Problem 2 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the equation of the tangent plane to the given surface at the indicated point. \(8 x^{2}+y^{2}+8 z^{2}=16 ;(1,2, \sqrt{2} / 2)\)

Step-by-Step Solution

Answer

The equation of the tangent plane is \(16x + 4y + 8\sqrt{2}z = 32\).

Step 1: Identify the Surface Function

The given surface is expressed by the equation \(8x^2 + y^2 + 8z^2 = 16\). Let's identify it as \(f(x, y, z) = 8x^2 + y^2 + 8z^2 - 16\).

Step 2: Calculate Partial Derivatives

To find the tangent plane, we'll calculate the gradient \(abla f\). Find the partial derivatives:\[\frac{\partial f}{\partial x} = 16x, \quad \frac{\partial f}{\partial y} = 2y, \quad \frac{\partial f}{\partial z} = 16z\]

Step 3: Evaluate Gradient at the Point

Now, evaluate each partial derivative at the point \((1, 2, \frac{\sqrt{2}}{2})\):\[\frac{\partial f}{\partial x} = 16(1) = 16, \quad \frac{\partial f}{\partial y} = 2(2) = 4, \quad \frac{\partial f}{\partial z} = 16\left(\frac{\sqrt{2}}{2}\right) = 8\sqrt{2}\]

Step 4: Write the Tangent Plane Equation

The equation of the tangent plane at \((x_0, y_0, z_0)\) is given by:\[\frac{\partial f}{\partial x}(x - x_0) + \frac{\partial f}{\partial y}(y - y_0) + \frac{\partial f}{\partial z}(z - z_0) = 0\]Substitute the evaluated derivatives and point \((1, 2, \frac{\sqrt{2}}{2})\):\[16(x - 1) + 4(y - 2) + 8\sqrt{2}\left(z - \frac{\sqrt{2}}{2}\right) = 0\]

Step 5: Simplify the Equation

Distribute and simplify the equation:\[16x - 16 + 4y - 8 + 8\sqrt{2}z - 8 = 0\]Combine and simplify further to get the final equation of the tangent plane:\[16x + 4y + 8\sqrt{2}z = 32\]

Key Concepts

Partial Derivatives

Partial derivatives play a significant role in multivariable calculus, especially when working with functions of multiple variables. A partial derivative represents how a function changes when only one of the variables is varied, while the others are held constant. This is similar to taking a regular derivative but tailored to functions with more than one variable.

For the given surface equation, we denote it as a function \( f(x, y, z) = 8x^2 + y^2 + 8z^2 - 16 \). Calculating the partial derivatives consists of taking the derivative with respect to each variable:\( x, y, \) and \( z \):

• \( \frac{\partial f}{\partial x} = 16x \)
• \( \frac{\partial f}{\partial y} = 2y \)
• \( \frac{\partial f}{\partial z} = 16z \)

These derivatives tell us how the function \( f \) changes as \( x, y, \) or \( z \) change. Partial derivatives form the basis for further operations like finding gradients and constructing tangent planes.

Gradient

In multivariable calculus, the gradient is a vector that contains all partial derivatives of a function. It is often denoted by \( abla f \) for a function \( f(x, y, z) \). The gradient points in the direction of the steepest increase of the function and is perpendicular to level surfaces.

For our function \( f(x, y, z) = 8x^2 + y^2 + 8z^2 - 16 \), the gradient \( abla f \) is:

• Gradient vector: \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) = (16x, 2y, 16z) \)

By evaluating the gradient at the given point \((1, 2, \frac{\sqrt{2}}{2})\), we obtain the normal to the tangent plane:

• \( abla f(1, 2, \frac{\sqrt{2}}{2}) = (16, 4, 8\sqrt{2}) \)

The gradient helps in finding equations of tangent planes, providing a vector which is orthogonal to the plane at the given point.

Multivariable Calculus

Multivariable calculus extends the concepts of single-variable calculus to functions with more than one variable. It involves understanding how changes in one variable affect the overall function, considering all variable interactions. This branch of calculus is vital when dealing with physical problems across different fields, like physics and engineering.

In the context of finding a tangent plane, multivariable calculus allows us to work with curves and surfaces in three dimensions. The process uses partial derivatives and gradients to explore how surfaces behave in space. By examining these changes, we can identify planes that "touch" the surface at specific points. This is crucial for understanding rates of change across multiple dimensions.

Surface Equation

Surface equations are mathematical representations of surfaces in three-dimensional space. They typically involve functions with two independent variables and one dependent variable. These equations allow us to define shapes and explore their properties geometrically and analytically.

The given surface equation \(8x^2 + y^2 + 8z^2 = 16\) is a type of ellipsoid. When expressed as \(f(x, y, z) = 8x^2 + y^2 + 8z^2 - 16 = 0\), it sets the stage for discovering tangent planes using calculus techniques. The surface itself is the set of points \((x, y, z)\) in space that satisfy the equation.

Understanding the shape of a surface helps in visualizing its geometry, guiding us in computations like finding tangent planes, which involve examining how the surface aligns or relates with linear planes at certain ways shown by the equation.