Problem 399 - Calculus Volume 3

Question

Sketch the function in one graph and, in a second, sketch several level curves. \(f(x, y)=x+4 y^{2}\)

Step-by-Step Solution

Answer

Graphic of surface: parabolic cylinder; Level curves: parabolas in \(xy\)-plane.

Step 1: Identify the Components of the Function

Understand that the function given is \[f(x, y) = x + 4y^2\]This is a function of two variables, \(x\) and \(y\). It combines linear and quadratic elements, where \(x\) is linear, and \(4y^2\) is quadratic in nature.

Step 2: Graph the Function in 3D

Plot the surface \(z = f(x, y) = x + 4y^2\) in three dimensions. This surface is a parabolic cylinder opening along the \(x\) axis, oriented along the \(z\) axis, with its minimum along the \(x\) axis at \(x = 0\) when \(y = 0\). The cross-sections parallel to the \(yz\) plane are parabolas.

Step 3: Select Level Values

Choose several values for the level \(c\) where the function has a constant value: \(c = 0, 1, 2, -1, -2\). These values will determine the level curves of the function \(x + 4y^2 = c\).

Step 4: Derive Equations of Level Curves

For each level value \(c\), the equation \(x + 4y^2 = c\) becomes \(x = c - 4y^2\). These are equations of parabolas opening to the left if \(c > 0\) and to the right if \(c < 0\) in the \(xy\)-plane.

Step 5: Sketch Level Curves in the 2D Plane

For each parabola,- The vertex is at \((c, 0)\).- For \(c = 0\): the line \(x = -4y^2\).- For \(c = 1\): the parabola \(x = 1 - 4y^2\), opening to the left.- For \(c = -1\): the parabola \(x = -1 - 4y^2\), opening to the right.Sketch these curves on the same 2D plot to show several level curves.

Key Concepts

Level Curves

Level curves, also known as contour lines, are essential for understanding the shape of a 3D surface in a 2D plane. They represent the set of points where a function of two variables, in this case \( f(x, y) = x + 4y^2 \), equals the same constant \( c \). By choosing different values of \( c \), we can explore different slices of the surface.

• For the function given, the level curves are defined by the equation \( x + 4y^2 = c \).
• Each level curve corresponds to a specific value of \( c \).
• By rearranging, we have \( x = c - 4y^2 \), which describes a series of parabolas.

These parabolas vary depending on the value of \( c \):

• If \( c > 0 \), the parabola opens to the left.
• If \( c < 0 \), it opens to the right.

The vertex of each parabola is at the point \((c, 0)\). Plotting these curves provides a 2D blueprint of the 3D surface's elevation at various levels.

3D Plotting

Creating a 3D plot for a function like \( f(x, y) = x + 4y^2 \) helps visualize how the surface behaves in space. Such a plot is crucial for understanding the distribution of the function's values over the plane.

• Here, the surface represented is \( z = x + 4y^2 \).
• This highlights how changes in \( x \) and \( y \) impact the value of \( z \).

The 3D plot for this function reveals a parabolic cylinder structure, which makes it easier to see the cross-sections and how they evolve:

• The cylinder opens along the \( x \)-axis.
• The surface stretches in the direction of the \( z \)-axis.
• The minimum point, \( x = 0 \) when \( y = 0 \), is crucial for understanding the orientation.

Such visualizations enable a deeper grasp of abstract concepts by providing a tangible layout of how the function unfolds in three dimensions.

Parabolic Cylinder

The idea of a parabolic cylinder may appear complex, but with a bit of clarity, it becomes easier to understand. In essence, a parabolic cylinder is a 3D shape formed by translating a parabola along an axis.

• For the function \( f(x, y) = x + 4y^2 \), it generates a parabolic cylinder when graphed.
• The cross-sections along planes parallel to the \( yz \)-plane are parabolas.

Visualizing this can be simplified:

• Imagine a curved wall shaped like a parabola. This is essentially a parabolic cylinder.
• The curve extends infinitely in one direction (in this case, along the \( x \)-axis).

The orientation of the surface is dictated by the quadratic component, \( 4y^2 \), while the linear component, \( x \), determines its translation. Understanding such shapes is key to mastering concepts in multivariable calculus, as they frequently appear in complex terrains and architectural designs.