangent planes.

Tangent Line

The concept of a tangent line extends from a curve to that of a tangent plane on a surface. A tangent lineat a point on a curve barely touches the curve at that point, providing an infinitesimally close approximation of the curve’s slope there. When we talk about surfaces, the tangent plane does a similar job for a multidimensional surface. It represents all the tangent lines at a point, capturing the essence of how the surface behaves around that point, \( P(3, 4) \).The tangent plane uses partial derivatives to find slopes along all directions in the \( xy \)-plane, permitting an accurate approximation of the surface near \( P \).This practical feature is useful in applications ranging from physics to data science, enhancing our ability to model and solve real-world problems.

Equations

Equations are at the heart of finding the tangent plane to a surface. The equation of a tangent plane helps encapsulate complex operations into an understandable relation betweenvariables.To derive the equation for the tangent plane:

• Find the partial derivatives, \( h_x \) and \( h_y \), which offer directional slopes.
• Identify the surface value \( z_0 \) at the point, \( P(3,4) \),by calculating \( h(3, 4) \).
• Apply the tangent plane equation formula:\[ z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \]

This provides a linear approximation of the surface in the vicinity of \( P \), effectively representing the surface's slope as a plane. Once assembled, this equation clarifies how the surface changes, making it essential for modeling and analytical purposes.