Level Curves and Partial Derivatives

• MHB
• Lancelot1
In summary, the conversation discusses the relationship between partial derivatives and level curves, specifically at the point P. It is mentioned that a partial derivative is the limit of a slope and that the slope can be determined by looking at the values of the level curves in the x and y directions. The discussion also covers how to determine whether a partial derivative is positive or negative based on the spacing of the level curves. The conversation ends with a question about the other partial derivatives and an explanation that the rate of change for the function increases as the level curves get closer together.
Lancelot1
Hello everyone,

I am trying to solve this wee problem regarding partial derivatives, and not sure how to do so.

The following image shows level curves of some function $z=f(x,y)$ :

View attachment 7998

I need to determine whether the following partial derivatives are positive or negative at the point P:

$f_{x} , f_{y} , f_{xx} , f_{yy} , f_{xy} , f_{yx}$

I am not sure how to relate the partial derivatives to the level curves. I know that partial derivatives at a point are slopes of a curve created when we fix a plane such as x=a or y=b. Where and how do I see it in level curves ?

Cheers !

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Hi Lancelot,

Remember that a partial derivative is the limit of a slope.
More specifically:
$$f_x(P)=\lim_{h\to 0}\frac{f(P+(h,0)) - f(P)}{h} \approx \frac{f(P+(h,0)) - f(P)}{h}$$
Let's start at P and make a step with size $h>0$ to the right.
To the right of P we see that it has value 4.
And at P itself we have value 6.
So we have:
$$f_x(P)\approx \frac{4 - 6}{h} < 0$$
That is, the surface slopes down in the x-direction at point P.For the second partial derivative with respect to x we have:
$$f_{xx}(P) \approx \frac{f_x(P+(h,0)) - f_x(P)}{h}$$
We see that to the left the level curves are closer together than to the right.
That means that at left the slope is steeper than at the right.
So we take a small negative slope (at the right) minus a bigger negative slope (at the left), and end up with a positive number.
Or put otherwise, we begin with a steep downward slope, and have to add something positive to it to get a less steep slope.
So:
$$f_{xx}(P) > 0$$

How far do you get with the other partial derivatives?

When I try to look at the derivate by y, I go up and down the y-axis, the level curves are constant there, am I wrong ?

Lancelot said:
When I try to look at the derivate by y, I go up and down the y-axis, the level curves are constant there, am I wrong ?

If we go up the y-axis starting from some arbitrary point, the level curve constants go up, don't they?
It means that $f_y > 0$.
That leaves the question how fast they go up.
Since the level curves are closer together for higher y, it means that the rate they go up accelerates.
Consequently $f_{yy} > 0$.

1. What are level curves?

Level curves are curves on a two-dimensional graph that represent the points where a function has the same output value. They are also known as contour lines and are used to visualize the behavior of a function.

2. How are level curves related to partial derivatives?

Level curves are closely related to partial derivatives because they show the rate of change of a function in a specific direction. The slope of a level curve is equal to the partial derivative of the function with respect to that direction.

3. How do you find the partial derivative of a multivariable function?

To find the partial derivative of a multivariable function, you hold all other variables constant and differentiate the function with respect to the variable of interest. This results in a new function that represents the rate of change of the original function in the direction of the variable of interest.

4. What is the significance of the gradient vector in level curves and partial derivatives?

The gradient vector is a vector that points in the direction of the greatest increase of a multivariable function. It is also perpendicular to the level curves of the function. Therefore, the gradient vector is used to find the direction in which a level curve will increase the fastest.

5. How are level curves and partial derivatives used in real-world applications?

Level curves and partial derivatives are used in many scientific fields, such as physics, economics, and engineering. They are used to analyze and optimize functions in these fields, such as finding the maximum or minimum values of a function. They are also used in data analysis and machine learning to model and predict relationships between variables.

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