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 equations of the level curves are of the form \(y = x - e^c\). As \(c\) ranges from a negative to a positive value, the level curves shift downward because the y-intercept decreases as \(c\) increases.

Step-by-step solution

Understand Level Curves

In a function with two variables, level curves are the collection of points where the function takes on a certain value. In this case, we set \( f(x, y) = c \) where \(c\) is the level value and solve the equation.

Set function equal to constant

Setting the function \( f(x, y) = \ln(x - y) \) equal to \(c\), we get the equation \(\ln(x - y) = c\). Since the base of the natural logarithm is \(e\), we can rewrite this equation as \(x - y = e^c\). Solving for \(y\), we get \(y = x - e^c\). This is the equation for the level curve.

Sketch Level Curves

For each given \(c\) value, substitute it into the equation \(y = x - e^c\) and sketch the resulting line. For example, when \(c = 0\), the equation becomes \(y = x - e^0 = x - 1\), which is a straight line with slope 1 and y-intercept of -1. Do this for all values of \(c\) to get the level curves.

Final considerations

Make sure each curve is labelled with the appropriate \(c\) value. Take note of how the level curves are changing as \(c\) increases. Realise that as \(c\) increases, the y-intercept is decreasing and the curves are moving downwards.