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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.
Link: http://imageshack.us/photo/my-images/840/64800067.png/
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)?
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)?
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.
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)!
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.
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.
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.
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.
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.