Curve sketching ~ Easy to understand maths

Curve sketching includes techniques that can be used to draw a rough idea of overall shape of a plane curve. For curve sketching we need to perform certain steps on the given equation of the curve.

For curve sketching we need to understand certain terms

Multiple points

A point through which two or more than two branches of a curve pass is called a multiple point.
In particular, if two branches of a curve pass through a point, the point is called a double point.

Classification of double points

There are three kinds of double points.

1. Node

A node is a point in the curve through which two real branches of the curve pass and two tangents at which are real and distinct. Thus P is a node.

2. Cusp

A double point on the curve through which two real branches of the curve pass and thw tangent at which are real and coincident is called cusp. Thus P is a cusp.

3. Conjugate point

A conjugate point on a curve is ap point in tbe neighbourhood of which there are no other real point of the curve.
The two tangent at a conjugate point are in general imaginary but sometimes they may be real.

Curve tracing

We shall keep in mind the following point for tracing the graph of the equation $f(x,y)=0$

1. Symmetry

Curve given by $f(x,y)=0$ is symmetric about

1. x-axis if it unchanged on changing y to -y i.e, if $f(x,-y)=f(x,y)$

2. y-axis if it unchanged on changing x to -x i.e, if $f(-x,y)=f(x,y)$

3. The origin if it is unchanged on changing x to -x and y to -y i.e, $f(-x,-y)=f(x,y)$

4. The line y=x if it is unchanged on changing x to y and y to x i.e, $f(x,y)=f(y,x)$

5. The line y=-x if it is unchanged on changing x to-y and y to -x i.e, $f(-y,-x)=f(x,y)$ .

2. Domain and range: find the domain and range.

3. Origin

Check whether origin lies on the curve. If curve passes through origin then find the tangent at the origin and also determine whether origin is node, cusp or an isolated point.

4. Asymptotes

Find all the asymptotes of the curve and the position of the curve relative to its asymptotes.

5. Point of intersection

Find the point of intersection of the curve with co-ordinate axis and obtain the equation of the tangent at these points. If any of these is a double point, then find the nature of the double point.

Also find some other points on the curve by giving suitable values to x.

6. Maxima and minima

Find the point where the function has maximum value or minimum value. Also find the maximum and minimum value at each point.

7. Points of inflexion

(a) find the interval of

(1) increase and decrease of an function

(2) concavity and convexity of the curve.

(b) Also find the points of inflexion, if any.

Examples

Q1. Trace the curve $y=x^{3}$ .

Sol.
The equation of the curve is   $y=x^{3}$

1. Symmetry : The curve is symmetrical about the origin as the equation of the curve does not change on changing x to -x and y to -y. The curve is symmetrical about the origin means the curve is symmetrical in opposite quadrant.

2. Origin : The curve passes through the origin
The tangent at the origin are y=0 i.e, x-axis

3. Domain: Here   $D_{f}=(-\infty ,\infty )$
Now   $x>0\, \, \Rightarrow \: \, y>0\: \, and\: \: x<0\: \: \Rightarrow y<0$
Curve lies either in the first quadrant or in the third quadrant.

4. Point of intersection : The curve meet x-axis where y=0. Putting y=0 in (1), we get x=0. Therefore curve meet x-axis in (0,0). Also the curve meet y-axis in (0,0).

