- #1
Thinkaholic
- 19
- 6
Hi! I know all of you might know what I'm about to post, but I just discovered it for myself, and I want to share my enthusiasm.
Let
and
(here, I'll be restricting the domain of f(x) to the positive real numbers.)
Here is a graph of the two, with f(x) in blue and F(x) in black:
1st question: Where does f(x) intersect with the line y=x?
you could write
squaring both sides of the equation, multiplying both sides by x-1, and subtracting x from both sides gives
Factoring x from the LHS and dividing both sides by x leaves you with
This is the minimal polynomial for the golden ratio, or φ, and the minimal polynomial for -φ^-1, or -Φ. This means that the quadratic above has two solutions at φ and -Φ. -Φ cannot be the solution we are looking for, as, as stated above, I am only dealing with f(x) within the domain of the positive real numbers (positive x values only). So, the intersection of f(x) and y=x is at (φ,φ)!
Question 2: What is the value of
?
Using L'Hospital's Rule, we obtain that
calling the limit as x approaches infinity of f(x) "L", then this becomes
and obviously L=1. So
.
Question 3: What is the derivative of f(x) at (φ,φ)?
If we take the derivative of f(x), plug in φ for x, and make sure to remember that φ-1=Φ and that φ^-1=Φ, we simplify:
Question 4 (finale): What is
?
From the fundamental theorem of calculus
So we could rewrite this as:
Hope I made no typos! Sorry if this is too long, but I want to share these interesting facts with y'all. Also, hopefully the type doesn't mess up, I used rendered LaTeX and pasted the images here. Also, the prefix is beginner, as most of the calculus stuff is taught in high school, but I really don't know what this is, so sorry if that is wrong.
Let
and
Here is a graph of the two, with f(x) in blue and F(x) in black:
1st question: Where does f(x) intersect with the line y=x?
you could write
squaring both sides of the equation, multiplying both sides by x-1, and subtracting x from both sides gives
Factoring x from the LHS and dividing both sides by x leaves you with
This is the minimal polynomial for the golden ratio, or φ, and the minimal polynomial for -φ^-1, or -Φ. This means that the quadratic above has two solutions at φ and -Φ. -Φ cannot be the solution we are looking for, as, as stated above, I am only dealing with f(x) within the domain of the positive real numbers (positive x values only). So, the intersection of f(x) and y=x is at (φ,φ)!
Question 2: What is the value of
Using L'Hospital's Rule, we obtain that
calling the limit as x approaches infinity of f(x) "L", then this becomes
and obviously L=1. So
Question 3: What is the derivative of f(x) at (φ,φ)?
If we take the derivative of f(x), plug in φ for x, and make sure to remember that φ-1=Φ and that φ^-1=Φ, we simplify:
From the fundamental theorem of calculus
So we could rewrite this as:
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