Problem 42 - Calculus with Concepts in Calculus

Question

Find an equation of the plane tangent to the given surface at the given point. $$ y e^{x y}+z^{2}=0 ;(0,-1,1) $$

Step-by-Step Solution

Answer

The tangent plane equation is \(x + y + 2z - 1 = 0\).

Step 1: Understand the Surface Equation

The surface equation is given by \(f(x, y, z) = y e^{xy} + z^2 = 0\). We need to find the tangent plane to this surface at the point \((0, -1, 1)\). To do this, we'll use the gradient of the function \(f(x, y, z)\).

Step 2: Calculate Partial Derivatives

Calculate the partial derivative of \(f\) with respect to \(x\):\[\frac{\partial f}{\partial x} = y^2 e^{xy}\]Calculate the partial derivative of \(f\) with respect to \(y\):\[\frac{\partial f}{\partial y} = e^{xy} + xy e^{xy}\]Calculate the partial derivative of \(f\) with respect to \(z\):\[\frac{\partial f}{\partial z} = 2z\]

Step 3: Evaluate Partial Derivatives at the Given Point

Substitute \((0, -1, 1)\) into the partial derivatives:\[\frac{\partial f}{\partial x} (0, -1, 1) = (-1)^2 e^{0} = 1\]\[\frac{\partial f}{\partial y} (0, -1, 1) = e^{0} + 0 \cdot (-1)e^{0} = 1\]\[\frac{\partial f}{\partial z} (0, -1, 1) = 2(1) = 2\]

Step 4: Form the Equation of the Tangent Plane

The general equation of the tangent plane to a surface \(f(x, y, z) = 0\) at a point \((x_0, y_0, z_0)\) is given by:\[\frac{\partial f}{\partial x}(x_0, y_0, z_0)(x - x_0) + \frac{\partial f}{\partial y}(x_0, y_0, z_0)(y - y_0) + \frac{\partial f}{\partial z}(x_0, y_0, z_0)(z - z_0) = 0\]Substitute the point \((0, -1, 1)\) and the evaluated partial derivatives:\[1(x - 0) + 1(y + 1) + 2(z - 1) = 0\]Simplify:\[x + y + 1 + 2z - 2 = 0\]\[x + y + 2z - 1 = 0\]

Step 5: Conclusion: Tangent Plane Equation

The equation of the plane tangent to the surface at the given point is:\[x + y + 2z - 1 = 0\].

Key Concepts

Surface Equation

In mathematics, the term "surface equation" refers to the algebraic representation of a surface in a three-dimensional space. For our exercise, the surface equation given is \( f(x, y, z) = y e^{xy} + z^2 = 0 \). This equation describes a surface that includes every point \((x, y, z)\) where the expression equals zero.
This particular form involves an exponential function and a polynomial term, making it slightly complex. Understanding how to analyze surfaces defined by such equations is crucial for tasks like finding tangent planes. A tangent plane lightly "touches" the surface at just one point, ensuring that its slope matches the surface's slope at that location. Having a solid grasp of the surface equation is a first step toward further calculations involving derivatives and gradients.

Partial Derivatives

Partial derivatives are derivatives of functions with several variables, showing how a function changes as just one variable changes while keeping the others constant. For a function \( f(x, y, z) \), the partial derivative with respect to \( x \) is denoted as \( \frac{\partial f}{\partial x} \), while keeping \( y \) and \( z \) constant.
In our exercise, we compute the partial derivatives of the surface equation with respect to \( x \), \( y \), and \( z \). They are:

• \( \frac{\partial f}{\partial x} = y^2 e^{xy} \)
• \( \frac{\partial f}{\partial y} = e^{xy} + xy e^{xy} \)
• \( \frac{\partial f}{\partial z} = 2z \)

These derivatives give us the rates at which the surface varies as each of the coordinates \( x \), \( y \), and \( z \) changes. They are vital for finding the gradient, which leads to the tangent plane equation.

Gradient

The gradient is a vector that bundles all the partial derivatives of a function, revealing the direction of steepest ascent for that function. For a three-variable function \( f(x, y, z) \), the gradient is denoted as \( abla f \) and is given by:
\[ abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \]
In our problem, the gradient at any point on the surface tells us the direction and rate of the fastest increase of the surface. The importance of the gradient vector lies in its use in formulating the equation of the tangent plane. While the surface equation provides the overall form, the gradient supplies the precise directional components necessary to define the tangent plane accurately at a specific point.

Evaluate at Point

To determine the tangent plane at a specific point on the surface, we evaluate the previously calculated partial derivatives at that point. In our case, the point is \((0, -1, 1)\). The step involves substitution into the partial derivatives:

• \( \frac{\partial f}{\partial x} (0, -1, 1) = 1 \)
• \( \frac{\partial f}{\partial y} (0, -1, 1) = 1 \)
• \( \frac{\partial f}{\partial z} (0, -1, 1) = 2 \)

This substitution process transforms the general gradient into specific values reflecting how each variable alters the surface at that precise point. These evaluated derivatives are crucial to formulating the tangent plane equation. They help position and orient the plane exactly at the desired location on the surface, ensuring the plane is a perfect "touchdown" at the point.