Problem 14 - Calculus with Concepts in Calculus

Question

Sketch the level curve \(f(x, y)=c\). \(f(x, y)=6 x^{2} ; c=6,24\)

Step-by-Step Solution

Answer

Level curves \( x = \pm 1 \) for \( c = 6 \) and \( x = \pm 2 \) for \( c = 24 \).

Step 1: Identify the function

The given function is a product of the variable squared and its coefficient, specifically, the function is \( f(x, y) = 6x^2 \). This implies that the function depends solely on the variable \( x \), and does not vary with \( y \).

Step 2: Set up equations for level curves

We need to find the level curves for different constant values \( c \). For \( c = 6 \) and \( c = 24 \), we set up the equations \( 6x^2 = 6 \) and \( 6x^2 = 24 \) respectively.

Step 3: Solve the first level curve equation

For the level curve when \( c = 6 \), solve the equation \( 6x^2 = 6 \) to find \( x \).Divide both sides by 6:\[ x^2 = 1 \]Solve for \( x \):\[ x = \pm 1 \].This implies that the level curve for \( c = 6 \) is a pair of vertical lines \( x = 1 \) and \( x = -1 \).

Step 4: Solve the second level curve equation

For the level curve when \( c = 24 \), solve the equation \( 6x^2 = 24 \).Divide both sides by 6:\[ x^2 = 4 \]Solve for \( x \):\[ x = \pm 2 \].Thus, the level curve for \( c = 24 \) is a pair of vertical lines \( x = 2 \) and \( x = -2 \).

Step 5: Sketch the level curves on the coordinate plane

To sketch the level curves:1. Draw the y-axis as a vertical line and the x-axis as a horizontal line.2. Draw vertical lines at \( x = 1 \) and \( x = -1 \) for \( c = 6 \).3. Draw vertical lines at \( x = 2 \) and \( x = -2 \) for \( c = 24 \).4. Since the function \( f(x, y) = 6x^2 \) is independent of \( y \), the level curves are parallel lines unaffected by the value of \( y \).

Key Concepts

Graphing Level Curves

Level curves are a great way to visualize functions of two variables. They show sets of points where the function takes the same constant value. For the function \( f(x, y) = 6x^2 \), we need to graph level curves for specific values of \( c \). The process starts by setting the function equal to a constant \( c \). Once you solve this equation, the level curves emerge.

• For \( c = 6 \), the equation becomes \( 6x^2 = 6 \). Solving for \( x \), you find \( x = \pm 1 \).
• For \( c = 24 \), the equation is \( 6x^2 = 24 \). Solving gives \( x = \pm 2 \).

In this exercise, the level curves for \( c = 6 \) and \( c = 24 \) are pairs of vertical lines because the function is only dependent on \( x \). This means that the value of \( y \) does not affect these curves, leading to simple, easy-to-interpret vertical lines.

Single-Variable Functions

The function in our exercise, \( f(x, y) = 6x^2 \), is an example of how specific calculations often revolve around just one variable, even in two-variable situations. Since there is no \( y \) variable in this function, it is effectively a single-variable function disguised as a two-variable one.

• This type of problem often simplifies complexity, focusing only on one key variable.
• Here, you can directly see that any change in \( y \) does not influence the function's value.
• The function boils down to finding where \( 6x^2 \) equals a constant.

By realizing this, you simplify the problem into a one-dimensional challenge, making calculations and graphs straightforward and less prone to mistakes.

Coordinate Plane Sketching

Sketching level curves on a coordinate plane involves identifying where those curves will lie and how they are positioned relating to the axes. Start with drawing your basic coordinate plane: - The \( x \)-axis is horizontal- The \( y \)-axis is verticalFor our task:

• Mark where \( x = 1 \) and \( x = -1 \) with vertical lines for the level curve with \( c = 6 \).
• Do the same for \( x = 2 \) and \( x = -2 \) for \( c = 24 \).

Because the function lacks a \( y \) term, these curves are unaffected by changes in \( y \). They are purely vertical, showing that \( x \) values, for the given constants \( c \), are unchanging along any \( y \). This results in distinct, parallel lines at constant intervals of \( x \), showing a clear and consistent visualization of the given function's behavior.