# MatheAss: News

MatheAss  is also revised from time to time between updates, mostly based on user instructions. With version 9.0 a new version is now available with many new functions.

## What's new in MatheAss 9.0?

The following program parts have been added:

# Algebra

Prime tuples
In an interval [a,b], all prime twins (p,p+2), prime cousins (p,p+4), sexy primes (p,p+6) and prime triplets are determined.
```Prime twins between 1 and 200

(3|5) (5|7) (11|13) (17|19) (29|31) (41|43) (59|61)
(71|73) (101|103) (107|109) (137|139) (149|151)
(179|181) (191|193) (197|199)

15 pairs of prime twins```
```Prime triplets between 1 and 100

(3|5|7) (5|7|11) [7|11|13] (11|13|17) [13|17|19]
(17|19|23) [37|41|43] (41|43|47) [67|71|73]

9 triplet prime triplets
4 of the form (p|p+2|p+6) and 4 of the form [p|p+4|p+6]```
Calculating percentages
The base value G, the percentage value W, the percentage p or p%, the growth factor q and the final value E are calculated if two independent values are entered.
```Given:
¯¯¯¯¯¯
basic value G = 150
percentage p% = 2.5% = 0.025 = 1/40

Results:
¯¯¯¯¯¯¯
percentage value W = 3.75
growth factor q = 102,5% = 1,025 = 41/40
final value E = 153.75```
```Given:
¯¯¯¯¯¯
percentage value W = -120
growth factor q = 95% = 0,95 = 19/20

Results:
¯¯¯¯¯¯¯
basic value G = 2400
percentage p% = -5% = -0.05 = -1/20
final value E = 2280  ```
Calculation with big integers
Calculation with two big integers  a  and  b  with a maximum of 10,000 digits. # Geometry

Special straight lines in a triangle
The program determines the equations of the perpendiculars, the bisectors of the sides, the bisectors of the angles and the heights of a triangle. In addition, the centers and radii of the circumference, the inscribed circle and the three excircles.
```Given:
¯¯¯¯¯¯
Edges:    A(1|0)   B(5|1)   C(3|6)

Results:
¯¯¯¯¯¯¯
Vertices:   a :  5·x + 2·y = 27
b :  3·x - y = 3
c :  x - 4·y = 1

Incircle:    Mi(3,119|1,962)          r i = 1,390

Excircles:    Ma(7,626|6,136)        ra = 4,346
Mb(-4,356|5,784)       rb = 6,910
Mc(3,248|-2,427)       rc = 2,900``` Mappings of Polygons
Displacement, straight line mirroring, point mirroring, rotation, centric stretching and shear can be applied to an n-gon.
The input has been made clearer and the construction lines can be drawn in the diagram.
```Counter image
A(1|1), B(5|1), C(5|5), D(3|7), E(1|5),

1. Translation: dx=2, dy=1  ☑
A(3|2), B(7|2), C(7|6), D(5|8), E(3|6),

2. Rotation: Z(2|-1), α=-60° ☑
A(5,0981|-0,36603), B(7,0981|-3,8301),
C(10,562|-1,8301), D(11,294|0,90192),
E(8,5622|1,634),``` Tangent lines to circles
The following tangents will be calculated:
• The tangent to a circle k at a point B.
• The tangents to a circle k through a point P outside the circle
• The tangents to a circle k parallel to a straight line g
• The tangents to two circles  k1  and  k2
```Given:
¯¯¯¯¯
k1 : M(5|8) ,   r =5
k2 : M(-1|2) ,   r =3

Outer tangents
¯¯¯¯¯¯¯¯¯¯¯¯
t1: -4,2923·x + 7,04104·y = -6,36427
t2: -7,04104·x + 4,29230·y = 40,3643

Inner tangents
¯¯¯¯¯¯¯¯¯¯¯¯
t3: 1,21895·x + 2,55228·y = 12,3709
t4: -2,55228·x − 1,21895·y = -8,3709``` # Analysis

Sequences and Series
The program determines the first n terms of a sequence  (ai)  and the associated series (sum of the sequence terms) if the first terms of the sequence and a recourse formula  ai=ƒ(a0, a1, ... , ai-1)  or an explicit function  ai = ƒ(i)  are given.
The sequence of odd numbers e.g. can be defined explicitly by  ai = 2·i + 1  or  recursively by  ai = ai-1 + 2  with  a0=1 .
```Folge
¯¯¯¯¯
( a[ i ] ) = (1; 3; 5; 7; 9; 11; 13; 15; 17; 19)

