Problem 19 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Sketch the level curves of the function corresponding to each value of \(z\). \(f(x, y)=2 x+3 y ; z=-2,-1,0,1,2\)

Step-by-Step Solution

Answer

The level curves of the function \(f(x, y) = 2x + 3y\) for the given values of \(z = -2, -1, 0, 1, 2\) are the lines \(y = -\frac{2}{3}x - \frac{2}{3}\), \(y = -\frac{2}{3}x + \frac{1}{3}\), \(y = -\frac{2}{3}x\), \(y = -\frac{2}{3}x + \frac{1}{3}\), and \(y = -\frac{2}{3}x + \frac{2}{3}\), respectively. These level curves are parallel lines with a slope of \(-\frac{2}{3}\) and constant vertical distances, which can be sketched on the coordinate plane and labeled with their corresponding z-values.

Step 1: Set function equal to z values and solve for y

For each value of z, set the function \(f(x, y) = 2x + 3y\) equal to z and solve for y in terms of x: 1. When \(z = -2\): \(2x + 3y = -2\) 2. When \(z = -1\): \(2x + 3y = -1\) 3. When \(z = 0\): \(2x + 3y = 0\) 4. When \(z = 1\): \(2x + 3y = 1\) 5. When \(z = 2\): \(2x + 3y = 2\) Now, let's solve each equation for y in terms of x: 1. \(y = -\frac{2}{3}x - \frac{2}{3}\) 2. \(y = -\frac{2}{3}x + \frac{1}{3}\) 3. \(y = -\frac{2}{3}x\) 4. \(y = -\frac{2}{3}x + \frac{1}{3}\) 5. \(y = -\frac{2}{3}x + \frac{2}{3}\)

Step 2: Sketch the level curves

Now we can sketch the level curves on the coordinate plane, using the equations obtained in Step 1: 1. When \(z = -2\), sketch the line \(y = -\frac{2}{3}x - \frac{2}{3}\) 2. When \(z = -1\), sketch the line \(y = -\frac{2}{3}x + \frac{1}{3}\) 3. When \(z = 0\), sketch the line \(y = -\frac{2}{3}x\) 4. When \(z = 1\), sketch the line \(y = -\frac{2}{3}x + \frac{1}{3}\) 5. When \(z = 2\), sketch the line \(y = -\frac{2}{3}x + \frac{2}{3}\) As you sketch, you should notice that the level curves are parallel lines with a slope of \(-\frac{2}{3}\), and their vertical distances are constant. These lines should appear in the coordinate plane, spaced apart and labeled with their corresponding z-values.

Key Concepts

Multivariable Calculus

Multivariable Calculus extends the concepts of calculus to functions of several variables. It deals with functions like \(f(x, y) = 2x + 3y\), which depend on more than one variable and allows us to explore how these functions behave. In this context, we use concepts like partial derivatives, multiple integrals, and level curves.

Level curves are crucial in multivariable calculus. They are slices of a three-dimensional surface at a constant value, usually expressed as \(f(x, y) = c\), where \(c\) is a constant. For each \(z\) value, we can see which combinations of \(x\) and \(y\) conform to this surface by examining level curves.

By sketching these curves, we're essentially showing how the function changes in the \(xy\)-plane at various heights or values, allowing us to simplify and understand complex functions.

Graphing Functions

Graphing functions is a powerful way to visualize mathematical equations, especially in mathematics and science. Specifically, in the context of functions like \(f(x, y) = 2x + 3y\), graphing helps us visualize how these equations manifest as linear forms, in this case, lines in a plane.

To graph a function of two variables, we take each value of \(z\) (our constants \(-2, -1, 0, 1, 2\)), solve the equation for \(y\) in terms of \(x\), and plot these graphs as lines. In our example, every \(z\) value results in a linear equation, allowing easy graphing in two dimensions.

• For \(z = -2\), plot \(y = -\frac{2}{3}x - \frac{2}{3}\).
• For \(z = -1\), plot \(y = -\frac{2}{3}x + \frac{1}{3}\).
• For \(z = 0\), plot \(y = -\frac{2}{3}x\).
• For \(z = 1\), plot \(y = -\frac{2}{3}x + \frac{1}{3}\).
• For \(z = 2\), plot \(y = -\frac{2}{3}x + \frac{2}{3}\).

The lines are parallel since their slopes are equal, which tells us about the function’s behavior—regular, predictable changes across different levels (or values of \(z\)).

Linear Equations

Linear equations represent relationships between variables where the variables are either directly or inversely proportional. These equations are graphically represented as a straight line in a two-dimensional space.

In the problem we're examining, the function \(f(x, y) = 2x + 3y\) is a linear equation of the form \(ax + by = c\). This equation format is significant because:

• The coefficients \(2\) and \(3\) decide the slope and orientation of the lines formed in the graph. It means the variable \(x\) affects the outcome twice as much as \(y\) per unit increase.
• The constant \(z\) shifts the line up or down on the graph. As this constant changes, it results in parallel shifts in the graph lines.
• The negative slope \(-\frac{2}{3}\) indicates that as \(x\) increases, \(y\) decreases, keeping the shape and orientation of the line constant.

Knowing the nature of linear equations allows us to predict the graphical shifts and understand the relationships and intersections between variables on a two-dimensional plane.