Draw a contour map of f(x, y) with an appropriate contour interval, showing at least six level c

Draw a contour map of f(x, y) with an appropriate contour...

Key Concepts

Level Curves

Level curves are the sets of points in the domain of a function where the function maintains a constant value. These curves, which are also known as iso-lines, help in understanding how the function changes over the domain and are fundamental in the analysis of multivariable functions.

Contour Interval

The contour interval is the constant difference in the function's value between adjacent contour lines on a contour map. Choosing an appropriate contour interval is crucial as it balances detail and clarity, allowing for effective visualization of the variations in the function's surface without cluttering the map.

Contour Map

A contour map is a two-dimensional graphical representation of a three-dimensional surface where curves, known as contour lines, connect points with the same function value. It is used to visualize the topography or gradient of a function of two variables, providing insights into the surface’s elevation or level without requiring a three-dimensional plot.

Transcript

00:01 We want to draw a contour map of f of xy is equal to x squared minus y with appropriate contour interval and showing at least six level curves.

00:14 So the first thing i'm going to do is go ahead and replace f of xy with c.

00:20 And then we can solve for y.

00:24 So we can add y over and then subtract c.

00:28 So we get y is equal to x squared minus c.

00:31 So now this will help us graph our contours a lot quicker, or our level curves a lot quicker.

00:40 So let's just go ahead and do the following.

00:43 So we can let c be in.

00:47 So let's do negative 2, negative 1, 0, 1, and 2, 1, and 2, and 2, and 3.

00:54 So 1, 2, 3, 4, 5, 6.

00:56 Yep, so that would be 6.

01:00 So when we have c is equal to zero, that would just give us the problem, x squared.

01:07 So that'll look something kind of like this here.

01:15 So something kind of like that.

01:20 And so this is when c is equal to zero, we get y is equal to x squared.

01:26 Now when c is equal to negative 1, we get x squared, minus 1, which would be x squared plus 1.

01:39 So it's going to look something kind of like this for that level curve.

01:47 And so this here is when c, actually let me go ahead and draw going up a little bit more on either side.

01:57 Just so i can draw above this.

02:00 So this is going to be c is equal to 1, or minus 1, i mean.

02:07 And so this is y is equal to x squared plus 1.

02:11 And then for when c is equal to negative 2, that would give us x squared plus 2...

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