User:ThaniosAkro/sandbox

=Big Matrix 64 by 65=

$$$$$$$$

A network containing 64 resistors is connected as shown in diagram.

What is resistance R between points P1, P2?

While this resistive circuit may be very theoretical, it lends itself well to the creation of a big matrix, 64 by 65, meaning that there are 64 resistors and 64 simultaneous equations to be solved.

Create matrix
An examination of the circuit produces the following conditions:

In iconditions and econditions there are 63 conditions that include all values of i and e. These 63 conditions are used to create matrix.

The 64th condition is added manually.

The matrix is: Bottom line of matrix is equivalent to: $$(1)i10 + (-1) = 0$$ or $$i10 = 1.$$

First result
The following conditions are included for testing: Produce results: $$\text{R}$$ is printed with precision of 60.

Matrix corrected
Original assumptions were that currrent flow through 64 resistors would be from bottom to top or from left to right.

Examination of results shows that flow through R70, R73 is from right to left. This is reflected in testing below.

Diagram shows positive direction of current through R70, R73.

$$\text{R}$$ is printed with precision of 56.

With precision of 56 all 64 results are equal.

=Conic sections generally= Within the two dimensional space of Cartesian Coordinate Geometry a conic section may be located anywhere and have any orientation.

This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of the section, and also how to calculate the foci and directrices given the equation.

Slope of curve
Given equation of conic section: $$Ax^2 + By^2 + Cxy + Dx + Ey + F = 0,$$

differentiate both sides with respect to $$x.$$

$$2Ax + B(2yy') + C(xy' + y) + D + Ey' = 0$$

$$2Ax + 2Byy' + Cxy' + Cy + D + Ey' = 0$$

$$2Byy' + Cxy' + Ey' + 2Ax + Cy + D = 0$$

$$y'(2By + Cx + E) = -(2Ax + Cy + D)$$

$$y' = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}$$

For slope horizontal: $$2Ax + Cy + D = 0.$$

For slope vertical: $$Cx + 2By + E = 0.$$

For given slope $$m = \frac{-(2Ax + Cy + D)}{Cx + 2By + E}$$

$$m(Cx + 2By + E) = -2Ax - Cy - D$$

$$mCx + 2Ax + m2By + Cy + mE + D = 0$$

$$(mC + 2A)x + (m2B + C)y + (mE + D) = 0.$$

$$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$

Quadratic function
$$y = \frac{x^2 - 14x - 39}{4}$$

$$\text{line 1:}\ x = 7$$

$$\text{line 2:}\ x = 17$$

$$$$

y = f(x)
Consider conic section: $$(-1)x^2 + (0)y^2 + (0)xy + (14)x + (4)y + (39) = 0.$$

This is quadratic function: $$y = \frac{x^2 - 14x - 39}{4}$$

Slope of this curve: $$m = y' = \frac{2x - 14}{4}$$

Produce values for slope horizontal, slope vertical and slope $$5:$$ $$$$$$$$$$$$$$$$$$$$ Check results:

$$$$ $$$$ $$$$ $$$$ $$$$ $$$$

x = f(y)
$$x = \frac{-(y^2 + 14y + 5)}{4}$$

$$\text{line 1:}\ y = -7$$

$$\text{line 2:}\ y = -11$$

Consider conic section: $$(0)x^2 + (-1)y^2 + (0)xy + (-4)x + (-14)y + (-5) = 0.$$

This is quadratic function: $$x = \frac{-(y^2 + 14y + 5)}{4}$$

Slope of this curve: $$\frac{dx}{dy} = \frac{-2y - 14}{4}$$

$$m = y' = \frac{dy}{dx} = \frac{-4}{2y + 14}$$

Produce values for slope horizontal, slope vertical and slope $$0.5:$$ $$$$$$$$$$$$$$$$$$$$ Check results:

$$$$ $$$$ $$$$ $$$$ $$$$ $$$$

Parabola
$$(9)x^2 + (16)y^2 + (-24)xy + (104)x + (28)y + (-144) = 0$$

$$\text{Line 1:}$$

$$(18)x + (-24)y + (104) = 0$$

$$\text{Line 2:}$$

$$(-24)x + (32)y + (28) = 0$$

$$\text{Line 3:}$$

$$(-30)x + (40)y + (160) = 0$$

$$$$$$$$

Consider conic section: $$(9)x^2 + (16)y^2 + (-24)xy + (104)x + (28)y + (-144) = 0.$$

This curve is a parabola.

