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Invfunct - a package to investigate functions

> with(plots):with(Invfunct):

example for a call of allsolve

> y:=tan(3*x+1);

y := tan(3*x+1)

> [fsolve(y,x=-4..4)];nops(%);[allsolve(y,x=-4..4)];nops(%);

[-.3333333333]

1

[-3.474925987, -2.427728436, -1.380530885, -.3333333333, .7138642179, 1.761061769, 2.808259320, 3.855456871]

8

> [allsolve(y,x=-4..4,maxsols=5,avoid={x=-1/3})];nops(%);

[-1.380530885, .7138642179, 1.761061769, 2.808259320, 3.855456871]

example for a call of ratsolve

> y:=x^3-2*x^2-3*x+1;

y := x^3-2*x^2-3*x+1

> [ratsolve(y,x)];evalf(%);nops(%);[fsolve(y,x)];nops(%);

[-1/12*104^(1/3)*13^(1/6)*cos(1/3*arctan(3*771^(1/2)/43))-1/312*104^(2/3)*13^(5/6)*cos(1/3*arctan(3*771^(1/2)/43))+2/3-1/2*3^(1/2)*(1/6*104^(1/3)*13^(1/6)*sin(1/3*arctan(3*771^(1/2)/43))+1/156*104^(2/...[-1/12*104^(1/3)*13^(1/6)*cos(1/3*arctan(3*771^(1/2)/43))-1/312*104^(2/3)*13^(5/6)*cos(1/3*arctan(3*771^(1/2)/43))+2/3-1/2*3^(1/2)*(1/6*104^(1/3)*13^(1/6)*sin(1/3*arctan(3*771^(1/2)/43))+1/156*104^(2/...[-1/12*104^(1/3)*13^(1/6)*cos(1/3*arctan(3*771^(1/2)/43))-1/312*104^(2/3)*13^(5/6)*cos(1/3*arctan(3*771^(1/2)/43))+2/3-1/2*3^(1/2)*(1/6*104^(1/3)*13^(1/6)*sin(1/3*arctan(3*771^(1/2)/43))+1/156*104^(2/...

[-1.198691244, .2864620652, 2.912229179]

3

[-1.198691244, .2864620650, 2.912229178]

3

example for a call of rasort

> L:=[infinity,Pi,exp(1),sin(3),cos(1/3*arctan(3/43*771^(1/2)))];

L := [infinity, Pi, exp(1), sin(3), cos(1/3*arctan(3*771^(1/2)/43))]

> ratsort(L);

[sin(3), cos(1/3*arctan(3*771^(1/2)/43)), exp(1), Pi, infinity]

> sort(L);

[infinity, Pi, exp(1), sin(3), cos(1/3*arctan(3*771^(1/2)/43))]

> sort(evalf(L));

[.1411200081, .9342104744, 2.718281828, 3.141592654, Float(infinity)]

example for a call of fdomain

> r:=3:f:=y^2+((x-2)*(x+1))^(-2)-r^2:

> fdomain(f,x=-6 .. 6,y,1000);evalf(op(2,op(2,%)));

[[-1, 2], [[(-3003)/500, (-279279)/250000], [(-219219)/250000, 117117/62500], [33033/15625, 3003/500]]]

[-.8768760000, 1.873872000]

> implicitplot(f,x=-2 ..3,y=-3 .. 3,numpoints=10000);

[Plot]

example for a call of curdisk

> X:=x-2:y:=(-X**8+3*X**6+4*X**4-2*X**2+3)/(X^4-4*X^2+3):

> curdisk(y,x=-1..5,yin=-40..10,sym=yes):

`This function is discussed over the range of real numbers`

f(x) = -(5-328*x+978*x^2-1280*x^3-412*x^5+109*x^6-16*x^7+x^8+936*x^4)/(x^4-8*x^3+20*x^2-16*x+3)

_________________________________

`Except of poles the domain of`*f(x)*`is the whole set of real numbers.`

______________

`The poles of`*f(x)*`are at`

2-3^(1/2) = .267949192

1

3

2+3^(1/2) = 3.732050808

______________

f(x)*`is symmetric to the axis`*x = 2

______________

`The zeros of`*f(x)*are

0.1599054702e-1

3.984009453

______________

f(x)*`intersects the y-axis at`

(-5)/3

______________

`The extremal points of`*f(x)*are

[.7577394485, -18.69415971]*maximum

[2.000000000, 1.000000000]*minimum

[3.242260552, -18.69427368]*maximum

______________

`The inflectional points of`*f(x)*are

[-.2591566374, -21.16232542]

[4.259156637, -21.16231038]

______________

f(x)*`does not have asymptotic lines non-perpendicular to the `*x*`-axis`

______________

`_______ END of the DISCUSSION ________`

> display(%);

[Plot]

example for a call of implicitan

> a:=20:b:=a/4:f:=a^2*x^2-(x-b)^2*(x^2+y^2):L:=[b-a,0,b,3/2*b,3/2*(b+a),b+a]:

> opt:=[printout=yes,Display=yes]:implicitan(f,x,y,L,opt);

The*poles*of*fcn = (400*x^2-(-5+x)^2*(x^2+y^2))*are

[5]

`Open intervals, except of poles, within the examined domain of the function under consideration`

[(-5989479)/400000, 9986481/400000]

Coordinates*of*points*with*a*perpendicular*tangent

[-15, 0], [0, 0], [25, 0]

Non*perpendicular*tangents*on*regular*points*of*the*curve

Pt*(1, 1) = [0, 0]*is*a*multipoint, number*of*branches = 2

Pt*(1, 2) = [0, 0]*is*a*multipoint, number*of*branches = 2

Pt*(2, 1) = [15/2, -45*7^(1/2)/2], tg*(2, 1) = 43*7^(1/2)*x/7-480*7^(1/2)/7

Pt*(2, 2) = [15/2, 45*7^(1/2)/2], tg*(2, 2) = -43*7^(1/2)*x/7+480*7^(1/2)/7

Tangents*on*singular*multipoints*of*the*curve

mPt*(1, 1) = [0, 0], mtg*(1, 1) = -15^(1/2)*x

mPt*(1, 2) = [0, 0], mtg*(1, 2) = 15^(1/2)*x

example for a call of sarea

> erg:=sarea(f,g,x=-2..2,1,0):

Results

> lower_bound:=op(1,erg);upper_bound:=op(2,erg);Sint:=evalf(op(3,erg));Sarea:=op(4,erg);

lower_bound := -2

upper_bound := 2

Sint := -.6866300533

Sarea := 3.134847678

> display(p,labels=[` `,` `]);

[Plot]