Problem 171 - Calculus 3

Question

For the following exercises, find the equation for the tangent plane to the surface at the indicated point. (Hint: Solve for \(z\) in terms of \(x\) and \(y . )\) $$z=-9 x^{2}-3 y^{2}, P(2,1,-39)$$

Step-by-Step Solution

Answer

The tangent plane equation at \( P(2,1,-39) \) is \( z = -36x - 6y + 39 \).

Step 1: Find the partial derivatives

To find the equation of the tangent plane, we need the partial derivatives of the function with respect to both variables. Given the function \( z = -9x^{2} - 3y^{2} \), calculate the partial derivatives. \( \frac{\partial z}{\partial x} = \frac{d}{dx}(-9x^{2} - 3y^{2}) = -18x \) and \( \frac{\partial z}{\partial y} = \frac{d}{dy}(-9x^{2} - 3y^{2}) = -6y \).

Step 2: Evaluate partial derivatives at the point

Evaluate the partial derivatives at the given point \( P(2,1,-39) \). For \( \frac{\partial z}{\partial x} \), substitute \( x = 2 \) and \( y = 1 \) to get \( -18(2) = -36 \). For \( \frac{\partial z}{\partial y} \), substitute \( x = 2 \) and \( y = 1 \) to get \( -6(1) = -6 \).

Step 3: Write the equation of the tangent plane

The equation of the tangent plane at a point \((x_0, y_0, z_0)\) is given by: \[ z - z_0 = \frac{\partial z}{\partial x}(x_0, y_0)(x - x_0) + \frac{\partial z}{\partial y}(x_0, y_0)(y - y_0) \]. Substitute \( (x_0, y_0, z_0) = (2,1,-39) \), \( \frac{\partial z}{\partial x} = -36 \), and \( \frac{\partial z}{\partial y} = -6 \) into the formula. This gives \( z + 39 = -36(x - 2) - 6(y - 1) \).

Step 4: Simplify the equation

Simplify the tangent plane equation. Start by distributing the partial derivatives: \( z + 39 = -36x + 72 - 6y + 6 \). Combine like terms to get \( z = -36x - 6y + 39 \). The simplified form of the tangent plane equation is \( z = -36x - 6y + 39 \).

Key Concepts

Partial Derivatives

When encountering functions with more than one variable, partial derivatives become essential. They help us understand how a function changes as each variable independently varies. In our exercise, the function is given as \[ z = -9x^{2} - 3y^{2} \]To find partial derivatives, we treat one variable as constant while differentiating with respect to another. This process gives us insights into the function's slope or rate of change along each axis:

• Partial derivative with respect to \(x\): Differentiate the expression treating \(y\) as a constant. The result is \(-18x\).
• Partial derivative with respect to \(y\): Here, \(x\) is treated as a constant. The derivative is \(-6y\).

These derivatives are crucial in finding adjustments at specific points on a surface, known as slope parameters. Evaluating these at a specific point such as \((2,1,-39)\) gives us

• For \(x = 2\) : \(-18(2) = -36\)
• For \(y = 1\) : \(-6(1) = -6\)

This shows how rapidly the surface changes with respect to each variable at that point.

Surface Equation

A surface equation represents a 3-dimensional object, often a curve or plane, in a structured mathematical form. For our example, the surface is shaped by \[ z = -9x^{2} - 3y^{2} \]This is a quadratic equation, which indicates that the surface is parabolic. The negative coefficients show that the parabola opens downwards. Understanding how to manipulate this equation allows us to interpret the curvature and behavior of surfaces:

• The term \(-9x^{2}\) suggests how steep the parabola is as \(x\) changes.
• The term \(-3y^{2}\) indicates a shallower curvature response as \(y\) changes.

When looking at any point on this surface, the equation defines where \(z\) is depending on the values of \(x\) and \(y\). This structure lays the groundwork for determining other features such as tangent planes.

Tangent Plane Formula

The tangent plane of a surface at a specific point serves as a plane that "touches" the surface at that point, offering the best linear approximation of the surface locally. The formula for the tangent plane is derived from the surface's function, ensuring it aligns with the surface's dimensions. The general formula is:\[ z - z_0 = \frac{\partial z}{\partial x}(x_0, y_0)(x - x_0) + \frac{\partial z}{\partial y}(x_0, y_0)(y - y_0)\]This equation uses:

• \(z_0\), \(x_0\), and \(y_0\) as coordinates of the point on the surface.
• Partial derivatives \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\), which we previously calculated.

By using these components, we obtain a plane that approximates the surface near the point \((2,1,-39)\) with the simplified equation:\[ z = -36x - 6y + 39 \]This form shows the linear relation and directly links the tangent plane to changes in 'x' and 'y', indicating the plane's slope based on the surface's behavior.