Problem 3 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Is the plane \(z=0\) a tangent plane to the graph of \(F(x, y)=\sqrt{x^{2}+y^{2}}\) shown in Figure 13.2D.

Step-by-Step Solution

Answer

No, the plane \(z=0\) is not a tangent plane to the graph at the origin.

Step 1: Determine the Point of Tangency

We need to identify if there is a point on the surface defined by \(F(x, y) = \sqrt{x^2 + y^2}\) where \(z = 0\) could be a tangent plane. The plane \(z = 0\) suggests examining points where \(F(x, y) = 0\). Since \(F(x, y) = \sqrt{x^2 + y^2}\), the condition \(F(x, y) = 0\) is met when \(x = 0\) and \(y = 0\), giving the point \((0, 0, 0)\).

Step 2: Compute the Gradient of F

The gradient vector of \(F(x, y) = \sqrt{x^2 + y^2}\) is the normal vector to the tangent plane. First, calculate the partial derivatives: \[ F_x = \frac{x}{\sqrt{x^2+y^2}}, \quad F_y = \frac{y}{\sqrt{x^2+y^2}} \]. Evaluating these at \((0, 0)\), we need to consider the limit, as direct substitution yields an indeterminate form (since denominator is zero).

Step 3: Analyze the Gradient at (0,0)

At the point \((0,0)\), both partial derivatives are undefined as they involve division by zero. To understand the tangent behavior at this point, consider approaching \((0,0)\) along different paths (e.g., along the x-axis or y-axis). Along any path, as you approach \(0,0\), the gradient components \(\frac{x}{\sqrt{x^2+y^2}}\) and \(\frac{y}{\sqrt{x^2+y^2}}\) tend to \(0\), suggesting the gradient direction is ambiguous and not well-defined.

Step 4: Reasoning with Ambiguity and Geometric Shape

Since there is no definitive gradient vector at \((0,0,0)\) to determine a unique tangent plane, consider the geometric shape. The surface \(z=\sqrt{x^2+y^2}\) forms a cone around the z-axis. At the origin, this surface forms a sharp point rather than a smooth surface. Hence, there is no tangent plane at the origin.

Key Concepts

Gradient Vector

The gradient vector is a vital concept when finding tangent planes, especially in multi-variable calculus. It's composed of partial derivatives, representing the direction of steepest ascent on a surface. This vector gives the normal line to the tangent plane at a given point.
To calculate the gradient vector \( abla F(x, y) \) for the function \( F(x, y) = \sqrt{x^2 + y^2} \), you find the partial derivatives with respect to \(x\) and \(y\). These are:

• \( F_x = \frac{x}{\sqrt{x^2+y^2}} \)
• \( F_y = \frac{y}{\sqrt{x^2+y^2}} \)

At the point \((0,0)\), the gradient is not well-defined because both partial derivatives result in an indeterminate form. This indeterminacy poses challenges in establishing the tangent plane at that point.

Partial Derivatives

Partial derivatives are the foundation for understanding changes in multi-variable functions. In the case of \( F(x, y) = \sqrt{x^2 + y^2} \), partial derivatives help determine the gradient vector needed for the tangent plane.
These derivatives are calculated as:

• The partial derivative with respect to \(x\), denoted \( F_x \), is \( \frac{x}{\sqrt{x^2+y^2}} \).
• The partial derivative with respect to \(y\), denoted \( F_y \), is \( \frac{y}{\sqrt{x^2+y^2}} \).

The derivatives indicate the rate of change of the function in the direction of each variable. However, at \((0,0)\), these expressions lead to an undefined form. This requires careful analysis when seeking to identify a tangent plane location.

3D Surfaces

In mathematics, 3D surfaces are visually represented as shapes extending in three dimensions. Consider our function \( F(x, y) = \sqrt{x^2 + y^2} \), forming a cone surface around the origin. Understanding this shape is essential when analyzing characteristics like tangent planes.
This surface implies:

• The \( z \)-value represents the distance from the point \((x, y)\) to the origin.
• Closer proximity to \((0, 0)\) forms sharp features at the origin.

The cone's sharp point at \((0,0,0)\) makes it challenging to determine a proper tangent plane, as a tangent plane assumes smoothness and continuity.

Point of Tangency

The point of tangency is where a plane just "touches" a surface without cutting through it. For the function \( F(x, y) = \sqrt{x^2 + y^2} \), identifying such a point is tricky.
Here's how it works:

• First, consider where \( z = 0 \), leading to the condition \( F(x, y) = 0 \).
• This occurs only when \( \sqrt{x^2 + y^2} = 0 \), at \((0, 0)\).

However, at this origin point, the surface forms a sharp cone, preventing an actual tangent plane from existing. Tangent planes typically require the surface to be flat and smooth locally. Thus, no such tangent exists at \((0, 0, 0)\) for this surface, illustrating the challenge of corner points on defined surfaces.