Sketch a contour diagram for the function with at least four labeled contours. Describe in words

Sketch a contour diagram for the function with at least four...

Key Concepts

Contour Diagram

A contour diagram is a two-dimensional representation of a three-dimensional function where curves connect points of equal function value. These curves, or level curves, provide insight into the function's behavior by revealing regions of constant values. In the context of multivariable functions, these diagrams help visualize gradients and the overall topography of the function.

Level Curves

Level curves are the sets of points in the domain where the function takes on a specific constant value. For a function f(x, y)=cos(sqrt(x²+y²)), each level curve corresponds to the equation cos(sqrt(x²+y²)) = c. This approach transforms a two-variable problem into a family of one-variable equations, simplifying the analysis of the function’s behavior across its domain.

Polar Coordinates

Many functions, especially those involving sqrt(x²+y²), naturally lend themselves to polar coordinates where r = sqrt(x²+y²) and ? is the angle. This conversion simplifies the description of level curves for radially symmetric functions. In this example, the function becomes f(r)=cos(r), turning the problem into understanding how the cosine function behaves as the radius changes.

Periodic Functions

The cosine function is periodic, meaning it repeats its values at regular intervals. This periodicity directly affects the spacing of the level curves in the contour diagram. For instance, the cosine function reaches the same value at regular intervals of its argument, which translates into regularly spaced circular contours for the original function when considered in terms of the radial coordinate.

Transcript

0:00 Hello there.

00:01 Okay, so for the following exercise, we have the following function that is defined as a cosine of the square root of x square plus y square.

00:14 And we need to sketch at least four of its contour for this counter diagram and then we need to describe in the word to what's corresponds to.

00:24 Well basically here, if we just to explain you why we have, what we are obtaining the and what for one curve for one for one value of contour we have at least two of these circles well basically let's remember that the cosine is an oscillating function it's going to be it's going to increase and decrease and that's why we have this kind of a structure because here we're taking the cosine of the square root of x squared plus y square it means that we are taking if we visualize this plot in three dimensions it will be it will look more or less like this okay like this in the three dimensions so when you pick some value for a contour you're going to obtain several circles that corresponds to these ones that appear here okay so that's more less the three -dimensional thinking now if we go to the analytical thinking what happened is that if we assign some value, so we have the following x square plus y square, it's equals to some value here, some value c, and it's important that this value z should be between one and minus one, because the co -osine is defined on that interval.

01:54 Then we take the arc cosine, so we obtain x square plus y square equals to the arc, cosine of c square and this is the equation of a circle now it's important that even though that we have whipped in some for some circles for some values of x and the y this determined value of the r cosine this can also take for this cosine here it is easier to explain here the cosine of the cosine of these this square root will be equal to c for some specific value but also if we take pi plus that value let's say here x tilda plus y tilde squared square and square roots okay so for these values and for if we add pi and some constants to that to that pie we're going to obtain the remaining circles okay, that's because of the nature of the cosine of this trigonometric equation.

03:09 But in essence we have some circles and i give you first the geometrical meaning its intersection of this kind of oscillating like a water drop wave that is constructed in three dimensions.

03:27 And yes, here we have these three diagrams...

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