Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

, plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts.. $$ x^{4}+y^{4}=1 $$

Step-by-Step Solution

Answer

The graph is a symmetric curve called a Lemniscate, with intercepts at (1,0), (-1,0), (0,1), and (0,-1).

Step 1: Check for Symmetry

To check for symmetry, substitute 1. - Replace \(x\) with \(-x\)\. Since the equation \(x^{4} + y^{4} = 1\) becomes \((-x)^{4} + y^{4} = 1\), which is the same after substitution, it is symmetric about the \text{y-axis}\. - Replace \(y\) with \(-y\)\. The equation \(x^{4} + (-y)^{4} = 1\) remains unchanged, so it is symmetric about the \text{x-axis}\. - Replace \(x\) with \(-x\)\ and \(y\) with \(-y\)\. The equation \((-x)^{4} + (-y)^{4} = 1\) is still the same. Therefore, it is symmetric about the \text{origin}\.

Step 2: Find x-intercepts

To find the x-intercepts, set \(y = 0\) in the equation \(x^{4} + y^{4} = 1\):\[ x^{4} + 0^{4} = 1 \Rightarrow x^{4} = 1\]Solving for \(x\),\[ x = \pm 1.\]Thus, the x-intercepts are \((1, 0)\) and \((-1, 0)\).

Step 3: Find y-intercepts

To find the y-intercepts, set \(x = 0\) in the equation \(x^{4} + y^{4} = 1\):\[ 0^{4} + y^{4} = 1 \Rightarrow y^{4} = 1\]Solving for \(y\),\[ y = \pm 1.\]Thus, the y-intercepts are \((0, 1)\) and \((0, -1)\).

Step 4: Plot the Curve

The equation \(x^{4} + y^{4} = 1\) represents a figure known as a \text{Lemniscate}\. Since the equation is symmetric about both the axes and the origin, it will be symmetric in all quadrants. The points we found, \((1,0), (-1,0), (0,1), (0,-1)\) and the symmetry suggest that the curve looks like a stretched circle in the shape of a \text{diamond}\, but with rounded corners at those points. Plot these points and draw a smooth curve through these to graph the equation.

Key Concepts

Symmetry in Graphs

In mathematics, understanding the symmetry of a graph can make analyzing and sketching it much easier. Symmetry implies that one part of the graph mirrors another, providing a neat and predictable structure to work with.
To determine symmetry, one can perform simple substitutions in the given equation.

• To check for y-axis symmetry, replace every occurrence of \(x\) with \(-x\). If the equation remains unchanged, it is symmetric about the y-axis.
• For x-axis symmetry, substitute \(y\) with \(-y\). If the equation holds, it's symmetric about the x-axis.
• To see if the graph is symmetric about the origin, replace both \(x\) with \(-x\) and \(y\) with \(-y\). If the equation remains the same, it has origin symmetry.

In the provided equation, \(x^4 + y^4 = 1\), all three substitutions keep the equation unchanged. This indicates symmetries about the y-axis, x-axis, and origin, suggesting a beautifully balanced graph shape.

X-intercepts and Y-intercepts

Finding x-intercepts and y-intercepts is a crucial step in graphing. These intercepts are points where the graph crosses the x-axis or y-axis.

- **X-intercepts:** To find these, we set \(y = 0\) in the equation and solve for \(x\). In the equation \(x^4 + y^4 = 1\), substituting \(y = 0\) gives \(x^4 = 1\). Solving this, we find \(x = \pm 1\). Hence, the x-intercepts are \((1, 0)\) and \((-1, 0)\).
- **Y-intercepts:** Here, set \(x = 0\) and solve for \(y\). Again, using \(x^4 + y^4 = 1\), we get \(y^4 = 1\), leading to \(y = \pm 1\). Thus, the y-intercepts are \((0, 1)\) and \((0, -1)\).

These intercepts serve as reference points when plotting and give insight into the overall behavior of the graph near the axes.

Even Function

In calculus, an even function is a function that satisfies a specific symmetry property. For a function \(f(x)\), it is considered even if \(f(-x) = f(x)\) for every \(x\) in its domain.

This property means that the graph of an even function is symmetric with respect to the y-axis. Essentially, the left and right sides of the graph mirror each other.

• In the equation \(x^4 + y^4 = 1\), each polynomial term \(x^4\) and \(y^4\) contributes to the evenness, as raising a negative number to an even power results in a positive number.
• This characteristic aligns with the symmetry tests for the graph, corroborating that it remains unchanged when \(x\) or \(y\) are negated.

Even functions offer predictability, making complex calculations more manageable and aiding in comprehensive graph analysis.

Lemniscate

The term "Lemniscate" refers to a specific type of curve that looks like a figure-eight or infinity symbol. It is a classic shape in mathematics, known for its symmetrical properties.

In the context of the equation \(x^4 + y^4 = 1\), the graph is a lemniscate but with a twist.

• The curve appears like a diamond with rounded points, ensuring smooth transitions between the axes and intercepts.
• This type of lemniscate is a result of the high symmetry the equation possesses, being symmetric around both axes and the origin.

When sketching this figure, the recognized intercept points \((1, 0)\), \((-1, 0)\), \((0, 1)\), and \((0, -1)\) guide us in shaping the curve accurately. The curve passing through these points straightforwardly portrays its unique, balanced form.