Problem 7 - Advanced Calculus

Question

Sketch the level curves for \(f\) when (a) \(f(x, y)=y^{2}-x\) (b) \(f(p)=|p|-1\) (c) \(f(p)=\left\\{\begin{array}{ll}1 & \text { when }|p|<1 \\ x-y & \text { when }|p| \geq 1\end{array}\right.\)

Step-by-Step Solution

Answer

1. For \(f(x, y)=y^{2}-x\), the level curves are families of parabolas. 2. For \(f(p)=|p|-1\), the level curves are lines parallel to the axes. There are no curves for values of < 0. For values > 0, the level curves are a set of two lines at \(c + 1\) and \(c - 1\). 3. For the defined function, the level curves are circles with radius < 1 for \(|p| < 1\) and lines with slope 1 for \(|p| \geq 1\), not including the area inside the circle with radius 1.

Step 1: Sketching Level Curves for \(f(x, y)=y^{2}-x\)

The level curves for \( f(x, y) = y^2 - x \) are found when we set \( f(x, y) \) equal to a constant, \( c \), which gives us \( y^2 - x = c \). Solving for \( x \), we get \( x = y^2 - c \). This equation represents a family of parabolas that open to the right if \( c \) is negative, and to the left if \( c \) is positive.

Step 2: Sketching Level Curves for \(f(p)=|p|-1\)

Since \( p \) is not clarified, let us consider \( p \) as a one-variable input. The level curves for \( f(p) = |p| - 1 \) are lines in the 2D-plane, parallel to the axes. When determining the level curves of \( f(p) = c \), for \( c < 0 \), there is no solution, as the absolute value of any real number cannot be less than one. For \( c = 0 \), the level curve will be the line \( p = 1 \) and \( p = -1 \). For \( c > 0 \), the level curves will be \( p = c + 1 \) and \( p = -c - 1 \)

Step 3: Sketching Level Curves for \( f(p) = \{1 \text{ when }|p|<1, \text{ and } x - y \text{ when }|p| \geq 1\} \)

For \( |p| < 1 \), the function \( f(p) \) is a constant and equal to 1. This represents level curves as circles around the origin with radius less than 1. For \( |p| \geq 1 \), the level curves of the function \( f(p) = x - y \) are lines with slope 1 in the 2D-plane. However, these lines will only exist outside the circle of radius 1 around the origin.

Key Concepts

Parabolas

Parabolas are a type of curve found in algebra and geometry, often shaped like a 'U' or an inverted 'U'. They can be described mathematically by quadratic equations. In our exercise, parabolas emerge when analyzing the function \( f(x, y) = y^2 - x \). The reason these curves are parabolas is that they fit the equation \( y^2 = x + c \), which is a standard form for a parabola.Parabolas have distinct features including:

• Vertex: The highest or lowest point on the curve, depending on its orientation.
• Axis of Symmetry: A line that splits the parabola into two mirror-image halves.
• Focus and Directrix: Used to precisely define the shape and position of the parabola.

In mathematical terms, moving from the general quadratic equation to understanding parabolas involves recognizing their standard form. Here, the level curves \( y^2 - x = c \) translate to horizontal or vertical "slices" of space, where \( x = y^2 - c \) gives us a series of parabolas. Pay attention to how changing the constant \( c \) will shift the parabola along the x-axis. This dynamic underpins their practical application in analyzing quadratic relationships.

Absolute Value Function

The absolute value function is a fundamental piece of algebra that turns any number into its non-negative equivalent. It’s represented as \( |p| \), and acts like a mirror of sorts, reflecting negative values into positive ones. In our context, we use the absolute value function in \( f(p) = |p| - 1 \).Understanding how this function works is key:

• Symmetry: The graph of an absolute value function is V-shaped and symmetric about the y-axis.
• No Negative Outputs: Since it measures the "distance" from zero, its result is never negative.
• Cases: We often break it into cases. For \( p \geq 0 \), \( |p| = p \), and for \( p < 0 \), \( |p| = -p \).

In the context of level curves, this function outlines lines parallel to the coordinate axes depending on the constant \( c \). It reflects the concept that only specific inputs—those falling on lines where the equation resolves to zero—will be level curves, such as \( p = 1 \) and \( p = -1 \) for \( c = 0 \). This approach aids in visualizing the 'steps' or segments, making absolute value a versatile tool in graphing scenarios.

Piecewise Functions

Piecewise functions let us describe situations where a function has different expressions based on the input value or range of the input. They are extremely useful in representing real-world scenarios or complex mathematical relationships. In the exercise, \( f(p) = \begin{cases} 1, & \text{if } |p| < 1 \ x-y, & \text{if } |p| \geq 1 \end{cases} \) is an example of a piecewise function.Here's how to understand and use these functions:

• Multiple Definitions: Define different outputs for different domains of the input.
• Graphical Implications: The graph will have disjoint sections, each reflecting its specific rule.
• Continuity: Piecewise functions can be continuous if the values meet or break at transition points smoothly.

In this context, the piecewise function splits the xy-plane into two regions. For \(|p| < 1\), the function is constant, meaning all outputs are 1, which graphically show as circles within a specific radius. For \(|p| \geq 1\), the function \(x-y\) introduces a linear relationship, forming level curves that are lines at different angles, specifically those with a slope of 1, but only outside the aforementioned circles. This versatility makes piecewise functions fundamental in depicting systems that change behavior at certain thresholds.