Loading DocumentVhince Yarzo
Non-Trigonometrical Derigrative Intersections proposes observations and analysis about the behaviours of the intersection points of derigrative functions, without involving trigonometrical relationships.
Derigratives are intersection points originating from the graphs of the derivative and the integrative (no constant) of a function.
There are 2 main postulates for the habit of derigratives in a function
In this observation, there were 2 theorems that have formulated:
Distance Derigrative Theorem
" For functions with m = b = p and/or with x2n−1, the distance between the two derigratives will always be parallel to each other while the graphs evolve overtime from the increase or decrease of p."
This theorem states that the two intersection points from the two graphs of the derivative and the integration - C of a function will always have the same distance from all evolutions of the graphs in any values of p, which is equal to the slope (m) and the y-intercept (b). The functions that makes this theorem true must have m = b = p or b = 0, and/or have x2n−1.
Proof:
"Distance Derigrative Notation"
∫f(x,p)dx−C=A(x,p),f′(x,p)=B(x,p)A(x,p)=B(x,p)⇒±x=x1,x2∀p∈R∣x1−x2∣=D
This equation states that "Regardless of any element of real number for p, the distance of the derigratives will always stabilize to D."
∫f(x,p)dx−C=A(x,p),f′(x,p)=B(x,p)
This term emphasizes that Graph A is equal to the integration - C of a function, and Graph B is equal to the derivative of the function.
A(x,p)=B(x,p)⟹±x=x1,x2
This term now presents the derigratives as actual intersection points, having respect to the subscripts of x in the initial form of a plus-minus value in terms of roots. Basically, this requires the value of x.
∀p∈R∣x1−x2∣=D
Lastly, this term now presents the distance of the derigratives, emphasizing that "For every instances p in the set of real numbers, the distance stabilizes to D." Our goal here is to prevent the term from ending in 0 (since the recent notation suggest that ±x relates to 2 sides of the derigratives)
Graph 1. Demonstrated the Distance Derigrative Theorem with f(x) = px + p
Example: Let f(x) turn to f(x,p) = px and find the derigrative functions
f(x,p)=pxf′(x,p)=p=B(x,p)∫f(x,p)dx−C=2px2=A(x,p)
Next, put all the derigrative functions inside the Distance Derigrative Notations, then solve for x and the distance.
B(x,p)=A(x,p)⟹2px2=p2px2=p→px2=2p→x2=2x2=2→x=±√2(x=x1,x2)∀p∈R∣∣∣√2−(−√2)∣∣∣=2√2
Take note: Most cases involve the use of the quadratic formula to solve for the value of x and find the subscripts, especially for cases where x2n−1,n>1∨b=p
Graph 2. The demonstration of the distance of the two derigratives being parallel to each other, with f(x,p) = px
Therefore, the distance between the derigratives of f(x,p) = px is equal to 2√2, regardless of ANY value of p as p evolves the curve overtime with the Curve Evolution Postulate.
Symmetrical Derigratives Theorem
"For functions with x2n, the distance between a derigrative and the origin (0,0) is always half the distance between the two derigratives with dependence to the changes from p"
This theorem states that when a function has its x to the power of an even number, the distance between one of the derigratives to the origin of the plane is always half the length of the distance between the two derigratives with effects from p. Basically, the distance here changes as p changes, and this goes to show how the y-axis bisects the "imaginary line" or bond between them.
𝗣𝗿𝗼𝗼𝗳:
"Symmetric Derigrative Notation"
∫f(x,p)dx=A(x,p)∧f′(x,p)=B(x,p)A(x,p)=B(x,p)⟹x=(x1,y1),(x2,y2)∀p∈R√(x1−x2)2+(y1−y2)2=dAB∀p∈R2√(x1−x2)2+(y1−y2)2=dAO
Now do not be overwhelmed, yet this is somewhat similar to the Distance Derigrative Notation. This whole notation implies that "For every evolution of real numbers to p, the distance between the two derigratives by their y position changes, but the distance between a derigrative and an origin by their y position will always be half."
∫f(x,p)dx=A(x,p)∧f′(x,p)=B(x,p)
This term again implies that Graph A is from the integrative - C of f(x,p) while Graph B is from the derivative of f(x,p).
A(x,p)=B(x,p)⟹x=(x1,y1),(x2,y2)
This term now states an intersection that forms Derigrative A and Derigrative B, saying "find the value of y and it's subscripts"
∀p∈R√(x1−x2)2+(y1−y2)2=dAB∀p∈R2√(x1−x2)2+(y1−y2)2=dAO
Lastly, this term implies that for every value of p which is an element of real numbers, the distance of position between derigrative A and derigrative B changes, and half of it is the distance between derigrative A or B and an origin.
Graph 3. Demonstration of the Symmetrical Derigrative Theorem with f(x,p) = px²
More theorems would be formulated as the observations evolve.