Question

Describe the level curves of the function. Sketch the level curves for the given c-values. $$ f(x, y)=\ln (x-y) \quad c=0, \pm \frac{1}{2}, \pm 1, \pm \frac{3}{2}, \pm 2 $$

Short answer

The level curves of the function \(f(x, y)=\ln(x-y)\) are lines of the form \(y=x-e^c\) for the given c-values. The function is not defined for \(x<y\), therefore those sections are excluded in the level curve.

Step-by-step solution

Understanding Level Curves

Firstly, it's important to understand what 'level curves' are. Level curves are curves on which the function takes constant values. In this case, we are given the values of c (c = 0, ± 1/2, ± 1, ± 3/2, ± 2). So we have to find the curves on which the function f(x, y) = ln(x-y) takes these constant values.

Solving the Equation for Given c-values

Taking the natural exponential on both sides, we can change the given equation \(f(x, y) = c\) to \(x-y = e^c\). So we get \(y = x - e^c\). This is the equation we'll use to plot our level curves.

Plotting the Level Curves

Now, for each given c-value, we have a curve. These curves will be parallel lines with slope 1 and varying y-intercepts (equal to -\(e^c\)). If \(e^c\) is already given, the y-intercept directly corresponds to the value of -\(e^c\). If only c is given (as in this case), the y-intercept can be determined by calculating \(e^c\) for each given c.

Finalizing the Level Curves

Each of these lines corresponds to a level curve. The level curves represent the lines of the form \(y = x - e^c\), for given values of c. Since for this exercise the function is not defined for \(x less than y\), those areas of the curve are not included.

Key concepts

Multivariable Calculus and Level Curves

In multivariable calculus, a level curve is a tool that helps us visualize a function of two variables by tracing the points where the function equals a constant value. Imagine 'slicing' the surface created by the function at different heights; the edges of these slices are the level curves.

For instance, in the exercise where we have the function \( f(x, y)=\text{ln}(x-y) \), finding the level curves entails determining the sets of points \((x, y)\) where the function equals certain specified constants. The provided values of \(c\) correspond to different 'heights' at which we are slicing the function's surface. This concept is pivotal as it allows us to capture and represent the behavior of a multivariable function on a two-dimensional plane, facilitating a better understanding of the function's properties.

In practice, constructing these level curves is usually the first step to analyzing a function's topography – the highs and lows, the peaks, and valleys – that define its shape. This represents a foundational skill in multivariable calculus, as it is widely applicable to disciplines like engineering, physics, and economics, where visualizing the impact of two different factors on a result is crucial.

The Natural Logarithm (\(\text{ln}\))

The natural logarithm, denoted as \(\text{ln}\), is the logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. This function is the inverse of the exponential function \(e^x\), which means that \(\text{ln}(e^x) = x\).

The natural logarithm plays a central role in various areas of mathematics and its applications due to its unique properties, such as the fact that the derivative of \(\text{ln}(x)\) is \(1/x\), making it invaluable when dealing with growth processes and time-related changes, such as in compound interest calculations and natural growth models.

In our specific exercise, the function \(f(x, y) = \text{ln}(x - y)\) contains a natural logarithm, which indicates a relationship based on proportional rates of change. By examining the level curves of this function, we can explore how the rate of change in the \(y\) value affects the output of \(f(x, y)\) as \(x\) and \(y\) change with respect to each other. Understanding the behavior of the natural logarithm within this context enhances our comprehension of how the function behaves throughout different sections of the plane, further expanding our calculus toolkit.

Graphing Functions and the Role of Level Curves

Graphing functions is a visual representation technique that provides insights into the nature of mathematical relationships. It's an essential aspect of mathematics, as it translates complex equations into understandable images – graphs that can be analyzed and interpreted.

The process of graphing involves plotting points on a coordinate plane to visualize the function’s behavior. The act of graphing gets more challenging as we move from single-variable to multi-variable functions. That's where level curves come into play; they serve as a bridge, allowing us to graph the otherwise complex surface of a function of two variables in two dimensions.

In our exercise, graphing the function \(f(x, y) = \text{ln}(x - y)\) by using level curves simplifies the three-dimensional representation down to a collection of two-dimensional slices. Specifically, by manipulating the equation to \(y = x - e^c\), we’re allowed to draw each level curve as a simple line on the xy-plane. Each line represents a different 'slice' of the function, corresponding to a different output value from the natural logarithm. One important note to highlight from the solution is that these lines are not defined where \(x < y\), which keeps in tune with the domain of the natural logarithm where the input must be positive. Graphing the level curves grants a better understanding of the function by showing us precisely where it rises and falls, how steeply it does so, and where it levels off, all of which are critical in understanding the landscape of \(f(x, y)\).