Produce values for slope horizontal, slope vertical and slope $$2:$$ $$$$$$$$$$$$$$$$$$$$ Because all 3 lines are parallel to axis, all 3 lines have slope $$\frac{3}{4}.$$

Produce values for slope horizontal, slope vertical and slope $$0.75:$$ $$$$$$$$$$$$$$$$$$$$ Axis has slope $$0.75$$ and curve is never parallel to axis.

$$$$ $$$$ $$$$ $$$$ $$$$ $$$$

Ellipse
$$(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0$$

$$\text{Line 1:}$$

$$(3542)x + (1944)y + (-44860) = 0$$

$$\text{Line 2:}$$

$$(1944)x + (2408)y + (-18520) = 0$$

$$\text{Line 3:}$$

$$(1598)x + (-464)y + (-26340) = 0$$

Consider conic section: $$(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0.$$

This curve is an ellipse.

Produce values for slope horizontal, slope vertical and slope $$-1:$$ $$$$$$$$$$$$$$$$$$$$ Because curve is closed loop, slope of curve may be any value including $$\frac{1}{0}.$$

If slope of curve is given as $$\frac{1}{0},$$ it means that curve is vertical at that point and tangent to curve has equation $$x = k.$$

For any given slope there are always 2 points on opposite sides of curve where tangent to curve at those points has the given slope. $$$$ $$$$ $$$$ $$$$ $$$$ $$$$

Hyperbola
$$(-351)x^2 + (176)y^2 + (-336)xy + (4182)x + (-3824)y + (-16231) = 0$$

$$\text{Line 1:}$$

$$(-702)x + (-336)y + (4182) = 0$$

$$\text{Line 2:}$$

$$(-336)x + (352)y + (-3824) = 0$$

$$\text{Line 3:}$$

$$(-1374)x + (368)y + (-3466) = 0$$

$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Consider conic section: $$(-351)x^2 + (176)y^2 + (-336)xy + (4182)x + (-3824)y + (-16231) = 0.$$

This curve is a hyperbola.

Produce values for slope horizontal, slope vertical and slope $$2:$$ $$$$$$$$$$$$$$$$$$$$

$$$$ $$$$ $$$$ $$$$ $$$$ $$$$

Asymptotes of Hyperbola
Let the conic section have the familiar equation: $$Ax^2 + By^2 + Cxy + Dx + Ey + F = 0.$$

Let a line have equation: $$y = f(x) = mx + c.$$

Let $$f(x)$$ intersect the conic section. For $$y$$ in equation of conic section substitute $$(mx + c).$$

$$y = g(x) = Ax^2 + B(mx+c)^2 + Cx(mx+c) + Dx + E(mx+c) + F = 0.$$

Expand $$g(x),$$ simplify, gather like terms and result is quadratic function in $$x:$$

$$h(x) = (a$$_$$)x^2 + (b$$_$$)x + (c$$_$$) = 0$$ where:

$$a$$_ $$ = A + Bm^2 + Cm$$

$$b$$_ $$ = 2Bcm + cC + D + Em$$

$$c$$_ $$ = Ec + F + Bc^2$$

If line $$f(x)$$ is an asymptote, then $$a$$_ $$ = b$$_ $$ = 0,$$ in which case:

$$j(m) =\ ($$_$$a)m^2 + ($$_$$b)m + ($$_$$c) = 0$$ where:

_$$a = B;\ $$_$$b = C;\ $$_$$c = A$$

and $$m_1,m_2$$ (roots of $$j(m)$$) are the slopes of the 2 asymptotes.

