Exploring Level Curves for f(x,y)=e^-(2x^2+2y^2)

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In summary, to calculate level curves for f(x,y)=e^-(2x^2+2y^2), set the equation equal to a constant, c, and solve for x or y. This will result in a general solution. It is important to note that c must be between 0 and 1 for the level curves to exist.
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cue928
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I am being asked to calculate level curves for the following equation:
f(x,y)=e^-(2x^2+2y^2) but I do not know where to start. Any advice on first steps would be greatly appreciated.
 
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Set e^-(2x^2+2y^2) = c, where c is a constant. Solve for x or y.

Once you do that, see if you can come up with a more general solution.
 
  • #3
One obvious point: if [itex]e^{-(2x^2+ 2y^2)}= c[/itex] then [itex]-(2x^2+ 2y^2)= ln(c)[/itex] so that [itex]x^2+ y^2= -ln(c)/2[/itex] which is only possible if -ln(c)< 0 which means 0< c< 1. What figures will level curves be?
 

1. What is a level curve?

A level curve is a curve on a graph that connects points with the same value. It is also known as an isopleth or contour line.

2. How do you calculate level curves?

The calculation of level curves involves setting a specific value and plotting all the points on the graph that have that same value. This can be done by finding the solution to a mathematical equation or by using software programs like MATLAB or Mathematica.

3. Why are level curves important in science?

Level curves are important in science because they help visualize and analyze data in a two-dimensional space. They can also provide insights into the relationships between different variables and help identify patterns or trends.

4. How do you interpret level curves?

The interpretation of level curves depends on the context and the type of graph being used. In general, level curves that are closer together indicate a steeper slope, while level curves that are further apart indicate a gentler slope. Additionally, the direction of the curves can also provide information about the data being represented.

5. What are some common uses of level curves in different fields of science?

Level curves have various applications in different fields of science. For example, in geography, they are used to represent elevation on topographic maps. In meteorology, they are used to show areas with the same air pressure. In economics, they can be used to illustrate production possibilities and utility curves. Additionally, level curves are also commonly used in physics, engineering, and biology to analyze and model various phenomena.

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