Reihe
¯¯¯¯¯
( Σ a[ i ] ) = (1; 4; 9; 16; 25; 36; 49; 64; 81; 100)```
Factoring Polynomials
The program calculates the rational zeros and the linear factorization of a polynomial.
```p(x) = x5 - 9·x4 - 82/9·x3 + 82·x2 + x - 9
= (1/9)·(9·x5 - 81·x4 - 82·x3 + 738·x2 + 9·x - 81)
= (1/9)·(3·x - 1)·(3·x + 1)·(x - 9)·(x - 3)·(x + 3)

Rational Zeros: 1/3, -1/3, 9, 3, -3```
Transforming Polynomials
A polynomial  p(x)  can be shifted or stretched in the x-direction and y-direction.
```ƒ(x) =  - 1/4·x4 + 2·x3 - 16·x + 21

Shifted by  dx = -2 ,  dy = 0

ƒ(x + 2) =  - 1/4·x4 + 6·x2 + 1```
Polynomial Functions
The program carries out the curve discussion for polynomial function. This means that the derivatives and the antiderivative are determined, the function is examined for rational zeros, for extremes, for inflection points and for symmetry.
```Function :
¯¯¯¯¯¯¯¯
ƒ(x) = 3·x4 - 82/3·x2 + 3
= 1/3·(9·x4 - 82·x2 + 9)
= 1/3·(3·x - 1)·(3·x + 1)·(x - 3)·(x + 3)

Derivations :
¯¯¯¯¯¯¯¯¯¯
ƒ'(x)  = 12·x3 - 164/3·x
ƒ"(x)  = 36·x2 - 164/3
ƒ'"(x) = 72·x

Antiderivative:
¯¯¯¯¯¯¯¯¯¯¯¯
ƒ(x) = 3/5·x5 - 82/9·x3 + 3·x + c

…``` Rational Functions
The program carries out the curve discussion for a rational function. That is, the derivatives, the definition gaps and the continuous continuation are determined. The function is examined for zeros, extrema, points of inflection: and the behavior for | x | → ∞.
```Function :
¯¯¯¯¯¯¯¯
3·x3 + x2 - 4         (x - 1)·(3·x2 + 4·x + 4)
ƒ(x) = —————— = ———————————
4·x2 - 16                4·(x - 2)·(x + 2)

Definition gaps :
¯¯¯¯¯¯¯¯¯¯¯¯¯
x = 2  Pol mit Vorzeichenwechsel
x =-2  Pol mit Vorzeichenwechsel

Derivations :
¯¯¯¯¯¯¯¯¯¯
3·(x4 - 12·x2)             3·(x2·(x2 - 12))
ƒ'(x) = ———————— = —————————
4·(x4 - 8·x2 + 16)       4·(x - 2)2·(x + 2)2

6·(x3 + 12·x)                6·(x·(x2 + 12))
ƒ"(x) = ——————————— = ————————
x6 - 12·x4 + 48·x2 - 64        (x - 2)3·(x + 2)3
…``` # Stochastics

Statistics
In the statistics section, the histogram was supplemented by a box plot. Logistic Regression
The program determines a curve fit for a series of measurements to the logistic function with the parameters    a1 = ƒ(0)·S ,  a2 = ƒ(0) ,  a3 = S - ƒ(0) ,  und  a4 = -k·S  and the saturation limit S .
```Data from:  "hopfenwachstum.csv"

Saturation limit:  6
Dark figure:  1

4,0189
ƒ(x) = ————————————————
0,66981 + 5,3302 · e^(-0,35622·t)

Inflection point W(5,8226/3)

Maximum growth rate ƒ'(xw) = 0,53433

8 Values
Coeff.of determin.   = 0,99383916
Correlation coeff.    = 0,99691482
Standard deviation = 0,16172584
``` Data series from Johns Hopkins University (JHU) on the corona pandemic are attached as CSV files.

# Registration

## How much costs MatheAss 9.0?

29 € for the private license

79 € for the school license

360 € for the extended school license, with which the serial number may be passed on to the pupils.

## How much is the update?

10 € for owners of a private license

30 € for owners of a school license

90 € for owners of an extended school license

## How can I pay?

Here by PayPal : 