# Level curves (or contour curves) of a certain 3 dimensional function.

• Almost935
In summary, the conversation discusses the concept of level curves for the function f(x,y)=xy, where a fixed number k is substituted for f(x,y). The resulting contour diagram shows that when k=0, the graph consists of two lines (y=0 and x=0) on the x and y axes, respectively. When k is not equal to 0, the graph forms hyperbolas with the x and y axes as asymptotes.
Almost935
I can't seem to quite comprehend the level curves for f(x,y)=xy. I realize this should be very simple yet the answer eludes me. f(x,y) for this the two dimensional representation of this function will be substituted with k(a fixed number). I would greatly appreciate somebody explaining my contour diagram to me and what the graph of k(fixed number)=xy even resembles in two dimensions and why.

$xy=k \Rightarrow y=\frac{k}{x}$
$k=0 \Rightarrow \left\{ \begin{array}{cc} y=0 \ \ \ x \ axis \\ x=0 \ \ \ y \ axis \end{array} \right.$
$k\neq 0 \Rightarrow$ hyperbolas with x and y axes as asymptotes.

## 1. What are level curves (or contour curves) of a certain 3-dimensional function?

Level curves, also known as contour curves, are the two-dimensional representations of a three-dimensional function. They are created by plotting points where the function has a constant output, resulting in a curved line on a 2D plane.

## 2. How are level curves useful in understanding a 3-dimensional function?

Level curves provide a visual representation of how the output of a 3-dimensional function changes as the input variables vary. They can help identify patterns and relationships within the function and aid in making predictions about its behavior.

## 3. What do the different levels on a contour map represent?

The different levels on a contour map represent the values of the output of the 3-dimensional function. Each level curve corresponds to a specific constant output value, with higher levels representing higher values and lower levels representing lower values.

## 4. How do you interpret the shape of a level curve?

The shape of a level curve can provide information about the behavior of the 3-dimensional function. For example, if the level curves are concentric circles, it indicates that the function is symmetric. If the level curves are close together, it suggests that the function is changing rapidly in that region.

## 5. Can level curves intersect?

No, level curves of a certain 3-dimensional function cannot intersect. This is because each level curve represents a specific constant output value, and the function can only have one output value for a given input. If two level curves were to intersect, it would mean that the function has two different output values for the same input, which is not possible.

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