If $$b$$_ $$ == 0:\ c = \frac{-(D + Em)}{2Bm + C}.$$

Implementation
$$(-351)x^2 + (176)y^2 + (-336)xy + (4182)x + (-3824)y + (-16231) = 0$$

Gallery
$$$$$$$$$$$$$$$$$$$$$$$$$$$$

An example
Consider conic section: $$(-351)x^2 + (176)y^2 + (-336)xy + (4182)x + (-3824)y + (-16231) = 0.$$

This curve is a hyperbola. $$$$$$$$$$$$$$$$$$$$

$$$$ $$$$ $$$$ $$$$ $$$$ $$$$

Latera recta et cetera
"Latus rectum" is a Latin expression meaning "straight side." According to Google, the Latin plural of "latus rectum" is "latera recta," but English allows "latus rectums" or possibly "lati rectums." The title of this section is poetry to the eyes and music to the ears of a Latin student and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic.

The translation of the title is "Latus rectums and other things." This section describes the calculation of interesting items associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines.

When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above. Then verify that the curve is in fact an ellipse.

From these values everything about the ellipse may be calculated. For example:

Consider conic section: $$1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.$$

This curve is ellipse with random orientation. $$$$ $$$$ $$$$ $$$$ $$$$ $$$$

Major axis
Techniques similar to above can be used to calculate points $$I2, ID2.$$

Latus rectums
Techniques similar to above can be used to calculate points $$R, S.$$

Checking
All interesting points have been calculated without using equations of any of the relevant lines.

However, equations of relevant lines are very useful for testing, for example:


 * Check that points $$ID2, I2, F2, M, F1, I1, ID1$$ are on axis.


 * Check that points $$R, F2, S$$ are on latus rectum through $$F2.$$


 * Check that points $$Q, M, T$$ are on minor axis through $$M.$$


 * Check that points $$P, F1, U$$ are on latus rectum through $$F1.$$

Test below checks that 8 points $$I1, I2, P, Q, R, S, T, U$$ are on ellipse and satisfy eccentricity $$e = 0.9.$$ $$$$ $$$$

Note the differences between "raw" values of $$e_1$$ and "clean" values of $$e_2.$$

$$$$ $$$$ $$$$ $$$$

Traditional definition of ellipse
Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant.

In the diagram the two fixed points are the foci, Focus 1 or F1 and Focus 2 or F2.

Distance between F1 and F2, distance_F1_F2, must be non-zero.

Point G on perimeter of ellipse moves so that sum of distance_F1_G and distance_F2_G is constant.

Points T1 and T2 are on axis of ellipse and the same rule applies to these points.

distance_F1_T1 + distance_T1_F2 is constant.

distance_F1_T1 + distance_T1_F2

$$=$$ distance_F1_G + distance_G_F2

$$=$$ distance_F2_T2 + distance_T1_F2

$$=$$ length_of_major_axis.

Therefore the constant is length_of_major_axis which must be greater than distance_F1_F2.

From information given, calculate eccentricity $$e$$ and equation of one directrix. Choose directrix 1 $$dx1$$ associated with focus F1.

$$$$

Properties of ellipse
In diagram:

Point $$M$$ is center of ellipse.

Line $$M F_1$$ is major axis. Length of major axis = 2$$(\text{distance}\ M I_1)$$.

Line $$M R$$ is minor axis. Length of minor axis = 2$$(\text{distance}\ M R)$$.

Line $$I_2 R_1$$ is Directrix 1.

By definition, $$\frac{\text{distance}\ R F_1} {\text{distance}\ R R_1 } = e = \frac{te}{t}$$. Let $$e$$ be positive.

Because$$\text{distance}\ M I_1 = \text{distance}\ R F_1$$, $$\frac{\text{distance}\ M I_1} {\text{distance}\ R R_1 } = e$$ $$= \frac{\text{length of major axis}} {\text{distance between directrices}}$$.

$$\text{distance}\ F_1 I_1 = te - h$$

$$\text{distance}\ I_1 I_2 = t - te$$

By definition, $$\frac{\text{distance}\ F_1 I_1 } {\text{distance}\ I_1 I_2 } = \frac{te - h}  {t - te} = e$$

$$te - h = te - tee$$

$$h = te^2$$

$$\frac{h}{te} = \frac{2h}{2te} = e = \frac{\text{distance between foci}}{\text{length of major axis}}$$

$$\frac{h}{t} = \frac{2h}{2t} = e^2 = \frac{\text{distance between foci}}{\text{distance between directrices}}$$

