Problem 40 - Calculus with Concepts in Calculus

Question

Find an equation of the plane tangent to the given surface at the given point. $$ \frac{x^{2}}{4}+\frac{y^{2}}{9}+\frac{z^{2}}{16}=1 ;(0,0,-4) $$

Step-by-Step Solution

Answer

The tangent plane equation is \( z = -4 \).

Step 1: Identify the surface

The given surface is the ellipsoid defined by the equation \( \frac{x^{2}}{4} + \frac{y^{2}}{9} + \frac{z^{2}}{16} = 1 \).

Step 2: Verify the point lies on the surface

Substitute the point \((0, 0, -4)\) into the surface equation. Calculate: \( \frac{0^2}{4} + \frac{0^2}{9} + \frac{(-4)^2}{16} = 0 + 0 + \frac{16}{16} = 1 \). Since the total equals 1, the point lies on the surface.

Step 3: Find the gradient of the surface equation

The gradient \( abla f(x, y, z) \) of the function \( f(x, y, z) = \frac{x^{2}}{4} + \frac{y^{2}}{9} + \frac{z^{2}}{16} - 1 \) is given by partial derivatives: \( \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \). Compute each partial derivative: \( \frac{\partial f}{\partial x} = \frac{x}{2} \), \( \frac{\partial f}{\partial y} = \frac{2y}{9} \), \( \frac{\partial f}{\partial z} = \frac{z}{8} \). Thus, \( abla f(x, y, z) = \left( \frac{x}{2}, \frac{2y}{9}, \frac{z}{8} \right) \).

Step 4: Evaluate the gradient at the given point

Substitute \((0, 0, -4)\) into the gradient \( abla f(x, y, z) \): \( abla f(0, 0, -4) = \left( \frac{0}{2}, \frac{2 \times 0}{9}, \frac{-4}{8} \right) = (0, 0, -\frac{1}{2}) \).

Step 5: Write the equation of the tangent plane

The equation of the tangent plane at a point \((x_0, y_0, z_0)\) on the surface with gradient \( (a, b, c) \) is \( a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 \). Use \((0, 0, -4)\) and the gradient \((0, 0, -\frac{1}{2})\): \( 0(x - 0) + 0(y - 0) + -\frac{1}{2}(z + 4) = 0 \). Simpify to \(-\frac{1}{2}z - 2 = 0\), which gives \( z = -4 \).

Step 6: Simplify the tangent plane equation

Rearrange the simplified equation \( z = -4 \) to emphasize its form and confirm that the plane is indeed the horizontal plane at \( z = -4 \).

Key Concepts

Tangent Plane

When discussing surfaces in calculus, the concept of a **tangent plane** is key. Imagine a flat surface that just "touches" a curve or a surface at a single point. This flat surface is what we call the tangent plane. It is used to approximate surfaces near that point, much like how a tangent line approximates a curve.
To find the equation of a tangent plane to a surface defined by some function at a given point:

• Identify the normal vector to the surface at that point. This vector is perpendicular to the tangent plane.
• Use this normal vector, along with the coordinates of the given point, to find the equation of the plane.

The equation of a tangent plane greatly relies on concepts like gradients and partial derivatives, which we'll explore further.

Partial Derivatives

Let's dive into **partial derivatives**. They play a significant role in finding tangent planes. Consider a function of several variables, such as \[f(x, y, z)\]. Partial derivatives involve differentiating with respect to one of these variables while keeping the others constant. For example, the partial derivative of \[f(x, y, z)\] with respect to \(x\) is noted as \( \frac{\partial f}{\partial x} \).
Partial derivatives are calculated similarly to normal derivatives, but each time we focus on a single variable:

• To compute \(\frac{\partial f}{\partial x}\), treat \(y\) and \(z\) as constants.
• Do similarly for other variables.

Partial derivatives form the components of the gradient vector, which is essential for determining tangent planes.

Gradient of a Function

The **gradient of a function** is a vector that carries significant information about the function's behavior. It combines all partial derivatives of a function \[f(x, y, z)\].
The gradient vector is expressed as:\[abla f(x, y, z) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)\]This vector points in the direction of the greatest rate of increase of the function.

• It is perpendicular to level surfaces of the function, which is why it serves as a normal vector to tangent planes.
• The tangent plane at any point on the surface can be described using the gradient, making it a fundamental component in such calculations.

By evaluating the gradient at a given point, you obtain the necessary normal vector for the tangent plane.

Ellipsoid

An **ellipsoid** is a 3D shape that resembles a stretched or squashed sphere. The general equation for an ellipsoid is:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\]This equation displays symmetry and shows how each axis (\(x, y, z\)) scales with constants \(a, b, \text{and } c\), determining the ellipsoid's dimensions. Each term in the equation relates to the radius along a specific axis.
This geometry alters how surfaces and their tangent planes behave compared to other shapes like planes and spheres. Understanding the specific stretch or squash along each axis helps solve exercises about tangent planes more accurately.

Surface Equation

The **surface equation** describes a 3D object's shape and form. In this case, we examined the equation \[\frac{x^{2}}{4} + \frac{y^{2}}{9} + \frac{z^{2}}{16} = 1\], which defines an ellipsoid. Surface equations are like blueprints for geometric figures, providing detail on how the shape extends through each axis.
Utilizing a surface equation, one can:

• Locate points on the surface by substituting values of \(x, y, ext{and } z\) and checking solutions.
• Verify if a particular point lies on the surface.

This is essential when finding tangent planes, as the point where the tangent plane touches the surface must satisfy the surface equation.