Easy to understand maths

Friday, 1 March 2019

Posted by Parth sharma on March 01, 2019 with 3 comments

Multiple point

It refers to a point on a curve through which two or more branches of the curve pass.

Singular point

A point on the curve at which the curve behaves in an extraordinary manner is called a singular point.
There are two types of singular points:
1. Points of inflection
2. Multiple point

Point of inflection:-A curve's Inflection point is the point at which the curve's concavity changes. A point that separates the convex part of the curve from part of the curve is called a point of inflection.

Multiple point:- A point on the curve through which more than one branches of the curve pass is called a multiple point.

Double point:- A point on the curve through which two branches of the curve pass 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.

Note 1. Conjugate point is also called isolated points.

Note 2. The determination of the nature of double point depends basically on the nature of the two branches of the curve passing through it and not on the tangents to the curve at that point. Generally when the tangent at a double point are real, the branches are also real. But there are cases, when the tangent may be real, yet the branches may be imaginary.

☆Working rule for finding the nature of origin which is a double point

Find the tangent at the origin by equating to zero the lowest degree terms in x and y of the equation of the curve. If the origin is a double point, then we shall get two tangents which may be real or imaginary.

1. If two tangent are imaginary, then origin is a conjugate point.

2. If two tangent are real and coincident, then origin is a cusp or a conjugate point.

3. If two tangent are real and distinct, then origin is a node or a conjugate point.

To be sure, examine the nature of curve in the neighbourhood of origin. If the curve has real branches through the origin, then it is a node, otherwise a conjugate point.

Example

Q1. Examine the nature of origin for the curve : $x^{4}-ax^{2}y +axy^{2}+a^{2}y^{2}=0,a>0$
Sol.
The equation of curve is

$x^{4}-ax^{2}y +axy^{2}+a^{2}y^{2}=0,a>0$

Equation to zero, the lowest degree terms, the tangent at the origin are given by

$a^{2}y^{2}=0\:\: i.e.\: \: y^{2}=0$ $i.e. \: \, y=0,\: \: y=0$
There are two real and coincident tangent at the origin
Origin is either a cusp or a conjugate point

To be sure, we study the nature of given curve near the origin.

Now: $x^{4}-ax^{2}y +axy^{2}+a^{2}y^{2}=0,a>0$
$\therefore y=\frac{ax^{2}\pm \sqrt{a^{2}x^{4}-4ax^{4}(x+a)}}{2a(x+a)}$

$\therefore y=\frac{ax^{2}\pm x^{2}\sqrt{-4ax-3a^{2}}}{2a(x+a)}$
Now, for small value of $x\neq 0,\: \: -4ax-3a^{2}$ is negative and so y is imaginary in the neighbourhood of origin.

Origin is a conjugate point.

☆Working rule for finding the position and nature of double points of the curve

Step1. Find $\frac{\partial f}{\partial x},\:\frac{\partial f}{\partial y},\: \frac{\partial^2 f}{\partial x^2},\: \frac{\partial^2f }{\partial x \partial y},\: \frac{\partial^2 f}{\partial y^2}$

Step2. Solve the equation $\frac{\partial f}{\partial x}=0\: \: and\: \: \frac{\partial f}{\partial y}=0$ to get possible double points
Reject those points which do not satisfied the equation f (x,y)=0 of the curve remaining are the double points.

Step3. At each double point, calculate $D=\left ( \frac{\partial^2 f}{\partial x^2 \partial y } \right )^2-\frac{\partial^2 f}{\partial x^2}\frac{\partial^2 f}{\partial y^2}$
(a) If D is positive, double point is a node or conjugate point.
(b) If D=0, double point is a cusp or conjugate point.

In these cases (a) and (b), find the nature by shifting the origin to the double points and then testing the nature of tangent and existence of the curve in the neighbourhood of new origin.

(c) If D is negative, double point is a conjugate point.

Example

Q2. Find the position and nature of the double points on the curve $\left ( x-2 \right )^{2}=y\left ( y-1 \right )^{2}$
Sol.
The equation of curve is $\left ( x-2 \right )^{2}=y\left ( y-1 \right )^{2}$ (1)
$\frac{\partial f}{\partial x}=2\left ( x-2 \right )$

