Question

Describe the level curves of the function. Sketch the level curves for the given \(c\) -values. $$ f(x, y)=e^{x y / 2}, \quad c=2,3,4, \frac{1}{2}, \frac{1}{3}, \frac{1}{4} $$

Short answer

Level curves of the given function \( f(x, y) = e^{xy/2} \) for the constants 2, 3, 4, 1/2, 1/3, and 1/4 correspond to the equations \(y = \frac{2\ln(2)}{x}\), \(y = \frac{2\ln(3)}{x}\), \(y = \frac{2\ln(4)}{x}\), \(y = \frac{2\ln(1/2)}{x}\), \(y = \frac{2\ln(1/3)}{x}\), and \(y = \frac{2\ln(1/4)}{x}\), respectively. When plotted, the curves become progressively steeper as the value of c increases.

Step-by-step solution

Set the Function Equal to Constant

First, set the function equal to each constant value in turn. This will give six separate equations of the form \(e^{xy/2} = c\). Express y in terms of c and x by taking the natural logarithm (ln) of both sides.

Solve for y in terms of x and c

After obtaining the equation \(e^{xy/2} = c\), take the natural logarithm of both sides of the equation to get \(xy/2 = \ln(c)\). Solving for y gives \(y = \frac{2\ln(c)}{x}\). Repeat these steps for each value of c.

Plot the Level Curves

Next, plot the separate equations obtained from step 2 to create the level curves for each value of c.

Analyze the Level Curves

Interpret the resulting plot. Consider the curvature of the levels, the direction they bend, and how they change with different constant values. Observe that as the value of c increases, the curves become steeper.