Triangle $$F_1 R_1 R:$$

$$(\text{distance}\ F_1 R)^2 + (\text{distance}\ F_1 R_1)^2$$

$$= (te)^2 + g^2 + (h - t)^2$$

$$= (te)^2 + g^2 + h^2 - 2ht + t^2$$

$$= (te)^2 + (te)^2 -2(te^2)t + t^2$$

$$= t^2 = (\text{distance}\ R R_1)^2$$

Triangle $$F_1 R_1 R$$ is right triangle.

$$\frac{ \text{distance} \ M F_1 } { \text{distance} \ R F_1 } = e = \cos \angle   M F_1 R  $$

$$\frac{ \text{distance} \ R F_1 } { \text{distance} \ R R_1 } = e = \cos \angle  F_1 R R_1     $$

Minor axis:

$$g^2 = (te)^2 - h^2$$

$$= (te)^2 - (te)^2 e^2$$

$$= (te)^2(1 - e^2)$$

$$\frac{g^2}{(te)^2} = (\frac{\text{length of minor axis}}{\text{length of major axis}})^2 = (1 - e^2)$$

Latus rectum:

$$\frac{\text{distance}\ P F_1 } {\text{distance}\ F_1 I_2 } = e$$

$$\text{distance}\ P F_1 = e (\text{distance}\ F_1 I_2)$$ $$= e(t - h)$$

$$\text{length of latus rectum} = 2e(t - h)$$ $$= 2e(t - te^2)$$ $$= 2et(1 - e^2)$$ $$= (\text{length of major axis})(1-e^2)$$

$$\frac{\text{length of latus rectum}} {\text{length of major axis}} = (1 - e^2)$$ $$$$

$$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$

Intercept form of equation
$$\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1$$ $$$$ $$$$

In diagram:

Intercept $$I_1$$ has coordinates $$(a,0).$$

Intercept $$I_2$$ has coordinates $$(-a,0).$$

Intercept $$A$$ has coordinates $$(0,b).$$

Intercept $$B$$ has coordinates $$(0,-b).$$

Focus $$F_1$$ has coordinates $$(f,0)$$ where $$f = ea.$$

Focus $$F_2$$ has coordinates $$(-f,0).$$

Curve has equation $$\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,$$ called intercept form of equation of ellipse because intercepts are apparent as the fractional value of each coefficient.

Standard form of this equation is: $$(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.$$ While the standard form is valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and intercepts are neatly contained within the equation.

Slope of curve:

$$b^2x^2 + a^2y^2 = a^2b^2$$

Derivative of both sides:

$$b^22x + a^22yy' = 0$$

$$y' = \frac{-xb^2}{ya^2}$$ $$= \frac{-x(1-e^2)}{y}$$

At point $$P$$ on latus rectum $$PQ:$$

$$m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e$$

Slope of line $$PD = m_2 = \frac{PF_1}{F_1D} = e$$

$$m_1 = m_2.$$ Line $$PD$$ is tangent to curve at latus rectum, point $$P.$$ $$\text{ }$$ $$\text{ }$$ $$\text{ }$$ $$\text{ }$$ $$\text{ }$$ $$\text{ }$$ $$\text{ }$$ $$\text{ }$$ $$\text{ }$$ $$\text{ }$$ $$\text{ }$$ $$\text{ }$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$ $$$$

Example
Given: Calculate equation of ellipse. Equation of ellipse in standard form:

$$(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0$$

For more insight into method of calculation and proof: $$(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0$$ $$$$ $$$$ $$$$ $$$$ $$$$

=quartic= A close examination of coefficients $$R, S$$ shows that both coefficients are always exactly divisible by $$4.$$

Therefore, all coefficients may be defined as follows:

$$P = 1$$

$$Q = A2$$

$$R = \frac{A2^2 - C}{4}$$

$$S = \frac{-B4^2}{4}$$ $$$$ $$$$

The value $$Rs - Sr$$ is in fact:

which, by removing values $$aa, ad$$ (common to all values), may be reduced to: If $$status == 0,$$ there are at least 2 equal roots which may be calculated as shown below.

