Partial Differentiation and Conic Asymptotes

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In summary, the conversation discusses the use of partial differentiation to find the equations of asymptotes for different conics. It is mentioned that for a hyperbola, the equations of its asymptotes are the lines y=(b/a)x and y=-(b/a)x. The concept of asymptotes as lines that the curve tends to for large x and y values is also discussed. The conversation ends with a request for clarification on the meaning of partial differentiation.
  • #1
chaoseverlasting
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If you partially differentiate the equation of a hyperbola w.r.t. x or y do you get the equation of its asymptotes? I know that if you do partially differentiate it, the two lines that you get, intersect at its center.

This is true for any conic and pair of straight lines. What about other conics? I.e, if you partially differentiate the equations for a parabola, ellipse, circle, what exactly do you get?
 
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  • #2
Investigate this link (and click 'next', too): http://www.maths.abdn.ac.uk/~igc/tch/ma1002/diff/node48.html".
 
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  • #3
It might help to know that "asymptote to a curve" is essentially the same thing as "tangent line to a curve at some point at infinity".

You can shift your perspective (so that the points at infinity become ordinary points) by using a transformation, such as

(s, t) = (y/x, 1/x).

How does your hyperbola, and its asymptotes, look in (s, t)-coordinates?


Oh, if it helps, the reverse transformation is:

(x, y) = (1/t, s/t)
 
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  • #4
chaoseverlasting said:
If you partially differentiate the equation of a hyperbola w.r.t. x or y do you get the equation of its asymptotes? I know that if you do partially differentiate it, the two lines that you get, intersect at its center.

This is true for any conic and pair of straight lines. What about other conics? I.e, if you partially differentiate the equations for a parabola, ellipse, circle, what exactly do you get?
It's not exactly clear what you mean. The equation of a hyperbola in "standard position" is
[tex]\frac{x^2}{a^2}- \frac{y^2}{b^2}= 1[/itex]
If you differentiate that with respect to x and y you get, respectively,
[tex]\frac{2x}{a}= 0[/tex] and
[tex]\frac{2y}{b}= 0[/tex]
or x= 0, y= 0, certainly NOT the equations of its asymptotes, which are
y= (b/a)x and y= -(b/a)x.

The asymptotes of a hyperbola are the lines the hyperbola tends to for large x and y. One way of finding them is to say that, for very large x, y, "1" is very small and can be neglected in comparison to the other terms: the curve will satisy, approximately, for large x, y
[tex]\frac{x^2}{a^2}- \frac{y^2}{b^2}= 0[/tex]
[tex]\frac{x^2}{a^2}= \frac{y^2}{b^2}[/tex]
[tex]\frac{x}{a}= \pm \frac{y}{b}[/itex]
the equations of the asymptotes.
 
  • #5
Thank you, that helps a lot. I am not familiar with partial differentiation, I just know certain specific applications of it, such as the ones I mentioned above, what does it mean really?
 

Related to Partial Differentiation and Conic Asymptotes

What is an asymptote of an hyperbola?

An asymptote of an hyperbola is a straight line that the hyperbola approaches but never touches.

How many asymptotes does a hyperbola have?

A hyperbola has two asymptotes.

How do you find the equations of the asymptotes of a hyperbola?

The equations of the asymptotes of a hyperbola can be found by taking the limits of the hyperbola's equations as the x and y values approach infinity.

What is the significance of asymptotes in hyperbolas?

Asymptotes help define the shape and behavior of a hyperbola. They also help determine the domain and range of the hyperbola's function.

Can a hyperbola have more than two asymptotes?

No, a hyperbola can only have two asymptotes since it is a conic section and follows the general equation of (x^2 / a^2) - (y^2 / b^2) = 1, which only has two variables for x and y.

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