Kategorier
Calculus Gymnasiematematik(high school math) matematik 3c matematik 4

Indefinite integrals

If you need to calculate the distance travelled when you know the velocity as a function of time , since s'(t) = v(t) you need to be able to perform antiderivation i.e. finding a function whose derivative equals your function.

This process is called integration.

Hence s(t) is the indefinite integral or primitive function, to v(t).

For an arbitrary function f(x) the primitive funktion is conventionally written as

F = \int f(x)\,dx. 
and  F’ (x) =  f(x). 
Table of indefinita integrals: 
formula                         
formula
formula
formula

formula

formula

formula

formula

formula

formula

formula

formula

formula

formula

formula

formula

formula

formula

formula

formula

formula

formula

Integrazione per sostituzione

Per il calcolo di integrali del tipo formula, talvolta può essere vantaggioso sostituire alla variabile d’integrazione x una funzione di un’altra variabile t, purché tale funzione sia derivabile e invertibile.

Ponendo formula, da cui deriva formula, si ha che:

formula di integrazione per sostituzione

Read moreIndefinite integrals – primitiva funktioner

p. 161-169

Kategorier
Calculus Gymnasiematematik(high school math) matematik 3c matematik 4

Differentiating the natural logarithm, products and quotients

In order to be able to deduce the derivative of the natural logarithm we resort to using implicit differentiation.

Let x= ey(x)

Differentiating both sides gives

dx/dx = d ey(x)/dx

1=ey(x) dy(x)/dx

Solving for dy(x)/dx one obtains

dy(x)/dx = 1/ey(x) = 1/x .

The product rule is given by

d f(x) g(x)/dx = df(x)/dxg(x) + f(x) dg(x)/dx

A beautiful proof for this theorem is given by G.W. Leibniz:

(u + du) *(v+dv) = u*v + u*dv + v* du + du*dv.

The last term is the product of two infinitesimals and can therefore be neglected. The differential of the product of the two functions u(x)*v(x) is thus equal to

u(x)*dv + v(x)*du.

Q.E.D.

To read more about this formula click here.

The rule for differentiation of the quotient of  two functions can be deduced from the product rule and is given by

df(x)/g(x) =( df(x)/dx g(x) – f(x) dg(x))/dx)/(g(x))2.

Kategorier
Geometri Gymnasiematematik(high school math)

Pi-day

March the 14th. has officially been named the international π-day to honour this magical number which equals the ratio of the circmference to the diameter for all circles.

http://www.wikihow.com/Celebrate-Pi-Day

In 1882 the german mathematician Ferdinand Lindemann showed that pi is a transcendental number http://en.wikipedia.org/wiki/Transcendental_number

Such numbers have an infinitesimal decimal expansion with no periodicity in the numbers.

It has been said that the decimalexpansion of pi is the perfect generator of random numbers.

The record is ten million decimals generated with a computer using the Taylor-series expansion of arcustangens π/4 =1 and then solving for pi.

Perhaps the strangest quality of pi is that it surfaces in many areas of science other than geometry.

Probability calculus, imaginary numbers, infinitesimal series, calculus e.g..

So in the infinite number of decimals of pi are hidden the answers to many scientific enigmas. http://en.wikipedia.org/wiki/Pi

Kategorier
Calculus Gymnasiematematik(high school math) matematik 4

Differentiation of the trigonometric functions

To be able to differentiate the trigonometric functions one needs some standard limits:

With the aid of these and the definition of the derivative

f'(x_0)= \lim_{h \to 0} \frac{f(x_0+h) - f(x_0)}{h}

it is possible to show that

f(x)= sin (x) implies  f ‘(x) = cos(x)

and

f(x) = cos(x) implies f ‘(x) = -sin (x).

Kategorier
Geometri Gymnasiematematik(high school math)

Radians and an attempt at squaring the circle

Katedralen i Siena

That one rotation equals 360 degrees is just a convention, There is nothing partcular about 360 except that it can be divided by many numbers.

Another, and a more fruitful, approach to measuring angles is to use the length of the arc cut out by the angle on the perimeter of the unit-circle.

Since the circumference of the unit-circle is 2π this corresponds to 360 degree

π RADIANS then equal 180degrees and hence 1 radian = 180/π degrees.

It follows that one revolution corresponds to 2π such radians.

http://en.wikipedia.org/wiki/Radian

This reasoning can be expanded to  arbitrary circles having radius r.

Their circumference being 2πr which means 2π radians proving that the angle is equal to 2π radians in this case as well.

To see an attempt to square the circle click on the figure below. Courtesy of Wiking Björkman (my grandfather):

To listen to an 1981 interview (Radio Kronoberg) with him click below
11 Spår 11_(new).mp3