# Level Curves Graph and Partial Derivatives.

• nikkihendrix
In summary, the conversation discussed the level curves of a function and arranging five quantities in ascending order based on their partial derivatives. The correct order is f_x (S), f_y(Q), 0, f_y(R), f_x(P). The rate of change, represented by the partial derivatives, can indicate if the function is increasing or decreasing and how quickly. Explicit values for the partial derivatives can help understand the behavior of the function.
nikkihendrix
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nikkihendrix said:
The level curves of a function z=f(x,y) are shown in the link provided.

f (sub x) (P), f (sub y) (Q), f (sub y) (R), f (sub x) (S), and the NUMBER 0.

Assume that the scales along the x and y axes are the same. Arrange the following five (5) quantities in ASCENDING order. Give a brief explanation for your reason.

I figured 0 is the least because there are no level curves.

I know that f_y (Q) and f_x (S) are negative. Also, f_x (P) and f_y (R) are positive.
But f_x (S) have more compact level curves therefore it's bigger than f_y (Q). Similarly the reason why f_y (R) is bigger, right?

Does it go: 0, f_y (Q), f_x (S), f_x (P), f_y (R)?

Hi nikkihendrix!

If fy(Q) and fx(P) are negative, then 0 is larger than these two values, right?? So 0 can't be the least.

Furthermore, saying that fx(S) has more compact level curves that fy(Q) means that it's absolute value is larger. That doesn't mean that fx(S) actually is larger...

nikkihendrix said:
How would I determine whether f_x (S) goes before or after f_y (Q)?

Or f_x (P) is greater or less than f_y (R)?

So, it's f(sub x) (P) < 0 < f(sub y) (Q) < f(sub y) (R) < f(sub x) (S)

f(sub x) (P) has negative slope.
f(sub y) (Q) has positive slope. (Flipping x and y can be seen)
f(sub y) (R) less than or equal to f(sub x) (S)?

The partial derivative indicates the rate of change. For example, a partial derivative of -1 means that I'm going downwards with a rate of 1. A partial derivative of -10 means that I'm going downwards with a rate of 10: this is much faster!
So I expect the level lines with a partial derivative of -10 to lie much closer together than with a partial derivative of -1.

The same way, I expect the level lines with a partial derivative of 10 to lie much closer together than with a partial derivative of 1.

If the partial derivative is 0, then you neither increase neither decrease in that direction.

So, in our case: in which case of fx(S) and fy(Q) are you going downwards the fastest?

In which case of fx(P) and fy(R) are you going upwards the fastest?

nikkihendrix said:
f_x (S) is going downwards faster than f_y (Q).

What does it mean by ascending order than? f_y(Q) < f_x (S) then.

If you calculate partial derivatives. Which ones is going downward faster: -10 or -1?

nikkihendrix said:
-10. oh!

okay so...it's actually suppose to be
f_x (S), f_y(Q), 0, f_y(R), f_x(P)!

Which one is increasing fastest? +10 or +1? What does that mean for your last two values?

Indeed!

When given a situation like in the picture. Try to give explicit values to the partial derivatives. It makes it easier to see what's going on!

Seems good!

Very Interesting !

## What are level curves graphs?

Level curves graphs, also known as contour plots, are two-dimensional representations of three-dimensional functions. They show how the output of a function varies at different input values, with each contour line representing a constant output value.

## How are level curves graphs useful?

Level curves graphs are useful for visualizing and understanding the behavior of multivariable functions. They can help identify critical points, such as maxima and minima, and provide insights into the overall shape of the function.

## What is a partial derivative?

A partial derivative is a type of derivative that measures the rate of change of a multivariable function with respect to one of its variables while holding all other variables constant. It is denoted by ∂f/∂x, where f is the function and x is the variable of interest.

## How are partial derivatives related to level curves graphs?

Partial derivatives are useful for determining the slope of a level curve at a given point. The direction of steepest ascent or descent at that point is perpendicular to the level curve, and the partial derivatives in each variable determine the slope in that direction.

## What are some real-world applications of level curves graphs and partial derivatives?

Level curves graphs and partial derivatives are used in various fields, such as economics, physics, and engineering, to model and analyze complex systems. They can help optimize processes, predict outcomes, and understand relationships between variables. For example, in economics, they can be used to analyze supply and demand curves, while in physics, they can be used to study electric and magnetic fields.

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