Problem 8 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Let \(F\) be defined by $$\begin{aligned}F(x, y) &=x^{2} \quad \text { for } \quad y>0 \\ &=0 \quad \text { for } \quad y \leq 0\end{aligned}$$ 1\. Sketch a graph of \(F\) in three dimensional space. 2\. Is \(F_{1}(x, y)\) continuous on the interior of a circle with center (0,0)\(?\) 3\. Let \(L(x, y)=0\) for all \((x, y)\). Is it true that $$\lim _{(x, y) \rightarrow(0,0)} \frac{F(x, y)-L(x, y)}{\sqrt{x^{2}+y^{2}}}=0 \quad ?$$ 4\. Are you willing to call the plane \(z=0\) a tangent plane to the graph of \(F ?\)

Step-by-Step Solution

Answer

1. Sketch shows a paraboloid/z-plane. 2. F is discontinuous. 3. Limit is 0. 4. Yes, z = 0 is a tangent plane.

Step 1: Understand the function F

The function \( F(x, y) \) is defined in piecewise form, where \( F(x, y) = x^2 \) when \( y > 0 \) and \( F(x, y) = 0 \) for \( y \leq 0 \). This indicates that we have a paraboloid shape for positive \( y \) values and a flat (zero height) plane for non-positive \( y \) values.

Step 2: Sketch the 3D Graph of F

To sketch the graph, consider two scenarios:- For \( y > 0 \), the surface is a paraboloid that opens upwards with a cross section of \( z = x^2 \).- For \( y \leq 0 \), the surface lies on the \( xy \)-plane (\( z = 0 \)).Draw the transition where the upper part of the paraboloid suddenly flattens out to the plane for \( y \leq 0 \).

Step 3: Analyzing Continuity of F on a Circle

The interior of the circle centered at (0,0) includes points where \( y > 0 \) and \( y \leq 0 \). On the line \( y = 0 \), \( F(x,y) = 0 \), while for \( y > 0 \), \( F(x,y) = x^2 \). This discontinuity at the line \( y = 0 \) implies that \( F \) is not continuous over the entire interior of the circle.

Step 4: Evaluating the Limit for the Given Function L

Given \( L(x, y) = 0 \), compute the limit:\[ \lim_{(x, y) \to (0,0)} \frac{F(x, y) - L(x, y)}{\sqrt{x^2 + y^2}} = \lim_{(x, y) \to (0,0)} \frac{F(x, y)}{\sqrt{x^2 + y^2}}. \]For \( y > 0 \), \( F(x, y) = x^2 \), which simplifies to:\[ \lim_{(x, y) \to (0,0)} \frac{x^2}{\sqrt{x^2 + y^2}} = 0, \]because \( x^2 \) approaches 0 faster than \( \sqrt{x^2 + y^2} \) as \( (x, y) \to (0,0) \).For \( y \leq 0 \), \( F(x, y) = 0 \) so the limit is trivially 0.

Step 5: Evaluating the Tangency of the Plane z=0

To determine if the plane \( z = 0 \) is a tangent plane at the origin, consider:1. The definition of a tangent plane involves the plane \( z = L(x, y) \) approximating \( F(x, y) \) near the origin better than any other linear function when \( (x, y) \) approaches (0,0).2. Since the limit from Step 4 is 0, \( z = 0 \) is indeed a tangent plane for \( F(x, y) \) at (0,0) because \( F(x, y) \) collapses to the plane for small changes around the origin.

Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions based on the values of its input. In this case, the function \( F(x, y) \) is defined differently depending on the value of \( y \). When \( y > 0 \), \( F(x, y) = x^2 \). This means that for positive \( y \) values, the graph of the function creates a paraboloid shape because it behaves like a parabolic curve in the \( z \)-direction, with the vertex at the origin.
For \( y \leq 0 \), \( F(x, y) = 0 \), resulting in a flat surface on the \( xy \)-plane. This split creates two different behaviors that are clearly distinguished in three-dimensional space, showcasing how piecewise functions can model complex surfaces by switching between expressions based on conditions.

3D Graphing

Three-dimensional graphing allows us to visualize functions of two variables. To sketch the function \( F(x, y) \), you'll need to plot it considering its piecewise nature. For \( y > 0 \), draw a surface resembling a paraboloid where each cross-section parallel to the \( yz \)-plane, for a fixed \( x \), follows a parabolic shape created by \( z = x^2 \).
For \( y \leq 0 \), the graph lies flat along the \( xy \)-plane, with the surface at \( z = 0 \). The challenge in 3D graphing is capturing the transition between these two sections, where the curved surface for positive \( y \) values transitions to a flat surface for \( y \leq 0 \). Understanding these transitions is crucial in graphing complex piecewise functions.

Continuity

Continuity in multivariable functions requires the function to be continuous at every point in its domain. For the function \( F(x, y) \), the main concern is the line where \( y = 0 \). Above this line, for \( y > 0 \), the function is \( x^2 \) and behaves continuously. However, directly along \( y = 0 \), \( F(x, y) = 0 \), creating a jump discontinuity between the paraboloid surface and the plane.
This discontinuity along the line \( y = 0 \) means the function is not continuous over any region that straddles this line, such as the interior of any circle centered at the origin that includes parts where \( y > 0 \) and \( y \leq 0 \). Understanding where and why a function is not continuous helps in applying multivariable calculus concepts practically.

Limits

Limits are essential in analyzing the behavior of functions near a point. Here, the question involves evaluating the limit: \[ \lim_{(x, y) \to (0,0)} \frac{F(x, y) - L(x, y)}{\sqrt{x^2 + y^2}} \] With \( L(x, y) = 0 \), this becomes: \[ \lim_{(x, y) \to (0,0)} \frac{F(x, y)}{\sqrt{x^2 + y^2}} \] For \( y > 0 \), where \( F(x, y) = x^2 \), this limit approaches zero as \( x^2 \) becomes insignificant compared to \( \sqrt{x^2 + y^2} \) near the origin. Similarly, for \( y \leq 0 \), \( F(x, y) = 0 \), making the entire fraction zero. The limit is zero from all directions in the plane, signifying that around this point, changes in \( F(x, y) \) are smoother than they might appear when examining the discontinuity directly. This analysis solidifies the understanding of continuity and neighboring behavior of functions.

Tangent Planes

The concept of a tangent plane extends the idea of a tangent line to three dimensions. The tangent plane to a surface at a given point is the plane that best approximates the surface near that point. For \( F(x, y) = x^2 \) when \( y > 0 \), the tangent plane approximation at the origin would be \( z = 0 \) because this plane closely approximates both the flat surface and the paraboloid near \((0,0)\). Determining a tangent plane involves ensuring that the approximation does not just need to visually touch the function but should mathematically represent the direction and slope as closely as possible around the point of tangency. Since the limit condition is satisfied, the plane \( z = 0 \) serves as the tangent plane for \( F(x, y) \) at the origin, simplifying the behavior of the surface at this point well. Understanding tangent planes is key in linear approximations and estimating surface behavior.