Problem 19 - Calculus 3

Question

For the following exercises, find the level curves of each function at the indicated value of \(c\) to visualize the given function. $$ h(x, y)=2 x-y ; c=0,-2,2 $$

Step-by-Step Solution

Answer

The level curves are lines: \( y=2x \), \( y=2x+2 \), \( y=2x-2 \).

Step 1: Understanding Level Curves

Level curves are the curves where a function of two variables is equal to a constant value. For the function \( h(x, y) = 2x - y \), we want to find curves where \( h(x, y) \) equals the given \( c \) values of 0, -2, and 2.

Step 2: Setting up Equations for c=0

Substitute \( c=0 \) into the equation \( h(x, y) = 2x - y \), obtaining \( 2x - y = 0 \). Rearrange this to show \( y = 2x \). This is the equation of the line representing the level curve for \( c=0 \).

Step 3: Setting up Equations for c=-2

Substitute \( c=-2 \) into \( h(x, y) = 2x - y \), giving \( 2x - y = -2 \). Rearrange this to write \( y = 2x + 2 \). This line is the level curve for \( c=-2 \).

Step 4: Setting up Equations for c=2

Substitute \( c=2 \) into \( h(x, y) = 2x - y \), resulting in \( 2x - y = 2 \). Rearrange to \( y = 2x - 2 \). This is the third level curve for \( c=2 \).

Key Concepts

Multivariable Calculus

Multivariable calculus extends the concepts of calculus, such as differentiation and integration, to functions of more than one variable.
In simpler terms, instead of dealing with functions that have a single input and output, multivariable calculus handles functions with multiple inputs and outputs, which can be represented as points in a plane or space. This branch of mathematics is crucial for understanding how different quantities change with respect to each other in real-world applications, like physics and engineering.

• In multivariable calculus, curves and surfaces aren't just lines on a flat surface but can take on complicated shapes in three-dimensional space.
• This allows for the design and analysis of more complex systems.

By understanding and using these mathematical tools, we can better predict and manipulate the behavior of the systems around us.

Functions of Two Variables

A function of two variables, commonly expressed as \( f(x, y) \), assigns every pair of \( x \) and \( y \) a single resulting value.
These functions provide a way to model situations where the output depends on two distinct inputs.
For example:

• The function \( h(x, y) = 2x - y \) tells us that the outcome is found by multiplying \( x \) by 2 and then subtracting \( y \) from the result.
• This forms a plane in three-dimensional space.

The value at each point corresponds to the height at that point in a plotted graph. Being adept with functions of two variables opens up a deeper understanding of how changes in inputs affect system responses.

Equation Rearrangement

Rearranging equations is an essential skill in algebra and calculus, especially when dealing with level curves.
The process involves solving the equation for a different variable and revealing another perspective of the same relationship.
For instance, given the equation from the exercise:

• Start with \( 2x - y = 0 \).
• Rearrange to find \( y \) as the subject: \( y = 2x \).

This makes it straightforward to graph or visualize the relationship, as it is expressed as a line with a specific slope. Rearranging equations simplifies complex systems, making them manageable and easy to interpret for visualization or solving problems.

Visualizing Curves

Visualizing curves, especially in the context of level curves, provides insight into understanding the behavior of functions.
Level curves, or contour lines, are where the function's value remains constant, illustrating how values change across a plane.
For the function \( h(x, y) = 2x - y \):

• The curve for \( c=0 \) is \( y = 2x \), representing a diagonal line through the origin.
• For \( c=-2 \), it’s \( y = 2x + 2 \), a parallel line shifted upwards.
• The curve for \( c=2 \) is \( y = 2x - 2 \), shifted downwards.

These visualizations help in grasping how changes in the parameters of the function can shift and transform the output across the plane. Such graphical representations are indispensable in fields like economics, engineering, and meteorology.