Problem 11 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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

Step-by-Step Solution

Answer

For the function \(f(x, y) = 2x + 3y\), we found the equation of the level curves to be \(y = \frac{z-2x}{3}\). By substituting different values of z (e.g., \(z_1, z_2, z_3\)), we obtain the equations for the individual level curves, which are lines with slope \(-\frac{2}{3}\) and y-intercepts \(\frac{z_1}{3}, \frac{z_2}{3}, \frac{z_3}{3}\). Sketch these lines and label them with their corresponding z-values.

Step 1: Set up the equation

Given the function f(x, y) = 2x + 3y, we want to find the level curves for different values of z. To do this, we'll set the function equal to a constant z and solve for y: f(x, y) = z 2x + 3y = z

Step 2: Solve for y

Next, we will solve for y to find the equation of the level curve for each value of z: 3y = z - 2x y = (z - 2x) / 3 Now we have an equation for the level curves in terms of x and z.

Step 3: Plug in different values of z

To sketch the level curves, we'll plug in different values of z into our equation for y, and then plot the curve for each value. Let's consider 3 values of z: z1, z2, and z3. For z1: y = (z1 - 2x) / 3 For z2: y = (z2 - 2x) / 3 For z3: y = (z3 - 2x) / 3

Step 4: Sketch the level curves

Now we will sketch the level curves using the equations we found for z1, z2, and z3. The level curves will be lines with slopes -2/3 and y-intercepts equal to z1/3, z2/3, and z3/3. 1. For z1: Plot a line with slope -2/3 and y-intercept z1/3 2. For z2: Plot a line with slope -2/3 and y-intercept z2/3 3. For z3: Plot a line with slope -2/3 and y-intercept z3/3 Finally, label the level curve with the corresponding value of z.

Key Concepts

Multivariable Functions

In mathematics, a multivariable function is a function that has more than one input variable. These are like the functions you're used to but a bit more extensive in scope, meaning they can take two, three, or even more variables as inputs. For example, in the exercise herein, our function is expressed as \(f(x, y) = 2x + 3y\), which takes both \(x\) and \(y\) as inputs. The output can change with different combinations of \(x\) and \(y\).

These types of functions are essential when dealing with scenarios where different factors affect the outcome. For instance, calculating a price based on both weight and distance might require using a multivariable function. With them, one can:

• Analyze how changes in input variables affect the output.
• Sketch shapes and surfaces that these functions create in space.
• Understand how different variables interact with each other.

By comprehending multivariable functions, you open up avenues for exploring more complex real-world situations.

Contour Plots

Contour plots are valuable tools in visualizing multivariable functions. They provide a two-dimensional view of a three-dimensional graph, where lines (called contours) represent points of equal function value, also known as level curves. In other words, each curve connects all points \(x, y\) where the function \(f(x, y)\) is constant.

In our exercise, level curves are lines drawn for different constant values of \(z\), where \(f(x, y) = z\). For example, by setting \(z_1\), \(z_2\), and \(z_3\), and solving for \(y\), we find different lines that represent these contours. The utility of contour plots includes:

• Helping identify the tendency of the function as it travels across a plane.
• Enabling prediction of how function changes between values.
• Allowing easier reading in scientific data sets and geographic maps.

Contour plots simplify multidimensional problems, making them a key tool in fields like meteorology and topography.

Linear Equations

Linear equations form the backbone of many mathematical concepts, including level curves and multivariable functions. A linear equation represents a straight line when plotted on a graph and can generally be expressed in the form \(ax + by = c\).

In the given example, our function \(2x + 3y = z\) rearranges into a linear equation for each fixed \(z\), like \(y = \frac{z - 2x}{3}\). Here, the slope is constant, and this constancy translates to a straight line on the graph. Characteristics of linear equations include:

• They graph as straight lines in two dimensions.
• Their solutions can be one, none, or infinitely many based on the system of equations.
• They help map relationships directly proportional to one another.

Using linear equations, one easily predicts data trends and calculates solutions across various fields. For students and engineers alike, mastering linear equations is crucial for tackling practical and theoretical challenges.