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Home » Math Homework Help » Calculus Homework Help » Cartesian Curve Tracing
Cartesian Curve Tracing
The following steps are very useful in tracing a cartesian curve ƒ(x, y) = 0.
   
1. Symmetry
   
(i) The curve is symmetrical about x-axis if all powers of y in the equation of the given curve are even [∵ƒ(x, y) = ƒ(x, -y)].
   
(ii) The curve is symmetrical about y-axis if all powers of x in the equation of the given curve are even [∵ƒ(x, y) = ƒ(-x, y)].
   
(iii) The curve is symmetrical about the line y = x if the equation of the given curve remains unchanged on interchanging x and y.
   
2. Origin

Find out if the origin lies on the curve. If it does, find out the tangent or tangents at the origin. In case the origin is a multiple point, find out its nature.
   
3. Intersection with the co-ordinate axes

Find out the points of intersection of the curve with co-ordinate axes and the tangents at such points.
   
4. Asymptotes

Find out the asymptotes of the curve.
   
5. Region

Find out the regions of the plane in which no part of the curve lies. To determine such regions we solve the given equation for y in terms of x or vice-versa. Suppose that y becomes imaginary for x > a, the curve does not lie in the region x > a.
   
6. Solving the equation

If possible, solve the equation of the given curve for y in terms of x and observe how y varies from –∞ to +∞.
   
7. Critical points

Find out the values of x at which dy/dt = 0.

At such points y generally changes its character from an increasing function of x to a decreasing function of x or vice-versa.
   
8. Points of inflexion

Find out the points of inflexion (given by ) and the regions of convexity and concavity of the curve.

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