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Cartesian Curve Tracing

The following steps are very useful in tracing a cartesian curve

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.

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

Find out the asymptotes of the curve.

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

If possible, solve the equation of the given curve for y in terms of x and observe how y varies from

Find out the values of

At such points y generally changes its character from an increasing function of

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

Natural Logarithms
First Principles Differentiation
Asymptotes
Tangents At The Origin
Radius Vector, Tangent Angle
Intersection Angle Two Curves
Arc Length Derivative
Area Bounded By Closed Curve
Area Under Cartesian Curve
Area Under Polar Curve
Algebraic Curves Asymptotes
Polar Curve Asymptotes
Cartesian Curve Tracing
Cauchy Mean Value Theorem
Centre Of Curvature
Centre Of Gravity
Centre Of Gravity Of Plane Area
Centre Of Gravity-Volume
Chord Of Curvature
Clairauts Equation
Concave Curve Concavity Test
Continuity Of A Function
Elementary Functions Continuity
Curvature
Definite Integral
Definite Integrals
Derivability And Continuity
Derivative Differentiation
Implicit Function Derivative
Parametric Derivative
Transformation Derivatives
Differential Equation
Differential First Order Degree
Homogeneous Reducible Equation
Equations Solvable For P
Equations Solvable For X
Equations Solvable For Y
Eulers Homogeneous Function
Exact Differential Equation
Exponential Function
Curves Family Envelope
Extreme Values Test
Differential Equation Formation
Chain Rule Function Derivative
Function Of Two Variables
Integral Fundamental Theorem
Higher Order Derivatives
Homogeneous Equations
Homogeneous Functions
Homogeneous Linear Differential
Hyperbolic Functions
Indefinite Integral
Infinite Limits
Inflexion Point
Integrals With Infinite Limits
Partial Fraction Integration
Integration By Parts
Integration By Substitution
Trigonometric Function Integration
Intermediate Forms
Intersection Of A Curve
Intervals
Intrinsic Equation
Function Inverse Derivative
Inverse Trigonometry Function
Irrational Functions
L Hospitals Rule
Lagrange Mean Value Theorem
Left, Right Hand Derivatives
Left, Right Hand Limits
Cartesian Curve Arc Length
Parametric Curve Arc Length
Function-Limit, Continuity
Limit Of A Function
Linear Differential Equations
Oblique Asymptotes
Linear Constant Coefficients
Maclaurin Mean Value Theorem
Maxima And Minima
Inertia Moment
Multiple Points
Curves One Parametric Family
Parametric Curve Tracing
Partial Derivatives
Particular Integrals
Pedal Equation
Point Neighbourhood In Plane
Polar Co-ordinates
Polar Curve Tracing
Double Points Position, Nature
Cartesian Curvature Radius
Rational Functions
Real Number Modulus
Real Numbers
Rigorous Second Derivatives
Rolle-Continuous Function
Extreme Value Second Derivative
Second Order Partial Derivative
Subtangent And Subnormal
Summation Of Series
Standard Integrals Table
Tangent Equation
Taylors Mean Value
The Operator D
Composite Function Derivative
Function Total Differentiation
Trigonometric Functions
Uniform Continuity
Revolution Solid Volume
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