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Magazine Chapter 9: Problem 4
Find an equation of the tangent plane to the given surface at the specified point. \(z=x e^{x y}, \quad(2,0,2)\)
Short Answer
Expert verified
The equation of the tangent plane is \( z = x + 4y \).
Step by step solution
01
Understand the problem
We need to find the equation of the tangent plane to the surface given by the equation \( z = xe^{xy} \) at the specific point \((2,0,2)\). The general formula for the equation of a tangent plane at point \((x_0, y_0, z_0)\) is \( z = z_0 + f_x(x_0, y_0)(x-x_0) + f_y(x_0, y_0)(y-y_0) \), where \( f_x \) and \( f_y \) are the partial derivatives of \( z \).
02
Calculate partial derivatives
Find the partial derivative of \( z \) with respect to \( x \) and \( y \). - \( f_x = \frac{\partial}{\partial x}(xe^{xy}) = e^{xy} + xye^{xy} \)- \( f_y = \frac{\partial}{\partial y}(xe^{xy}) = x^2e^{xy} \).
03
Evaluate partial derivatives at the point (2,0)
Substitute \( x = 2 \) and \( y = 0 \) into the partial derivatives.- \( f_x(2,0) = e^{2 \cdot 0} + 2 \cdot 0 \cdot e^{2 \cdot 0} = 1 \)- \( f_y(2,0) = 2^2e^{2 \cdot 0} = 4 \).
04
Write the equation of the tangent plane
Plug the values of \( f_x(2,0) = 1 \), \( f_y(2,0) = 4 \), and the point \((2,0,2)\) into the tangent plane equation: \[ z = 2 + 1(x-2) + 4(y-0) \]\[ z = 2 + x - 2 + 4y \]\[ z = x + 4y \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Partial Derivatives
Partial derivatives are a crucial concept in calculus, especially when dealing with functions of several variables. They represent the rate at which a function changes as one of its variables changes while keeping the other variables constant. In our exercise, we're dealing with the function \( z = x e^{xy} \), which has two variables: \( x \) and \( y \).
- Partial Derivative with respect to \( x \): This is denoted as \( f_x \) and shows how \( z \) changes as \( x \) changes, with \( y \) held constant. Using the power and chain rules of differentiation, we calculated \( f_x = e^{xy} + xye^{xy} \).
- Partial Derivative with respect to \( y \): This is denoted as \( f_y \), representing the rate of change of \( z \) as \( y \) varies, with \( x \) constant. We derived \( f_y = x^2e^{xy} \) using similar differentiation rules.
Calculus
Calculus is the mathematical study of continuous change. It's divided into two main branches: differential calculus and integral calculus. Our focus here is on differential calculus, particularly the application of derivatives to understand surface behavior.
- Derivatives: The derivative represents an instantaneous rate of change. In terms of surface analysis, derivatives help us understand how a function's output (like the height \( z \) of a surface above each point \( (x, y) \)) changes as we move around the plane.
- Chain Rule and Product Rule: These are powerful techniques in calculus for taking derivatives of more complex functions. The chain rule helps differentiate composite functions like \( e^{xy} \), while the product rule is useful for functions like \( xe^{xy} \).
Tangent Plane Equation
The tangent plane at a point on a surface provides a linear approximation to the surface near that point. The equation we use for the tangent plane is derived from the concept of differentials.For a surface described by \( z = f(x, y) \), the tangent plane at a point \((x_0, y_0, z_0)\) has the equation:\[ z = z_0 + f_x(x_0, y_0)(x-x_0) + f_y(x_0, y_0)(y-y_0) \]
- Determining \( f_x \) and \( f_y \): These are the partial derivatives we calculated. They tell us how the surface's height changes in the x and y directions, respectively.
- Point-specific Evaluation: By plugging in the coordinates of the point of interest, \( (2,0,2) \), into these derivatives, we determine how the function behaves at that exact point.
- Substituting and Simplifying: Once we have \( f_x \) and \( f_y \), and the original point, we substitute these values into the tangent plane equation to find the explicit form of the tangent plane, which in this exercise is \( z = x + 4y \).
Surface Analysis
Surface analysis involves examining the properties and behavior of a surface, typically represented by a function of two variables, in our case \( z = xe^{xy} \).
- Understanding the Surface: The given function represents a surface in a three-dimensional space. Analyzing this surface involves understanding how it behaves at different points, as influenced by changes in \( x \) and \( y \).
- Tangent Planes as Linear Approximations: The tangent plane gives a simplified, linear representation of the surface at a specific point, making it easier to analyze and visualize near that point.
- Application: Surface analysis is essential for applications ranging from physics to engineering, where understanding the behavior of fields or surfaces at a local level is important for predictive modeling and design.
