A graphing software is available at padowan.dk

Another software is winplot available at math.exeter.edu/rparris/

An interactive geometry graphing system where input can be in the form of equations or co ordinates which can be manipulated even after the graph has been drawn is available at geogebra.org

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## Friday, February 27, 2009

## Monday, February 23, 2009

### proving a trigonometry identity

show that [( tanx -sinx ) / (2tanx) ] = sin² (x/2)

divide term by term and change

every thing in terms of sinx and cosx and

then use half angle formulae

use this link for trigonometric identities formulae

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divide term by term and change

every thing in terms of sinx and cosx and

then use half angle formulae

use this link for trigonometric identities formulae

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### trigonometric identities

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### Integration formulae

note that ln(x) refers to natural log, that is log to the base e

Integration formulae of functions like

x^n , e^(ax ) exp(ax), sin(ax), cos(ax), tan(x), 1 / sqrt(a^2 - x^2), 1 / (a^2 - x^2) ...

integration guide on certain topics

PAGE 1 BASIC INTEGRATION

PAGE 2 INTEGRATION BY SUBSTITUTION

PAGE 3 INTEGRATION BY COMPLETION OF SQUARES

PAGE 4 INTEGRATION BY PARTS

PAGE 5 INTEGRATION BY MANIPULATION OF

NUMERATOR IN TERMS OF DENOMINATOR

PAGE 6 INTEGRATION USING PARTIAL FRACTIONS

PAGE 7 INTEGRATION OF 1 / [ACOSX +BSINX +C]

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## Saturday, February 21, 2009

### equation of tangent at (e,e) on y = x lnx

y = x lnx

dy/dx = x(1/x) + lnx (1) = 1 + lnx

slope of tangent , m = (dy/dx) at (e,e) = 1 + ln(e) = 1 + 1 = 2

equation of tangent at (e,e) is

(y - y1) = m (x-x1)

y - e = 2(x-e)

or y = 2x - e

other examples

equation of tangent to y =x^4 at the point where x=2

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dy/dx = x(1/x) + lnx (1) = 1 + lnx

slope of tangent , m = (dy/dx) at (e,e) = 1 + ln(e) = 1 + 1 = 2

equation of tangent at (e,e) is

(y - y1) = m (x-x1)

y - e = 2(x-e)

or y = 2x - e

other examples

equation of tangent to y =x^4 at the point where x=2

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## Thursday, February 19, 2009

### problem on automorphism in a group

If G is a group and f :G -->G is defined as f(x) = a ֿ¹ x a for x in G and a is a fixed element of G

then show that f is an isomorphism of G onto G

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then show that f is an isomorphism of G onto G

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### sum of n terms of a geometric progression

find the nth term of the series 1 + 2 + 4 + 8 + 16 + ...

Also find the sum of the first n terms

The sequence 1, 2 , 4 , 8 , 16 , ... is a geometric progression (G.P.) with

first term a=1 and common ratio, r = 2

since 2/1 = 4/2 = 8/4 =16/8 = ...

nth term = a {r ^ ( n-1) } = 2^(n-1)

try the following page for formulae on sequences and series AP and GP

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Also find the sum of the first n terms

The sequence 1, 2 , 4 , 8 , 16 , ... is a geometric progression (G.P.) with

first term a=1 and common ratio, r = 2

since 2/1 = 4/2 = 8/4 =16/8 = ...

nth term = a {r ^ ( n-1) } = 2^(n-1)

try the following page for formulae on sequences and series AP and GP

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## Wednesday, February 18, 2009

### idempotent law in a boolean algebra

show that a + a = a in a boolean algebra

a = a + 0

= a + (a. a' )

= (a + a ). (a + a ' )

= (a + a ). 1

= (a + a )

therefore a + a = a

note that the dual is a . a = a

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a = a + 0

= a + (a. a' )

= (a + a ). (a + a ' )

= (a + a ). 1

= (a + a )

therefore a + a = a

note that the dual is a . a = a

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### boundedness law in a boolean algebra

Show that a +1 = 1 in a boolean algebra

1 = a + a'

= a + (a' .1 )

= (a + a' ) . ( a+1) distributive law

= 1 . ( a+1)

=a +1

therefore a + 1 = 1

note that the dual is a . 0 = 0

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1 = a + a'

= a + (a' .1 )

= (a + a' ) . ( a+1) distributive law

= 1 . ( a+1)

=a +1

therefore a + 1 = 1

note that the dual is a . 0 = 0

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## Friday, February 13, 2009

### problem on integration using trigonometry formulae

evaluate integral of sin(5x)sin(8x)

first use the trigonometric formulae for sinAsinB to split the integral into two terms

it is easy to use the bigger angle as A

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first use the trigonometric formulae for sinAsinB to split the integral into two terms

it is easy to use the bigger angle as A

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## Thursday, February 12, 2009

### telescoping series

simplify 1/12 + 1/20 +1/30 +1/42 +1/9900 using partial fraction

try to express in the form 1 / n(n+1)

use partial fraction to split into two terms

then sum

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try to express in the form 1 / n(n+1)

use partial fraction to split into two terms

then sum

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## Friday, February 6, 2009

### trigonometric identities

trigonometric identities

addition formulae , double angle formulae, half angle formulae , formulae for removing square and cube from sin and cos

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addition formulae , double angle formulae, half angle formulae , formulae for removing square and cube from sin and cos

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### integration by trigonometric substitution

evaluate integral of 1 /sqrt(1-9x^2) using trigonometric substitution

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## Thursday, February 5, 2009

### real and imaginary parts of ln(x+iy)

find the real and imaginary parts of ln(x+iy) or logarithm of a complex number,

first express x+iy in modulus amplitude form

then take log

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first express x+iy in modulus amplitude form

then take log

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**basic result of complex number for plus two student**disclaimer:

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## Monday, February 2, 2009

### to construct truth tables for [~ (p ^ q ) ]v (~r)

since there are three variables there will be 2^3 =8 rows

start the first column with 4 T's and 4F's

second column with 2 T's and 2F's ...

third column T and F alternatively

then use the definition of ^ , ~ to get the truth table of the components and finally that of the question

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start the first column with 4 T's and 4F's

second column with 2 T's and 2F's ...

third column T and F alternatively

then use the definition of ^ , ~ to get the truth table of the components and finally that of the question

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## Sunday, February 1, 2009

### solving a fourth degree equation given one of its complex roots

solve 2x^4-8x^3+19x^2-22x+5=0 given that 1+2i is one of its roots

complex roots occur in conjugate pairs if the coefficients are real

so 1-2i is also a root

{x - (1+2i)}{x - (1-2i)} is a factor

multiply into a quadratic

divide the given equation by this quadratic factor to extract the other quadratic factor and solve that

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complex roots occur in conjugate pairs if the coefficients are real

so 1-2i is also a root

{x - (1+2i)}{x - (1-2i)} is a factor

multiply into a quadratic

divide the given equation by this quadratic factor to extract the other quadratic factor and solve that

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### problem on sequences

If Sn is the sum of the cubes of the first n natural numbers, find the sum of 1 / sqrt(Sn) where n varies from 1 to 2000

first use the formula for sum of the cubes of the first n natural numbers

take the square root and the reciprocal then adjust into a telescoping series

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first use the formula for sum of the cubes of the first n natural numbers

take the square root and the reciprocal then adjust into a telescoping series

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