Problem 34 - College Algebra and Calculus: An Applied Approach

Question

Describe the level curves of the function. Sketch the level curves for the given c-values. $$ z=6-2 x-3 y \quad c=0,2,4,6,8,10 $$

Step-by-Step Solution

Answer

The equations for the level curves for the given c-values are: For \(c = 0\), \(y = 2 - \frac{2x}{3}\); For \(c = 2\), \(y = \frac{4 - 2x}{3}\); For \(c = 4\), \(y = \frac{2 - 2x}{3}\); For \(c = 6\), \(y = -\frac{2x}{3}\); For \(c = 8\), \(y = \frac{-2x - 2}{3}\); For \(c = 10\), \(y = \frac{-2x - 4}{3}\). These are lines with different slopes and intercepts on the (x, y) plane.

Step 1: Write Down the Function and Determine Level Curves

The function given is \(z = 6 - 2x - 3y\). To determine the level curves, we must find the set of points \((x,y)\) that makes this function equals to the given c-values (0, 2, 4, 6, 8, 10). The resulting equations will be the level curves.

Step 2: Calculate and Express the Level Curves

At each value of c, we solve \(6 - 2x - 3y = c\). This gives us the equations of the different level curves at each specified c-value: At \(c = 0\), the level curve equation is: \(6 - 2x - 3y = 0\). At \(c = 2\), the level curve equation is: \(6 - 2x - 3y = 2\). At \(c = 4\), the level curve equation is: \(6 - 2x - 3y = 4\). At \(c = 6\), the level curve equation is: \(6 - 2x - 3y = 6\). At \(c = 8\), the level curve equation is: \(6 - 2x - 3y = 8\). At \(c = 10\), the level curve equation is: \(6 - 2x - 3y = 10\). These equations can be rearranged to \(y\) as subject, which will be easier for plotting the curves.

Step 3: Sketch the Level Curves

Use these equations obtained in step 2 to sketch the level curves on the xy-plane. Each level curve corresponds to a line on the (x,y)-plane. Plot each line for each respective c-value.

Key Concepts

Functions and Their Role in Mathematics

In mathematics, functions are fundamental building blocks that help depict relationships between different quantities. A function provides a systematic way to map each input from a set to a single output value. In simpler terms, it is like a machine where you input a value, and it processes this value to give an output. Consider the example of the function given in the exercise: - This is expressed as \( z = 6 - 2x - 3y \), showcasing a linear relationship.- Here, \( z \) is the output, calculated based on input values \( x \) and \( y \).- The equation represents a plane in three-dimensional space.Functions are essential because they allow us to visualize and analyze the behavior of mathematical models. By understanding how changes in \( x \) and \( y \) affect \( z \), we can make predictions about real-world phenomena.

Understanding Graphing Techniques

Graphing is a powerful tool that assists in visualizing mathematical concepts. It transforms arithmetic expressions into geometric representations, making abstract ideas tangible. In the context of our exercise:- We focus on representing the function's output on a graph via level curves.- Each level curve equation such as \( 6 - 2x - 3y = c \) is linear and forms a specific line on the plane.- The primary aim of graphing here is to identify and sketch these lines corresponding to different \( c \)-values.The process of graphing these lines enhances comprehension of how the function behaves across different inputs. It helps in determining where the function maintains consistent outputs as \( x \) and \( y \) vary.

Coordinate System Essentials

A coordinate system is a framework that uses coordinates to uniquely specify a point in space. In most mathematical problems, including our exercise, the Cartesian coordinate system is employed. This consists of:- Two perpendicular axes: the x-axis (horizontal) and y-axis (vertical).- Each point on the plane is given by an \((x, y)\)-pair, showing positions relative to these axes.In this specific exercise:- The level curves are sketched on the \( xy \)-plane.- Each curve represents points satisfying \( 6 - 2x - 3y = c \) for different values of \( c \).- Understanding this system is crucial for accurately placing and interpreting the lines corresponding to the level curves in a coordinate context. The coordinate system provides a structured visual aid, making abstract mathematical concepts easier to comprehend and analyze.