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Magazine Chapter 4: Problem 213
Find the equation for the tangent plane to the surface at the indicated point, and graph the surface and the tangent plane: \(z=\ln \left(10 x^{2}+2 y^{2}+1\right), P(0,0,0).\)
Short Answer
Expert verified
The equation of the tangent plane at the point is \\(z = 0\\).
Step by step solution
01
Understand the Tangent Plane Equation
The equation for a tangent plane to a surface 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 the function with respect to \(x\) and \(y\), respectively.
02
Find Partial Derivatives
Given \(z = \ln(10x^2 + 2y^2 + 1)\), we will find the partial derivatives \(f_x\) and \(f_y\).- Compute \(f_x\): \ f_x = \frac{d}{dx} \ln(10x^2 + 2y^2 + 1) = \frac{20x}{10x^2 + 2y^2 + 1}\- Compute \(f_y\): \(f_y = \frac{d}{dy} \ln(10x^2 + 2y^2 + 1) = \frac{4y}{10x^2 + 2y^2 + 1}\)
03
Evaluate Partial Derivatives at the Point
Evaluate the partial derivatives at the point \(P(0,0,0)\):- \(f_x(0,0) = \frac{20 \cdot 0}{10 \cdot 0^2 + 2 \cdot 0^2 + 1} = 0\) - \(f_y(0,0) = \frac{4 \cdot 0}{10 \cdot 0^2 + 2 \cdot 0^2 + 1} = 0\)
04
Determine the Equation of the Tangent Plane
Substitute \(x_0 = 0\), \(y_0 = 0\), \(z_0 = 0\), \(f_x(0,0) = 0\), and \(f_y(0,0) = 0\) into the tangent plane equation:\[ z - 0 = 0 \cdot (x - 0) + 0 \cdot (y - 0) \]Simplifies to: \[ z = 0 \]
05
Sketch the Graphs
The surface is \(z = \ln(10x^2 + 2y^2 + 1)\) and the tangent plane is \(z = 0\).- The surface is a 3D plot where \(z\) increases as \(x\) and \(y\) move away from the origin.- The tangent plane, \(z = 0\), is the XY-plane across the origin (at \(P(0,0,0)\)).In the graph, the surface should touch the XY-plane exactly at the point \(P(0,0,0)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Partial Derivatives
Partial derivatives help us understand how a function changes as one of its variables changes, keeping others constant. For a function like \(z = \ln(10x^2 + 2y^2 + 1)\), we want to find how \(z\) varies with \(x\) and how it varies with \(y\). This is where partial derivatives, \(f_x\) and \(f_y\), come into play.
- The partial derivative with respect to \(x\), denoted \(f_x\), represents the rate of change of \(z\) as \(x\) changes.
- The partial derivative with respect to \(y\), denoted \(f_y\), represents the rate of change of \(z\) as \(y\) changes.
Surface Equations
The equation \(z = \ln(10x^2 + 2y^2 + 1)\) describes a surface in three-dimensional space. Such an equation tells us the height \(z\) of the surface above every point \((x, y)\) on the XY-plane.
- Logarithmic surfaces tend to rise slowly, and this one is impacted by both \(x\) and \(y\) squared terms, which stretch the surface symmetrically around the origin.
- The equation ensures that no matter the values of \(x\) or \(y\), the result inside the logarithm is non-negative, maintaining a real value for \(z\).
3D Graphing
3D graphing allows us to visualize complex surfaces and their behaviors in space. When graphing \(z = \ln(10x^2 + 2y^2 + 1)\) along with its tangent plane, we observe a 3D representation of the relationship between \(x\), \(y\), and \(z\).
- 3D plots show how surfaces bend and twist, helping to better understand their contours and intersections.
- The graph of the surface reveals that as \(x\) or \(y\) increases, the height \(z\) also increases, forming a smooth dome-like shape controlled by the squared terms in the equation.
Tangent Plane Equation
The tangent plane equation approximates a surface near a particular point. It gives us a flat surface that just touches the curve of the original surface at point \(P(x_0, y_0, z_0)\). For our exercise, it is defined by: \[ z - z_0 = f_x(x_0, y_0) (x - x_0) + f_y(x_0, y_0) (y - y_0) \]
- This equation involves evaluating the partial derivatives \(f_x\) and \(f_y\), which measure the slope of the surface in the \(x\) and \(y\) directions, respectively.
- At point \((0,0,0)\), using the derivatives found earlier, the equation simplifies greatly: \(z = 0\), resulting in the tangent plane coinciding with the XY-plane.
- Understanding tangent planes is crucial for analyzing surfaces in calculus, as they provide linear approximations that can be used for various applications like optimizations and solving differential equations.