If coefficient $$d$$ is non-zero, it is not necessary to calculate $$status.$$

If coefficient $$d == 0,$$ verify that $$status = 0$$ before proceeding.

Examples
$$y = f(x) = x^4 + 6x^3 - 48x^2 - 182x + 735$$

$$y' = g(x) = 4x^3 + 18x^2 - 96x - 182$$

$$y       =        -182x^3        -        4032x^2      -     4494x      +      103684$$

$$y = -12852x^2 - 35448x + 381612$$

$$y =  -381612x^2  -  1132488x  +  10771572$$

$$y = 7191475200x + 50340326400$$

$$y = -1027353600x - 7191475200$$

$$$$

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Python function  below implements   as presented under Equal roots above. When description contains note $$B4 = 0,$$ depressed quartic was processed as quadratic in $$t^2.$$

$$$$ $$$$ $$$$ $$$$

Two real and two complex roots
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gallery
C $$$$ $$$$ $$$$ $$$$

$$y = \frac{x^5 + 13x^4 + 25x^3 - 165x^2 - 306x + 432}{915.2}$$

$$$$ $$$$ $$$$ $$$$

=allEqual=

$$y = f(x) = x^3$$

$$y = f(-x)$$

$$y = f(x) = x^3 + x$$

$$x = p$$

$$y = f(x) = (x-5)^3 - 4(x-5) + 7$$

Wikiversity is a Wikimedia Foundation project devoted to learning resources, learning projects, and research for use in all levels, types, and styles of education from pre-school to university, including professional training and informal learning. We invite teachers, students, and researchers to join us in creating open educational resources and collaborative learning communities. To learn more about Wikiversity, try a guided tour, learn about adding content, or start editing now.

Welcomee
 Wikiversity is a Wikimedia Foundation project devoted to learning resources, learning projects, and research for use in all levels, types, and styles of education from pre-school to university, including professional training and informal learning. We invite teachers, students, and researchers to join us in creating open educational resources and collaborative learning communities. To learn more about Wikiversity, try a guided tour, learn about adding content, or start editing now.

Welcomen
 Wikiversity is a Wikimedia Foundation project devoted to learning resources, learning projects, and research for use in all levels, types, and styles of education from pre-school to university, including professional training and informal learning. We invite teachers, students, and researchers to join us in creating open educational resources and collaborative learning communities. To learn more about Wikiversity, try a guided tour, learn about adding content, or start editing now.

=Testing=

table1
Coefficient $$a$$ may be negative as shown in diagram.

As $$abs(x)$$ increases, the value of $$f(x)$$ is dominated by the term $$-ax^3.$$

When $$x$$ has a very large negative value, $$f(x)$$ is always positive.

When $$x$$ has a very large positive value, $$f(x)$$ is always negative.

Unless stated otherwise, any reference to "cubic function" on this page will assume coefficient $$a$$ positive.

$$x_{poi} = -1$$ $$$$ $$$$ $$$$ $$$$

Various planes in 3 dimensions
$$\theta_1$$



$$O\ (0,0,0)$$

$$M\ (A_1,B_1,C_1)$$

$$N\ (A_2,B_2,C_2)$$

$$\theta$$

$$\ \ \ \ \ \ \ \ $$


 * $$\begin{align}

(6) - (7),\ 4Apq + 2Bq =&\ 0\\ 2Ap + B =&\ 0\\ 2Ap =&\ - B\\ \\                    p =&\ \frac{-B}{2A}\ \dots\ (8) \end{align}$$

$$\ \ \ \ \ \ \ \ $$


 * $$\begin{align}

1.&4141475869yugh\\ &2645er3423231sgdtrf\\ &dhcgfyrt45erwesd \end{align}$$

$$\ \ \ \ \ \ \ \ $$



4\sin 18^\circ = \sqrt{2(3 - \sqrt 5)} = \sqrt 5 - 1 $$