Use the graphs of the level curves ( c -values evenly spaced) of the function f to write a descr

Use the graphs of the level curves ( c -values evenly...

Key Concepts

Level Curves

Level curves are the sets of points in the domain of a function where the function takes on a constant value. They give a two?dimensional projection of the behavior of the function, allowing one to see how the function values change and how the domain is partitioned by these constant levels.

Graph Construction from Level Curves

Using level curves, one can imagine constructing a three-dimensional graph by assigning a z-value to each curve equal to its constant value. This process creates a surface where each horizontal 'slice' or cross-section of the surface corresponds to one of the given level curves, thus providing a visual representation of the function's behavior in space.

Non-Uniqueness of the Graph

Knowing only the level curves of a function does not uniquely determine the function’s graph because there are infinitely many functions that share the same level sets. Any function that is a monotonic transformation of the original will have the same level curves but a different vertical scaling, indicating that the exact shape and steepness of the graph in the z-direction are not fixed by the level curves alone.

Monotonic Transformations

Monotonic transformations refer to functions applied to the output of an original function that preserve the order of the values. Since level curves depend only on where the function’s value is constant, applying any strictly increasing or decreasing function to the output will yield the same level sets, thereby illustrating why the graph is not unique when generated solely from level curve data.

Transcript

00:01 We're given some information about the graph of y prime and y double prime.

00:05 We want to determine what the graph of y could look like based on that information.

00:11 What i'm going to do is identify the points on here where either y prime is zero or were y double prime is zero.

00:21 I've drawn y prime in green and y double prime in blue.

00:27 On the y prime graph, we can see that it crosses.

00:31 The x -axis at a few points.

00:34 And i'm just going to kind of give some estimates.

00:37 At about negative 5, 6 it crosses, at around a quarter it crosses, and around 1 .5 it crosses.

00:53 So above my number line, i'm going to say at those points, y prime is zero.

00:59 For y double prime, it crosses the x -axis at around negative 1 -half, and it also crosses at about 1.

01:16 Now, let's film in the sign of y -prime and y -double prime at the missing points.

01:24 For negative 1 -half, the green graph has a positive y coordinate.

01:29 So, y -prime is positive there, and at y -equals 1, the green graph is below.

01:35 The x -axis, so y prime is negative.

01:39 At the x value of negative 5 .6, y double prime has a positive, well, the coordinate there for the y -coordinate is a positive, so y -double prime is positive.

01:50 At x equals a quarter, y double prime is a negative coordinate, and at one and a half, it is a positive...

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