   $\frac{\partial f}{\partial x}=-2y\left ( y-1 \right )-\left ( y-1 \right )^{2}=-\left ( y-1 \right )\left ( 3y-1 \right )$
For the double points    $\frac{\partial f}{\partial x}=0, \: \: \frac{\partial f}{\partial y},\: \: f\left ( x,y \right ) =0$
Now   $\frac{\partial f}{\partial x}=0\: \Rightarrow 2\left ( x-2 \right )=0\, \, \Rightarrow x=2$
And $\frac{\partial f}{\partial y}= 0\: \: \Rightarrow -\left ( y-1 \right )\left ( 3y-1 \right )=0\, \, \Rightarrow y=1,\frac{1}{3}$
The possible double points are $\left ( 2,1 \right )\left ( 2,\frac{1}{3} \right )$
Now   $\left ( 2,\frac{1}{3} \right )$ does not satisfy (1)

(2,1) is the only double point
$\frac{\partial^2 f}{\partial x^2}=2,\: \, \frac{\partial^2 f}{\partial y^2}=-3\left ( y-1 \right )-\left ( 3y-1 \right ),\frac{\partial^2 f}{\partial x \partial y }=0$

$\frac{\partial^2 f}{\partial x^2}=2,\: \, \frac{\partial^2 f}{\partial y^2}=-3\left ( 1-1 \right )-\left ( 3-1 \right )=-2,\frac{\partial^2 f}{\partial x \partial y }=0$
$\left ( \frac{\partial^2f }{\partial x \partial y } \right )^{2}-\frac{\partial^2 f}{\partial x^2}.\frac{\partial^2 f}{\partial x y^2}=\left ( 0 \right )^{2}-\left ( 2 \right )\left ( -2 \right )=4>0$
There is a node at the point (2,1).

Saturday, 23 February 2019

Posted by Parth sharma on February 23, 2019 with No comments

Calculus

Inflection point

A curve's Inflection point is the point at which the curve's concavity changes. A point that separates the convex part of the curve from part of the curve is called a point of inflexion.

Consider the curve y=f(x) in [ a,b ]. Let it be continuous and possessing tangent at every point in ( a,b ).
Draw a tangent at any point P ( c,f(c) ) on the curve. Let us assume that this tangent is not parallel to Y-axis so that f'(c) is some finite number.

Point Of Inflection

The portion of the curve on the two side of P lies on different side of the tangent at P i.e, the curve crosses the tangent at P. In this case we say that P is a point of inflection on the curve.
So, at a point of inflection the curve change from concave upward so convex downward or vice-versa.

So at a point of inflexion f''(x)=0.

☆Working Method Of Find The Point Of inflection

(1). Evaluate $svg$

(2). Find the value of x which satisfy $svg$ and also the value of x where   $svg$ does not exist.
Such values x=a,b,c,....(say) are the possible point of inflection.

(3). x=0 will be point of inflection
If (1). Either   $svg$ change sign at x=a
or(2). $svg$ exist and is non-zero at x=a.

Note1.    $svg$ is not a sufficient condition for graph of f to have a point of inflection.

Note2. If at a point, x=c, $svg$ when n is even, then x=c is not a point of inflection.

Note3. If at a point, x=c, $svg$ for some even n and   $svg$ , then the curve has a point of inflection at x=c.

Example

Q1. Find the point of inflection on the graphs of the function $svg.latex?y=x^{4}$
Sol.
$svg$

   $svg$
   $svg$
Now $svg$
   $svg$

x=0 is not a point of inflection.

Wednesday, 20 February 2019

Posted by Parth sharma on February 20, 2019 with 1 comment

Calculus

Concavity and convexity

When you plot a function in the graph, if the curve opens towards the positive Y-axis then it is said to be concave up (or convex down) function while if it opens downwards then the curve is said to be concave down (or convex up ) function.

Concavity and convexity

1. Concave upward

A portion of the curve on both side of P, however small it may be, lies above the tangent at P ( i.e. towards the +ve direction of Y-axis ). In this case, we say that the curve is concave upward or convex downward at P. Such curve "hold water".
As x-increase, f'(x) is either of the same sign and increasing or change sign from -ve to +ve. In either case, the slop f'(x) is increasing and f''(x) >0. Such graphs are bending upward or the portion lies above the chord.

2. Concave downward

A portion of the curve on both side of P, however small it may be, lies below the tangent at P ( i.e. towards the -ve direction of Y-axis ). In this case, we say that the curve is convex upward or concave downward at P.
As x-increase, f'(x) is either of the same sign and decreasing or change sign from +ve to -ve. In either case, the slop f'(x) is decreasing and f''(x) <0. Such graphs are bending downward or the portion lies below the chord.

3. Point of inflection

☆Definition of concavity and convexity

A curve is said to be concave downward ( or convex upward ) on the interval ( a,b ) if all the points of the curve lies below any tangent to it on that interval. It is said to be concave upward ( or convex downwards ) on the interval ( a,b ) if all the points of the curve lies above any tangent to it on that interval.

