Equation of Level Curve

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In summary, at the point (2,1), the equation of the level curve for f(x,y) = yx^2 - y^2 is f(x,y) = yx^2 - y^2 = 3. The equation of the tangent line at this point is 10 = 4x + 2y.
  • #1
DeadxBunny
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Original question: Find an equation of the level curve of f(x,y) = yx^2 – y^2 through the point (2,1) and find an equation of the tangent line to the level curve at this point.

I think I have found the equation of the tangent line to the level curve: 10 = 4x + 2y (is this correct??), but I have no idea how to find the equation of the level curve. Please help! Thank you!
 
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  • #2
You've done the hard part!

at (2,1), x= 2, y= 1 so f(x,y)= f(2,1)= 1(4)- 1= 3. The whole point of a "level" curve is that the function stays at the same "level", i.e. the same value.
The level curve is f(x,y)= yx2- y2= 3.

Yes, your tangent line is correct.
 
  • #3


To find the equation of the level curve, we need to set the function f(x,y) equal to a constant value, say k. This will give us a curve that represents all the points (x,y) where the function has the same value, or in other words, the level curve.

In this case, we have f(x,y) = yx^2 - y^2. Setting this equal to a constant k, we get yx^2 - y^2 = k. We can rearrange this equation to get y(x^2 - 1) = k, and then solve for y to get y = k/(x^2 - 1).

Now, we can substitute the given point (2,1) in this equation to find the specific value of k. Plugging in x=2 and y=1, we get 1 = k/(4-1), which gives us k = 3. So the equation of the level curve is y = 3/(x^2 - 1).

To find the equation of the tangent line to this level curve at the point (2,1), we can use the derivative of the function f(x,y) with respect to x. This will give us the slope of the tangent line at any given point on the curve.

The derivative of f(x,y) with respect to x is 2yx, so at the point (2,1), the slope of the tangent line is 2(1)(2) = 4. We can use this slope and the given point (2,1) to find the equation of the tangent line using the point-slope form: y - y1 = m(x - x1). Plugging in the values, we get y - 1 = 4(x - 2), which simplifies to y = 4x - 7.

So the equation of the tangent line to the level curve at the point (2,1) is y = 4x - 7. Your answer of 10 = 4x + 2y is incorrect. I hope this helps clarify the process for finding both the equation of the level curve and the tangent line.
 

1. What is the equation of a level curve?

The equation of a level curve is a mathematical representation of points on a two-dimensional surface that have the same value for a given function. It can also be described as a contour line or isoline.

2. How is the equation of a level curve derived?

The equation of a level curve is derived by setting the given function equal to a constant value and solving for the variables. This results in a curve on the surface where all points have the same function value.

3. What is the purpose of the equation of a level curve?

The equation of a level curve is used to visualize and analyze a function on a two-dimensional surface. It can help identify areas of constant change, critical points, and overall patterns of the function.

4. Can the equation of a level curve be used for any function?

Yes, the equation of a level curve can be used for any function that has multiple inputs and outputs. It is commonly used in fields such as mathematics, physics, and engineering to understand and model complex systems.

5. How does the equation of a level curve relate to contour maps?

The equation of a level curve is essentially a contour line on a two-dimensional surface, similar to contour lines on a topographic map. Contour maps use a series of level curves to represent changes in elevation, while the equation of a level curve uses a single curve to represent changes in a function value.

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