Problem 45 - Calculus for Biology and Medicine

Question

Let $$f_{a}(x, y)=a x^{2}+y^{2}$$ for \((x, y) \in \mathbf{R}\), where \(a\) is a positive constant. (a) Assume that \(a=1\) and describe the level curves of \(f_{1}\). The graph of \(f_{1}(x, y)\) intersects both the \(x-z\) and the \(y-z\) planes; show that these two curves of intersection are parabolas. (b) Assume that \(a=4\). Then $$f_{4}(x, y)=4 x^{2}+y^{2}$$ and the level curves satisfy $$4 x^{2}+y^{2}=c$$ Use a graphing calculator to sketch the level curves for \(c=\) \(0,1,2,3\), and \(4 .\) These curves are ellipses. Find the curves of intersection of \(f_{4}(x, y)\) with the \(x-z\) and the \(y-z\) planes. (c) Repeat (b) for \(a=1 / 4\). (d) Explain in words how the surfaces of \(f_{a}(x, y)\) change when \(a\) changes.

Step-by-Step Solution

Answer

The curves of intersection are parabolas for all \(a\). As \(a\) changes, the level curves transition from circles to ellipses.

Step 1: Analyzing Level Curves for a=1

First, substitute \( a = 1 \) into the function, so \( f_1(x, y) = x^2 + y^2 \). The level curves for this function are described by \( x^2 + y^2 = c \), which are circles centered at the origin with radius \( \sqrt{c} \). These are typical level curves of a paraboloid as they depict concentric circles.

Step 2: Intersection with Planes for a=1

Consider the intersection of the graph \( f_1(x, y) = x^2 + y^2 \) with the \( x-z \) plane (\( y = 0 \)). In this plane, \( z = x^2 \), which is a parabola. Similarly, consider the \( y-z \) plane (\( x = 0 \)), which gives \( z = y^2 \), another parabola. Thus, the intersections are parabolas in both planes.

Step 3: Analyzing Level Curves for a=4

Now, consider \( a = 4 \) with \( f_4(x, y) = 4x^2 + y^2 \). The level curves are given by \( 4x^2 + y^2 = c \). Dividing by \( c \) gives the equation \( \frac{x^2}{c/4} + \frac{y^2}{c} = 1 \), which is in the form of an ellipse centered at the origin with semi-axes \( \sqrt{c/4} \) and \( \sqrt{c} \).

Step 4: Intersection with Planes for a=4

The intersection with the \( x-z \) plane (\( y = 0 \)) results in \( z = 4x^2 \), a vertical parabola. The intersection with the \( y-z \) plane (\( x = 0 \)) results in \( z = y^2 \), again a vertical parabola.

Step 5: Analyzing Level Curves for a=1/4

Consider \( a = 1/4 \) with \( f_{1/4}(x, y) = \frac{1}{4}x^2 + y^2 \). The level curves satisfy \( \frac{1}{4}x^2 + y^2 = c \), which simplifies to \( \frac{x^2}{4c} + \frac{y^2}{c} = 1 \). These are ellipses centered at the origin with semi-axes \( \sqrt{4c} \) and \( \sqrt{c} \).

Step 6: Intersection with Planes for a=1/4

For \( y = 0 \), \( z = \frac{1}{4}x^2 \) describes a vertical parabola. For \( x = 0 \), \( z = y^2 \) is another vertical parabola.

Step 7: Effect of a on the Surface

Varying \( a \) in the function \( f_a(x, y) = ax^2 + y^2 \) alters the shape of the level curves from circles to ellipses. As \( a \) increases, the ellipses become more elongated along the y-axis. When \( a \) decreases, they are more elongated along the x-axis. At \( a = 1 \), the level curves are circles, indicating a symmetric paraboloid.

Key Concepts

Level Curves

Level curves are a way to visualize a function of two variables, like the function given in this problem, on a flat surface. When you have a function, say \( f(x, y) \), level curves represent the set of points for which \( f(x, y) = c \), where \( c \) is a constant. In essence, they show where the function has the same value. For the function \( f_{1}(x, y) = x^2 + y^2 \), the level curves are circles centered at the origin with a radius \( \sqrt{c} \). Here, \( c \) is varied to show different levels, creating concentric circles that represent the shape of a typical paraboloid. This concept helps in understanding the topography of the surface as you move along different constant values of the function. By sketching these curves, you get a clearer picture of how the function behaves over the plane.

Parabolas

A parabola is a symmetrical open plane curve, which can be described algebraically as a quadratic function. In the context of this exercise, we encounter parabolas as the curves of intersection when the graph of the function intersects different planes. When you consider \( f_1(x, y) = x^2 + y^2 \), setting \( y=0 \) for the \( x-z \) plane gives the equation \( z = x^2 \). This equation is a classic parabola opening upwards in the \( x-z \) plane. Similarly, setting \( x=0 \) for the \( y-z \) plane results in \( z = y^2 \), presenting another parabola. Understanding these intersections illuminates how the function behaves along specific axes, providing insight into the geometrical structure of the function's surface in three-dimensional space.

Ellipses

When dealing with the function \( f_{4}(x, y) = 4x^2 + y^2 \), the level curves transform into ellipses. The general equation for these ellipses is \( \frac{x^2}{c/4} + \frac{y^2}{c} = 1 \). These ellipses are centered at the origin and can be described by their semi-axes \( \sqrt{c/4} \) along the x-axis, and \( \sqrt{c} \) along the y-axis. Ellipses represent a stretched version of circles along one or both axes. When \( a \) increases, as in this case, the ellipses become stretched along the y-axis. This elongation reflects the changes in how the function's surface plots as you vary \( a \), demonstrating the diverse geometric formations the function can adopt.

Intersection with Planes

Intersection with planes is a method used to better understand the position and shape of a surface in three-dimensional space. By slicing through the surface with a plane, you can observe the curve of intersection. For \( f(x, y) \) functions, these slices usually yield other geometric figures such as parabolas or ellipses. When the surface \( f_4(x, y) = 4x^2 + y^2 \) intersects the \( x-z \) plane (\( y=0 \)), the resulting curve is \( z=4x^2 \), which is a parabola. Similarly, the intersection along the \( y-z \) plane (\( x=0 \)) produces \( z=y^2 \), another parabola. These intersections help to pinpoint the overall surface shape in three dimensions, offering insight into the orientation and structure of the parabolic features on different cross-sections of the surface.