5. Asymptotes : The curve has no asymptotes.

6. Increasing and decreasing :   $\frac{\mathrm{d} y}{\mathrm{d} x}=3x^{2}$
   $\frac{\mathrm{d}^2 y}{\mathrm{d} x^2}=6x,\, \, \frac{\mathrm{d}^3 y}{\mathrm{d} x^3}=6$
Now $\frac{\mathrm{d} y}{\mathrm{d} x}=0\: \, \, \, \, \Rightarrow \, \, 3x^{2}=0\, \,\: \Rightarrow x=0$
Tangent at x=0 is parallel to x-axis
Now $\frac{\mathrm{d} y}{\mathrm{d} x}=3x^{2}>0\, \, \: \forall \, \, real\, \, \, x$
Curve is increasing for all real x. $\frac{\mathrm{d}^2 y}{\mathrm{d} x^2}=6x>0\: \, when \: \, x>0$

Curve is concave upward in $(0,\infty )$
Similarly curve is concave downward in $(-\infty ,0)$

Again   $\frac{\mathrm{d}^2 y}{\mathrm{d} x^2}=0\, \, at\, \, \, (0,0)$ , but   $\frac{\mathrm{d} ^3y}{\mathrm{d} x^3}=6\neq 0\, \, at\: (0,0)$
(0,0) is a point of inflexion.

7. Additional points : Now   $y\rightarrow \infty \:\, as\, \: x\rightarrow \infty$ and also   $y\rightarrow- \infty \:\, as\, \: x\rightarrow -\infty$
A rough sketch of the curve is shown in the figure.

Q2. Trace the curve $x^{3}+y^{3}=3axy,\: \: a\geq 0$ .
Sol.
The equation of curve is $x^{3}+y^{3}=3axy,\: \: a\geq 0$ (1)

1. Symmetry: The given equation (1) does not change when x is changed to y and y is changed to x.
Curve is symmetrical about the line y=x.

2. Origin: The curve passes through the origin.
The tangent at origin are given by xy=0
i.e, x=0, y=0. There tangent are different.
Origin is node.

3. Asymptotes: (1) can be written as
$(x+y)(x^{2}-xy+y^{2})-3axy=0$
Asymptote (if any) parallel to x+y=0 is given by $x+y-\lim_{x\rightarrow \infty }\frac{3axy}{x^{2}-xy+y^{2}}=0,\: where\, \, y=-x$
$x+y-\lim_{x\rightarrow \infty }\frac{-3ax^{2}}{x^{2}-x^{2}+x^{2}}=0$

$x+y+\lim_{x\rightarrow \infty }\frac{3ax^{2}}{3x^{2}}=0$
$x+y+a=0$
This is the only asymptote of the curve.

4. Point of intersection with axes

Putting x=0 in (1), we get y=0
Putting y=0 in (1), we get x=0
Curve meets axes in (0,0) only.

Putting y=x in (1), we get,
$x^{3}+x^{3}=3ax^{2}\: \, or \: \: x^{2}\left ( 2x-3a \right )=0$
$\therefore \, \: \: \: \: \: \: \: x=0,\frac{3a}{2}$

$\therefore \, \: \: \: \: \: \: \: y=0,\frac{3a}{2}$
Line y=x meet the curve in (0,0) and $\left ( \frac{3a}{2},\frac{3a}{2} \right )$

5. Region
From (1), it is clear that x and y both cannot be negative as in that case L.H.S of (1) is negative whereas R.H.S of (1) is positive.
No portion of the curve lies in the third quadrant.
A rough sketch of the curve is shown in the figure.

Here A is $\left ( \frac{3a}{2},\frac{3a}{2} \right )$ and blue curve is (1).

☆working method for tracing parametric curve

Case1. Eliminate the parameter if possible and get the corresponding cartesian equation of the curve which can be traced as done earlier.
Case2. If the parameter cannot be easily eliminated from the given equation, then we proceed like this :

1. Symmetry

1.If   $x=f(t)$ is an even function of t and   $y=\phi (t)$ is an odd function of t, then the curve is symmetrical about x-axis.

2.If   $x=f(t)$ is an odd function of t and   $y=\phi (t)$ is an even function of t, then the curve is symmetrical about y-axis.

3.If   $x=f(t)$ and   $y=\phi (t)$ are both function of t, then the curve is symmetrical in opposite quadrant.

2. Origin

If putting x=0, we get a real value of t, which makes y equal zero, then the curve passes through the origin.

