Problem 213 - Calculus Volume 3

Question

Find the equation for the tangent plane to the surface at the indicated point, and graph the surface and the tangent plane: \(z=\ln \left(10 x^{2}+2 y^{2}+1\right), P(0,0,0)\)

Step-by-Step Solution

Answer

The equation of the tangent plane is \( z = 0 \).

Step 1: Understand the Equation of the Tangent Plane

The general formula for the tangent plane to a surface defined by the function \( z = f(x, y) \) at a point \( P(x_0, y_0, z_0) \) is given by \( z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \), where \( f_x \) and \( f_y \) are the partial derivatives of \( f(x, y) \) with respect to \( x \) and \( y \), respectively.

Step 2: Calculate the Partial Derivatives

First, find the partial derivatives \( f_x \) and \( f_y \) of \( f(x, y) = \ln(10x^2 + 2y^2 + 1) \). The derivative \( f_x \) is \( \frac{20x}{10x^2 + 2y^2 + 1} \) and \( f_y \) is \( \frac{4y}{10x^2 + 2y^2 + 1} \).

Step 3: Evaluate the Partial Derivatives at the Point

Next, evaluate \( f_x(0, 0) \) and \( f_y(0, 0) \). Substituting \( x = 0 \) and \( y = 0 \) into the derivatives, you get \( f_x(0, 0) = \frac{20(0)}{1} = 0 \) and \( f_y(0, 0) = \frac{4(0)}{1} = 0 \).

Step 4: Plug Values into Tangent Plane Equation

Using the point \( P(0, 0, 0) \) and the values from the partial derivatives, substitute into the tangent plane equation: \( z - 0 = 0(x - 0) + 0(y - 0) \). This simplifies to \( z = 0 \).

Step 5: Describe the Tangent Plane and Surface

The equation \( z = 0 \) represents a horizontal plane in the \( xy \)-plane, implying that the tangent plane at point \( P(0, 0, 0) \) is a flat plane that just touches the surface \( z = \ln(10x^2 + 2y^2 + 1) \). To visualize, graph both the surface and tangent plane using a plotting tool to see the point of tangency at \( (0,0,0) \).

Key Concepts

Partial Derivatives

Partial derivatives are a fundamental concept in multivariable calculus. They are used to understand how a function changes as each variable is varied, while keeping the other variables constant.
For a function of two variables, say \( f(x, y) \), the partial derivative with respect to \( x \) is denoted by \( f_x \). It captures the rate of change of the function with changes in \( x \), keeping \( y \) constant.
Similarly, the partial derivative with respect to \( y \), denoted \( f_y \), represents the rate of change with respect to \( y \), keeping \( x \) constant.

• The formula for \( f_x \) is determined by treating \( y \) as a constant and differentiating \( f \) with respect to \( x \).
• For \( f_y \), treat \( x \) as a constant and differentiate with respect to \( y \).

In our problem, the function is \( z = \ln(10x^2 + 2y^2 + 1) \). Calculating the partial derivatives, we get:

• \( f_x = \frac{20x}{10x^2 + 2y^2 + 1} \)
• \( f_y = \frac{4y}{10x^2 + 2y^2 + 1} \)

These derivatives help us build the tangent plane at a specific point on the surface.

Surface Equation

The surface equation describes a three-dimensional shape in space. In this context, the equation is \( z = \ln(10x^2 + 2y^2 + 1) \).
This function determines a surface where each point \((x, y)\) has a corresponding \( z \)-value calculated by plugging \( x \) and \( y \) into the equation.

• The equation involves a logarithmic function, which impacts the shape of the surface, making it smooth and continuous.
• The presence of the terms \(10x^2\) and \(2y^2\) suggests that the surface will tend to rise steeply if either \( x \) or \( y \) is increased.

Understanding a surface equation allows us to predict the behavior of the function in a 3D space and is crucial for calculating derivatives and constructing tangent planes.

Tangent Plane Equation

The tangent plane equation is a tool for approximating the surface at a specific point. It provides a flat surface that just touches the original surface at one point, known as the tangent point.
For function \( z = f(x, y) \) at a point \( P(x_0, y_0, z_0) \), the equation is given by: \[ z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)\]

• \( f_x(x_0, y_0) \) and \( f_y(x_0, y_0) \) are the partial derivatives evaluated at \( P \).
• These partial derivatives act as the slopes, representing how tilted the tangent plane is in relation to the axes.

In our specific exercise, since the partial derivatives at \( P(0,0,0) \) are both zero, the equation reduces to \( z = 0 \).
This tells us the tangent plane is a horizontal plane, effectively forming a simple flat plane in the \( xy \)-plane at the point of tangency.

Graphing Surfaces

Graphing surfaces involves visualizing the three-dimensional shape described by a surface equation. It helps in understanding the geometric relationship between variables.
To graph the surface \( z = \ln(10x^2 + 2y^2 + 1) \), use software or graphing calculators capable of 3D rendering.

• The graph shows how \( z \) values vary as \( x \) and \( y \) change.
• Typically results in a smooth shape due to the logarithmic function and squared terms.

By plotting the surface alongside the tangent plane \( z = 0 \), you can visually inspect the point of tangency at \((0,0,0)\).
Graphing is essential for a hands-on understanding, revealing the interplay between different elements in multivariable calculus.