finding the derivative from first principles

find the derivative of sqrt(3x-2) from first principles.




another example

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orthogonal trajectory of y = (k/x)

find the orthogonal trajectory of y = (k/x)

first we find the diff. equation of the given family
y=(k/x) -----------(1)
diff. w.r.t. x
y' = -k / (x²)---------(2)

eliminating k , divide (2) by (1)

y' / y = -1/x ---------(3)

(3) is the diff. equation of the given family.
replace y with -1/y' in (3) to get the diff. equation of the orthogonal trajectory

diff. equation of the orthogonal trajectory is

-yy' = -1/x
y(dy/dx) = 1/x

ydy =xdx
integrating and simplifying, we see that the orthogonal trajectory is

x² - y² = C


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to find the equation of a circle with center (2,-6); tangent to the y-axis

Find the equation of the circle with center(2,-6) and tangential to the y-axis.

Since the y-axis is a tangent to the circle,
radius of the circle,r =|x-coordinate of the centre| = 2

equation of the circle is

(x-2)² +(y+6)² = 2²


formulae on circles



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beware of drm

If you buy music loaded with drm, you might face problems later on ,if the company maintaining the drm servers decides to shut down the servers due to decreasing profits or for any other silly reason.If the servers are shut down, then "if you do not back up your files before this date, you will no longer be able to transfer your songs to other computers or access your songs after changing or reinstalling your operating system or in the event of a system crash".

interesting example:

http://www.boingboing.net/2008/09/26/walmart-shutting-dow.html



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to find the square root of a cormplex number

to find sqrt(3-4i)

let sqrt(3-4i)= x+iy

squaring 3-4i = x² -y² +2ixy

equating real parts , x² -y² =3
equating imaginary parts 2xy = -4

(x² +y²)² =(x² - y²)² + (2xy)² = 9 +16 = 25
therefore
(x² +y²) = 5 ----------(1)
also
x² -y² =3-----------(2)

(1) +(2) implies

2 x² = 8 or x² = 4 or x = ± 2
using 2xy = -4
if x=2 , y = -1
if x = -2 , y =1

so sqrt(3-4i) = 2-1 i or-2 + 1i


sqrt(3-4i) = ± (2- i )

some notes on complex numbers
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to find cot(x ) if you are given that sin(x)=2/3 by drawing a triangle

find cot(x ) if you are given that sin(x)=2/3 and that x is acute.

draw a rt. triangle with one angle as x (not the right angle)



choose the opp. side as 2 and hypotenuse as 3 so that the third side is
sqrt(5) (using pythagoras theorem)

so cot(x) = (sqrt 5) /2


some trigonometry formulae

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If n is an integer, show that one of the integers n, n+2, or n+4 is divisible by 3.

If n is an integer, show that one of the integers n, n+2, or n+4 is divisible by 3.

solution:
If n is a multiple of 3, then there is nothing more to prove.

Suppose n is not a multiple of 3, then n must be of the form
n =3k+1 or n = 3k+2, for some integer k

if n=3k+1,
then n+2 = (3k+1)+2 = 3k+3 = 3(k+1) = multiple of 3

if n = 3k+2,
then n+4 = 3k+6 = 3(k+2) =multiple of 3


hence the result

some other math problems

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