The Distance & Section Formulae
Submitted by Dev on Sun, 03/29/2009 - 05:59.These two formulae form the basis of coordinate geometry. The distance formula enables us to find the distance between two points a & b, and the section forula helps us to find the coordinated of a point P which divides a line segment AB by some ratio.
The Section Formula
Submitted by Dev on Sun, 03/29/2009 - 05:15.Here, we have a point P which lies on the line joining A & B. It also divides the line segment AB in some ratio.
Lets assume that the ratio here is m:n. The section formula gives us the scope of finding the coordinates of this point P using the coordinates of A & B. We need the ratio m:n and the coordinates of the points A & B to find the coordinates of P.
We observe in the above figure that AHP & PCB are similar triangles,
Hence, we have AH/PJ=PH/BJ=AP/AB
The Distance Formula
Submitted by Dev on Sun, 03/29/2009 - 04:53.The distance formula gives us the distance between two points in space. We establish the distance formula using the cartesian system here. It is a direct result of the Pythagorean theorem. It is the most important formula and forms the basis of coordinate geometry.
The Co-ordinate Systems
Submitted by Dev on Sun, 03/29/2009 - 04:26.The two major co-ordinate systems are The Cartesian & The Polar co-ordinate systems.
When using the cartesian system, we need to specify the abscissa(x) & the ordinate(y) to denote a particular point while in case of the polar system, we need to specify the distance of a point from the origin & the slope of the line joining the origin to that point(the position vector of that point).
So, we have two axes(OX & OY) in case of the cartesian co-ordinates & we have a pole(O) & an initial line(OX) in case of the polar co-ordinates.
The Most Beautiful Questions from Integration
Submitted by Dev on Sun, 03/29/2009 - 03:44.Mathematics Millionaire
Submitted by Dev on Sat, 03/28/2009 - 11:17.
So who wants to be a math millionare? Put to test & develop your math skills with this game.
It will follow a patterm similar to the millionare shows with questions from various fields in mathematics.
Questions will be from general mathematics of high school level.
Major Topics : Algebra, Geometry , Trigonometry & Calculus
There will first be a random question from any of the above topics & the one who answers first will get to play the game for a million dollars(fake, as of now)
Differentiation Rules
Submitted by Dev on Sat, 03/21/2009 - 06:46.If two vector functions are in the sum or difference, the sum property holds true, same as what we have in single variable scalar functions.
| (u + v)' = u' + v' |
To find the derivative of the product of a scalar & a vector function.
[Here, f is a scalar function while u is a vector function]
| (ƒu)' = ƒu' + u dƒ/dt |
If we are given two vector functions, & we need to find the derivative of their dot product.
u(t)=(5t2)i + (2t)j + k
Examples of Implicit Differentiation
Submitted by Dev on Sat, 03/21/2009 - 06:24.1. sin(x+y)=x
cos(x+y)(1+y')=1
[Applying the chain rule, since, y=ƒ(x)
y'{cos(x+y)}=1-cos(x+y)
y'=[1-cos(x+y)]cos(x+y)-1
2. exy=(x-y)
exy[x(dy/dx)+y] = 1-(dy/dx)
xexy(dy/dx) + yexy = 1-(dy/dx)
(dy/dx)[1+xexy]=1-yexy
dy/dx=[1-yexy]{1+xexy}-1
3. x2+y3=siny
2x + 3y2(dy/dx) = cosy(dy/dx)
2x = (dy/dx){cosy-3y2)
dy/dx = 2x{cosy-3y2)-1
The Arc Length
Submitted by Dev on Wed, 03/18/2009 - 10:53.We define the arc length for vector functions to be the total length of the arc from the corner point a to b ie the interval for which the curve is defined. It is calculated for both the 2d & 3d vector functions in the same manner.
The Product & Quotient Rules
Submitted by Dev on Wed, 03/18/2009 - 07:46.If we have two or more functions in the product & we need to find the derivative, we use the product rule.
The rule says:
(uv)' = uv' + vu'
& If we have two functions in the form (u/v), we need the quotient rule or we can simply use the product rule replacing (u/v) by u.(1/v)which is now a procut of two functions u & 1/v.
Both the rules have been explained with examples in the videos...
