Problem 3
Question
The tangent plane at the indicated poini \(\left(x_{0}, y_{0}, z_{0}\right)\) exists. Find its equation. \(f(x, y)=x y ;(-1,2,-2)\)
Step-by-Step Solution
Verified Answer
The equation of the tangent plane is \(z = 2x - y + 2\).
1Step 1: Identify the Function and the Point
The function given is \(f(x, y) = xy\). The point at which we're finding the tangent plane is \((-1, 2, -2)\). Here, \(x_0 = -1\), \(y_0 = 2\), and \(z_0 = -2\).
2Step 2: Compute Partial Derivatives
Calculate the partial derivative of \(f\) with respect to \(x\): \(f_x(x, y) = y\). Calculate the partial derivative with respect to \(y\): \(f_y(x, y) = x\).
3Step 3: Evaluate Partial Derivatives at the Point
Substitute \(x_0 = -1\) and \(y_0 = 2\) to find \(f_x(-1, 2) = 2\) and \(f_y(-1, 2) = -1\).
4Step 4: Write the Equation of the Tangent Plane
The formula for the tangent plane is given by \(z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)\). Substitute \(z_0 = -2\), \(f_x(-1, 2) = 2\), and \(f_y(-1, 2) = -1\).
5Step 5: Simplify the Tangent Plane Equation
Substitute the values into the tangent plane equation: \(z + 2 = 2(x + 1) - 1(y - 2)\). Simplify this to obtain \(z = 2x - y + 2\).
Key Concepts
Partial DerivativesFunction EvaluationTangent Plane Equation
Partial Derivatives
In mathematics, partial derivatives are used to measure how a function changes as its variables change. When a function has more than one variable, like our function here, \( f(x, y) = xy \), we can still find out how the function behaves as each variable changes on its own. This is where partial derivatives shine!
For instance, the partial derivative with respect to \( x \), denoted by \( f_x(x, y) \), tells us how the function changes as \( x \) changes, keeping \( y \) constant. For the function \( f(x, y) = xy \), taking the partial derivative with respect to \( x \) gives us \( f_x(x, y) = y \). This means that for any fixed \( y \), the function's rate of change with respect to \( x \) is equal to \( y \).
Similarly, the partial derivative with respect to \( y \), denoted by \( f_y(x, y) \), tells us how the function changes as \( y \) changes, keeping \( x \) constant. For our function, \( f_y(x, y) = x \), meaning the rate of change with respect to \( y \) is \( x \).
Partial derivatives are vital in multivariable calculus and help us understand the complete behavior of functions of multiple variables.
For instance, the partial derivative with respect to \( x \), denoted by \( f_x(x, y) \), tells us how the function changes as \( x \) changes, keeping \( y \) constant. For the function \( f(x, y) = xy \), taking the partial derivative with respect to \( x \) gives us \( f_x(x, y) = y \). This means that for any fixed \( y \), the function's rate of change with respect to \( x \) is equal to \( y \).
Similarly, the partial derivative with respect to \( y \), denoted by \( f_y(x, y) \), tells us how the function changes as \( y \) changes, keeping \( x \) constant. For our function, \( f_y(x, y) = x \), meaning the rate of change with respect to \( y \) is \( x \).
Partial derivatives are vital in multivariable calculus and help us understand the complete behavior of functions of multiple variables.
Function Evaluation
Once we have the partial derivatives, the next step is to evaluate them at specific points. In our example, we're focusing on the point \((-1, 2, -2)\). Evaluating the partial derivatives at the given point helps us understand precisely how the function \( f(x, y) = xy \) behaves around that point.
In practical terms, we substitute \( x_0 = -1 \) and \( y_0 = 2 \) into the partial derivatives to obtain their values at this point:
This step is crucial because it lays the groundwork for calculating the tangent plane equation.
In practical terms, we substitute \( x_0 = -1 \) and \( y_0 = 2 \) into the partial derivatives to obtain their values at this point:
- For the partial derivative with respect to \( x \), \( f_x(-1, 2) = 2 \).
- For the partial derivative with respect to \( y \), \( f_y(-1, 2) = -1 \).
This step is crucial because it lays the groundwork for calculating the tangent plane equation.
Tangent Plane Equation
The tangent plane is a powerful concept in calculus that provides the best linear approximation of a surface at a given point. The equation of a tangent plane can be derived using the point of interest and the evaluated partial derivatives.
To find the tangent plane to a function \( f(x, y) \) at the point \((x_0, y_0, z_0)\), we use the formula: \[ z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \] Here, \( f_x \) and \( f_y \) are the partial derivatives evaluated at the point \((x_0, y_0)\).
For our specific case:
The end product gives us the plane that just "touches" the surface at our point of interest, providing great insight into the surface's behavior around that point.
To find the tangent plane to a function \( f(x, y) \) at the point \((x_0, y_0, z_0)\), we use the formula: \[ z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \] Here, \( f_x \) and \( f_y \) are the partial derivatives evaluated at the point \((x_0, y_0)\).
For our specific case:
- We substitute \( z_0 = -2 \),
- \( f_x(-1, 2) = 2 \),
- \( f_y(-1, 2) = -1 \).
The end product gives us the plane that just "touches" the surface at our point of interest, providing great insight into the surface's behavior around that point.
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