Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the level curve \(z=k\) for the indicated values of \(k\). \(z=\frac{1}{2}\left(x^{2}+y^{2}\right), k=0,2,4,6,8\)

Step-by-Step Solution

Answer

The level curves are concentric circles centered at the origin with radii 0, 2, \(2\sqrt{2}\), \(2\sqrt{3}\), and 4 for \(k = 0, 2, 4, 6, 8\) respectively.

Step 1: Understand the Equation

The equation given is \( z = \frac{1}{2}(x^2 + y^2) \). A level curve is a set of points \((x, y)\) where the function \(z(x, y)\) is constant at a particular value of \(k\). Thus, we have \( \frac{1}{2}(x^2 + y^2) = k \). We need to simplify this equation to sketch the level curves for different \(k\) values.

Step 2: Solve for the Level Curve Equation

Start by solving the equation \( \frac{1}{2}(x^2 + y^2) = k \) for \(x\) and \(y\). Multiply both sides by 2 to clear the fraction: \( x^2 + y^2 = 2k \). This represents a circle with radius \( \sqrt{2k} \) centered at the origin \((0, 0)\).

Step 3: Determine Radius for Different k Values

For each given value of \(k\), compute the radius of the circle:- \(k = 0\): \( x^2 + y^2 = 0 \Rightarrow \) Radius = 0 (single point at origin).- \(k = 2\): \( x^2 + y^2 = 4 \Rightarrow \) Radius = 2.- \(k = 4\): \( x^2 + y^2 = 8 \Rightarrow \) Radius = 2\(\sqrt{2}\).- \(k = 6\): \( x^2 + y^2 = 12 \Rightarrow \) Radius = 2\(\sqrt{3}\).- \(k = 8\): \( x^2 + y^2 = 16 \Rightarrow \) Radius = 4.

Step 4: Sketch the Circles for Each k

Draw each circle on the \(xy\)-plane:- For \(k = 0\), draw a point at the origin.- For \(k = 2\), draw a circle with radius 2 centered at (0,0).- For \(k = 4\), draw a circle with radius \(2\sqrt{2}\) centered at (0,0).- For \(k = 6\), draw a circle with radius \(2\sqrt{3}\) centered at (0,0).- For \(k = 8\), draw a circle with radius 4 centered at (0,0).

Key Concepts

Equation of a Circle

The concept of the equation of a circle is fundamental when analyzing level curves like those described in the problem. A circle in the Cartesian coordinate system can be defined by the equation \[x^2 + y^2 = r^2\]where \(r\) is the radius of the circle and the center is at the origin, \((0, 0)\). In essence, any point \((x, y)\) that satisfies this equation lies on the circle.
In the problem, the level curve is given by the equation \[z = \frac{1}{2}(x^2 + y^2) = k\]By simplifying, this becomes \[x^2 + y^2 = 2k\]which is essentially the equation of a circle centered at the origin. For each value of \(k\), this forms a set of circles with different radii.

Radius Calculation

Calculating the radius of a circle based on its equation requires understanding the mathematical structure of the circle's formula. Once you've set the equation in its simplest form, \[x^2 + y^2 = 2k\],comparing this to the standard form \[x^2 + y^2 = r^2\] enables us to solve for the radius, \(r\).
In our particular example, the radius is given by \[r = \sqrt{2k}\].

• For \(k = 0\), the radius is 0. This means the circle is a single point at the origin.
• For \(k = 2\), \(r = \sqrt{4} = 2\).
• For \(k = 4\), \(r = \sqrt{8} = 2\sqrt{2}\).
• For \(k = 6\), \(r = \sqrt{12} = 2\sqrt{3}\).
• For \(k = 8\), \(r = \sqrt{16} = 4\).

Each value of \(k\) gives a distinct circle with its own unique radius.

Function Analysis

Functional analysis in mathematics often involves solving equations and visualizing them, like finding and sketching level curves. The function in our original problem is \[z = \frac{1}{2}(x^2 + y^2)\]with different values of \(k\) representing different level curves.
By analyzing this function, we convert it into the form \[x^2 + y^2 = 2k\]which is much easier to understand and sketch. This transformation is central to interpreting how the function behaves as the parameters change. For each value of \(k\), the equation defines new circles, allowing us to see how the circles expand as \(k\) increases.

• Understanding how changes in \(k\) affect the size of circles helps us see the relationship between the function's output and its geometric representation.
• Each circle corresponds to points where the function's value is constant.

XY-Plane

The \(xy\)-plane is a two-dimensional coordinate system where geometric shapes and curves are plotted. It comprises axes, \(x\) and \(y\), which intersect at the origin point \((0, 0)\). Understanding the \(xy\)-plane is crucial when sketching shapes like circles or any curve.
In our problem, the level curves for different \(k\) values are drawn on the \(xy\)-plane. This visual representation makes it easier to grasp the relationship between algebraic equations and geometric figures.

• For \(k = 0\), a point at the origin is noted.
• For \(k > 0\), circles with varying radii are sketched.

These visualizations highlight how the mathematical concept of the equation relates to tangible graphics on the \(xy\)-plane, showcasing the area's versatility for seeing and understanding algebraic problems.