☆Working method for concavity and convexity

(1). Evaluate $\frac{d^2 y}{d x^2}$

(2). Find the interval ( a,b ) for which $\frac{d^2 y}{d x^2}>0$ .
Then ( a,b ) is the interval of being convex downward.

(3). Find the interval ( a,b ) for which $\frac{d^2 y}{d x^2}<0$ .
Then ( a,b ) is the interval of being convex upward.

Examples

Q1. Prove that the curve $y=e^x$ is concave upward for all x∈ℝ.
Sol.
Here $y=e^x$
$\therefore \: \: \: \: \: \: \frac{dy}{dx}=e^x$
and $\frac{d^2y}{dx^2}=e^x$

$\because \: \: \frac{d^2y}{dx^2}>0 \: for \: all\; x\epsilon \mathbb{R}$
The curve $y=e^x$ is concave upward $\forall \: \: x\: \epsilon \: \mathbb{R}.$

Q2. Prove that the curve $y=\log x$ is everywhere concave downward for x >0.
Sol.
Here $y=\log x$ , x >0
$\therefore \: \: \: \; \frac{dy}{dx}=\frac{1}{x},\: x> 0$
$\frac{d^2y}{dx^2}=-\frac{1}{x^2},\: x> 0$
Now $\frac{d^2y}{dx^2}> 0\: \; \forall \: x>0$
Hence the curve is concave downward or convex upward for x >0.

Saturday, 16 February 2019

Posted by Parth sharma on February 16, 2019 with No comments

Calculus

Intervals

Interval is a way to describe continuous set of real number by the number that bound them. However, they are not meant to denote a specific point. Rather, they are meant to describe an inequality or system of inequalities.

Let a and b be two distinct real number with a<b then,

1. Open interval

The set of all real number between a and b said to form an open interval from a to b denoted by (a,b). In symbols

$\left ( a,b \right )=\left \{ x: \: a< x< b\: \: and\: x\epsilon \mathbb{R} \right \}.$
Geometrically the open interval (a,b) is represented on the real line as

For example= Inequality: -1<x<5
Interval : (-1,5)
In this case, x does not equal -1 and 5. When both of the end point are excluded from the interval, the interval is open interval.

2. Closed interval

The set of all real number between a and b including the end points and b is said to form a closed interval and is denoted by [a,b].
In symbols $\left [ a,b \right ]=\left \{ x:\: a\leq x\leq b\: and \: x\epsilon \mathbb{R} \right \}.$
Geometrically, the closed interval [a,b] is represented on the real line as

For example= inequality: 3≤x≤9
interval : [3,9]
In this case, x could equal 3 or 9 when both of the end point are included in the interval, the interval is a closed interval.

3. Half-open intervals

An interval in which one end point is included are the other end is excluded is called half-open interval.
In symbols, $\left [ a,b \right )=\left \{ x:\: a\leq x< b\: and \: x\epsilon \mathbb{R} \right \}.$
Geometrically, [a,b) is represented on the line as

For example= inequality: -3≤x<5
interval : [-3,5]
In this case, x could equal -3 but it cannot equal 5. When one of the end points is included in the interval but the other is not, the the interval is a half-open interval.

Similarly $\left ( a,b \right ]=\left \{ x:\: a< x\leq b\: and \: x\epsilon \mathbb{R} \right \}.$
Geometrically, (a,b] is represented on the real line as

The interval defined above are called finite intervals. Now we define infinite intervals.

Infinite interval. The set of all real number x such that x>a forms an infinite set and is denoted by (a,∞).
Symbolically $\left ( a,\infty \right )=\left \{ x:x>a\: \: and \: \: x\epsilon \mathbb{R} \right \}.$ geometrically

If an interval has no lower bound or upper bound then the -∞ and ∞ symbol are used. These symbols are always used with a parentheses bracket, because infinity is not a number that can be included in a set

For example. inequality: x≤7
interval : (-∞,7)

inequality: x>3
interval : (3,∞).

Friday, 1 March 2019

Multiple point

Singular point

Classification of double points.

1. Node

2. Cusp

3. Conjugate point

Example

☆Working rule for finding the position and nature of double points of the curve

Example

Saturday, 23 February 2019

Inflection point

Point Of Inflection

☆Working Method Of Find The Point Of inflection

Example

Wednesday, 20 February 2019

Concavity and convexity

Concavity and convexity

1. Concave upward

2. Concave downward

3. Point of inflection

☆Definition of concavity and convexity

☆Working method for concavity and convexity

Examples

Saturday, 16 February 2019

Intervals

1. Open interval

2. Closed interval

3. Half-open intervals

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