3. Axis intersection : Find the point of intersection of the curve and coordinate axis.

4. Limitation

If possible, find the greatest and the least values of x and y which give us lines parallel to axis between which the curve lies or does not lie.

5. Points

Find the point where $\frac{\mathrm{d} y}{\mathrm{d} x}=0,\: \frac{\mathrm{d} y}{\mathrm{d} x}\rightarrow \infty$ .

6.Region

Find the region in which curve does not lie.

Consider the sign of $\frac{\mathrm{d} y}{\mathrm{d} t} \: \, and\, \: \frac{\mathrm{d} y}{\mathrm{d} t}$ .

Consider the value of $x,y,\frac{\mathrm{d} x}{\mathrm{d} t},\frac{\mathrm{d} y}{\mathrm{d} t},\frac{\mathrm{d} y}{\mathrm{d} x}$ .

7. Asymptotes: Find the asymptotes, if any.

Example

Q3. Trace the curve $x=a\left ( \Theta +\sin \Theta \right ),\: y=a\left ( 1+\cos \Theta \right )$
Sol.
The equation of the curve are
$x=a\left ( \Theta +\sin \Theta \right ),\: y=a\left ( 1+\cos \Theta \right )$
Here the parameter Θ cannot be easily eliminated.

1. Symmetry. The curve is symmetrical about the axis of y for $(\Theta +\sin \Theta )$ is an odd function of θ and (1+cosθ) is an even function of θ.

2. origin. The curve does not pass through the origin.

3. Intercepts. It meet the x-axis when
y=0 i.e, 1+cosθ=0
or cosθ=-1 i.e, θ=π,-π
the point of intersection with the x-axis are

$A(a\pi ,0),\: A'(-a\pi ,0)$ . Again it meet the y-axis when x=0
i.e, θ+sinθ=0 or sinθ=-θ or θ=0
it meets the axis of y at B(0,2a).

4. Asymptotes. There are no asymptotes.

5. Points. We have $\frac{\mathrm{d} x}{\mathrm{d} \Theta }=a(1+\cos \Theta );\: \frac{\mathrm{d} y}{\mathrm{d} \Theta }=-a\sin \Theta$
   $\therefore \: \: \frac{\mathrm{d} y }{\mathrm{d} x} =\frac{-a\sin \Theta }{a(1+\cos \Theta )}=-\frac{2\sin \frac{\Theta }{2}\cos \frac{\Theta }{2}}{2\cos ^2\frac{\Theta }{2}}=-\tan \frac{\Theta }{2}$
$\therefore \, \, \frac{\mathrm{d} y}{\mathrm{d} x}\, \, when \, \: \Theta =0$
i.e, at (0,2a), the tangent is parallel to the x-axis.
Also $\frac{\mathrm{d} y}{\mathrm{d} x}\rightarrow \infty \: \: \: when\: \Theta =\pi,- \pi$
At $A(a\pi ,0),\: A'(-a\pi ,0)$ , the tangent is perpendicular to the axis of x.

6. Region. For all value of   $\Theta ,\: \frac{\mathrm{d} x}{\mathrm{d} \Theta }$ is +ve

x is always increases with θ
   $\frac{\mathrm{d} y}{\mathrm{d} \Theta }\: is \: +ve\: for\: -\pi \leq \Theta \leq \pi .$
Hence y increases when θ increases from -π to 0 and y decreases when θ increases from 0 to π.

Hence approximately, the shape of the curve is as shown in the diagram.

Tuesday 12 March 2019

Curve sketching

Classification of double points

1. Node

2. Cusp

3. Conjugate point

Curve tracing

1. Symmetry

3. Origin

4. Asymptotes

5. Point of intersection

6. Maxima and minima

7. Points of inflexion

Examples

☆working method for tracing parametric curve

1. Symmetry

2. Origin

4. Limitation

5. Points

6.Region

Example

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