Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 17-22, sketch the level curve \(z=k\) for the indicated values of \(k\). $$ z=\frac{x}{y}, k=-2,-1,0,1,2 $$

Step-by-Step Solution

Answer

Sketch the lines for \( x = ky \) with \( k = -2, -1, 0, 1, 2 \) as their respective level curves.

Step 1: Understanding the Level Curve Equation

The expression given is \( z = \frac{x}{y} \). For a level curve, we fix \( z = k \) where \( k \) is a constant. Thus, the equation becomes \( \frac{x}{y} = k \). Rearranging this, we find \( x = ky \). For each value of \( k \), this represents a different line through the origin in the xy-plane.

Step 2: Sketch for k = -2

Substituting \( k = -2 \) into \( x = ky \), we get \( x = -2y \). This equation represents a line through the origin with a slope of -2. The line will pass through points such as (0, 0), (-2, 1), and (2, -1). Sketch this line on a graph.

Step 3: Sketch for k = -1

Substituting \( k = -1 \) into \( x = ky \), we get \( x = -y \). This represents a line with a slope of -1. The line will pass through points such as (0, 0), (-1, 1), and (1, -1). Sketch this line on the same graph.

Step 4: Sketch for k = 0

Substituting \( k = 0 \) into \( x = ky \), we get \( x = 0 \). This is the y-axis itself. The line represents all points where \( x = 0 \), like (0, 1), (0, -1), and (0, 0). Sketch this vertical line.

Step 5: Sketch for k = 1

Substituting \( k = 1 \) into \( x = ky \), we get \( x = y \). This is a line with a slope of 1. The line will pass through points such as (0, 0), (1, 1), and (-1, -1). Sketch this line on the same graph.

Step 6: Sketch for k = 2

Substituting \( k = 2 \) into \( x = ky \), we get \( x = 2y \). This is a line with a slope of 2. The line will pass through points such as (0, 0), (2, 1), and (-2, -1). Sketch this line on the same graph.

Key Concepts

Multivariable Calculus

Multivariable calculus extends the concepts of calculus from dealing with functions of a single variable to functions of multiple variables. In this context, the function given is \( z = \frac{x}{y} \), where both \( x \) and \( y \) are independent variables. The variable \( z \) depends on both \( x \) and \( y \), creating a surface in three-dimensional space.

Level curves are a vital concept in multivariable calculus. They represent the intersection of a surface in three dimensions with a horizontal plane, resulting in a curve or a set of curves on a two-dimensional plane. For our function \( z = \frac{x}{y} \), level curves are found by setting \( z \) to a constant, \( k \), which simplifies the three-dimensional problem to two dimensions, as in \( \frac{x}{y} = k \).With this approach,

1. We can directly visualize changes in \( z \).
2. We gain insights into the behavior of the function on the xy-plane.

These level curves help us understand the relationships and gradients in multivariable functions.

Coordinate Geometry

Coordinate geometry provides a bridge between algebraic equations and geometric figures through the coordinate plane. In this exercise, we use the equation \( x = ky \) to represent lines on the coordinate plane. The equation is derived from fixing \( z = k \) in the expression \( z = \frac{x}{y} \).

• Each value of \( k \) gives us a linear equation \( x = ky \).
• Geometrically, this represents a line through the origin (0,0), with slope \( k \).

Understanding the slope is essential in coordinate geometry. The slope \( k \) determines the steepness and direction of the line:- Positive slopes (\( k \gt 0 \)) create lines that angle upwards from left to right.- Negative slopes (\( k \lt 0 \)) angle downwards.- A slope of 0 results in a horizontal line.Coordinate geometry enables us to plot these results on a plane, illustrating how algebraic relationships manifest geometrically.

Graphing Techniques

Graphing techniques simplify the visualization of mathematical relationships. For the given function \( z = \frac{x}{y} \), each level curve \( x = ky \) is effectively a straight line graph.By plotting these:- Start by choosing specific points that satisfy the equation for each \( k \).- Typical points include the origin and points where either \( x \) or \( y \) is set to 1, making calculations straightforward.- For instance, \( k = -2 \) leads to points like (0, 0), (-2, 1), and (2, -1).Once points are determined, you connect them to form the line. Repeating for every \( k \) value results in a series of lines

• Through the origin.
• With varying slopes depending on \( k \).

This procedure solidifies the understanding of how changing \( k \) affects the curve's orientation.Graphing lines effectively uses basic techniques like selecting easy-to-calculate intercepts and slopes, allowing students to visualize equations intuitively.

Linear Equations

Linear equations are straightforward polynomial equations of the first degree, typically written as \( ax + by = c \). For our function \( z = \frac{x}{y} \), by setting \( z = k \), we express it in the form \( x = ky \), a linear relation.These particular equations of the form \( x = ky \):

• Describe lines through the origin on the coordinate plane.
• Reveal the slope directly as \( k \), the coefficient of \( y \).

Understanding linear equations helps simplify complex problems. They tell us:- How two variables relate directly to each other, with a non-changing rate (slope).- How the graph will behave for different values of \( k \), giving insight into the behavior of more complex mathematical models.Linear equations are fundamental in mathematics, providing a basis for modeling everything from simple scenarios to more complicated systems involving several variables.