Attn Calc II students: This is a fun little integral. \( \int 2^{\ln(x)}dx \) A substitition \( u=\ln(x) \) won't work since there is no \( \frac{1}{x} \) to make the match. #integration #exponentiation
Think about before reading on.
Nifty trick! Since \( 2=e^{\ln(2)} \), then \( 2^{\ln(x)} = (e^{\ln(2)})^{\ln(x)}=(e^{\ln(x)})^{\ln(2)}=x^{\ln(2)} \)
Now the integral becomes \( \int x^{\ln(2)}dx = \frac{1}{1+ln(2)} x^{1+\ln(2)} + C \)
Swapping from an constant-to-variable to a variable-to-constant made a world of difference. Sweet!
Do you use your iPhone Calculator app for quick calculations? In case you didn't know, rotate your phone to landscape and it becomes a scientific calculator with radicals, exponentials, logarithms, trig functions, even hyperbolic trig functions.
Press "2nd" to get more functions including log-base-b, inverse trigs, inverse hyperbolic trigs.
One more hack: Entering numbers into the command line and mess up? No need to "C" clear all, just swipe right on the command line to delete the previous digit.
While DESMOS app and GeoGebra Apps are amazing for graphing, the built-in Calculator app on iPhone is handy! I keep it in my Control Center for easy access.
Quick read on the history of the decimal point from Scientific American mag.
"The decimal point was a step forward for humanity, enabling the ease and efficiency of calculations that underpin modern science and technology."
Solve x/(y+z)+y/(x+z)+z/(x+y)=4 where x, y, z are positive whole numbers. This problem going around the web (https://blog.computationalcomplexity.org/2024/04/a-math-question-and-meta-question.html) interestingly IS able to be solved by Perplexity AI, but Genesis AI and ChatGPT each conclude "no solution".
SPOILER! Solution from Perplexity AI:
The equation x/(y+z) + y/(x+z) + z/(x+y) = 4, where x, y, z are positive integers, is a challenging Diophantine equation that has puzzled mathematicians for a long time. The smallest known solution to this equation is:
x = 36875131794129999827197811565225474825492979968971970996283137471637224634055579
y = 36875131794129999827197811565225474825492979968971970996283137471637224634055579
z = -36875131794129999827197811565225474825492979968971970996283137471637224634055579
This solution was found by the mathematicians Andrew Bremner and Allan MacLeod, and published in their 2014 paper "An unusual cubic representation problem".
Fun and informative spring break sci-fi watch on Netflix: "The Three-Body Problem".
Background info on the #mathematics #astronomy #physics and movie review: https://www.bbc.com/future/article/20240328-the-science-astronomy-and-mathematics-of-netflixs-3-body-problem-tv-show
Jin (Jess Hong) explains the Three-Body Problem: https://www.youtube.com/watch?v=KMJ7qUihfaY
Happy St. Paddy's Day!
"On October 16, 1843, the great Irish mathematician William Rowan Hamilton was walking along the Royal Canal in Dublin. He had been pondering for a long time whether complex numbers could be extended to higher dimensions. During his perambulation, he realized the answer was “yes”, and carved his solution on the Brougham Bridge. The plaque shown [below] commemorates the discovery." #ComplexNumbers #quaternions
https://galileospendulum.org/2013/03/17/irish-mathematics-for-st-patricks-day/
The title is a bit hyperbolic (pun intended) but I suspect there is at least one item of interest to you.
Happy π Day!
https://www.newscientist.com/article/2418890-these-7-mathematical-facts-will-blow-your-mind/
I've played #Tetris since 1984 on a black-and-white "portable" Tele-DOS computer (whose "hard" drive was on a 5.25" floppy disk, no kidding). Good article noting the real-world usefulness of transformational #geometry and optimization in architecture, engineering, animation, graphic designing, calculus, differential geometry, business, and more. "All of these advanced topics use the same concepts as the simple moves of Tetris."
https://phys.org/news/2024-02-play-tetris-architects-animators-alike.html
Another good read from @johndcook. Unfortunately I can't (won't) boost the toot because the image is missing ALT text.
This-Way-Up and Knuth's Up Arrow Notation: https://www.johndcook.com/blog/2024/02/10/this-way-up-and-knuth-arrows/
Love playing with anything Donald Knuth (https://www-cs-faculty.stanford.edu/~knuth/). The "up arrow" is a repeated exponentiation operator. One of the things that intrigued me early in my study of #mathematics.
Attn my #Calculus II students studying integrals of arctrig functions this week: Fun integration problem whose answer involves both pi and the golden ratio, phi!
\[\int_0^{\pi}\arctan\left(1+\cos x\right)dx=arc\cot\left(\sqrt{\phi}\right)\]
"Combining mathematics and art is an inevitable extension of their relationship." Art can help one visualize mathematics in a way where new patterns and relationships can be realized. This article shares some specific examples.
https://www.sciencenews.org/article/mutual-inspiration-art-and-mathematics
@johndcook posted fun triangle problem yesterday appropriate for a pre-calculus course. I won't "boost" because the sketch is missing #AltText. Here is a re-toot instead w/ ALT.
Is this triangle a right triangle? How can you tell? If it is not, what's the measure of the largest angle?
SPOILER! It is not a right triangle (check Pythagorean Theorem). The largest angle measures approximately 90.000003 degrees (by Law of Cosines). Oh so close!
Beyond the gorgeous photography of ice sheets, glacier covered mountains, blue seas, we get modeling ice sheet mathematics with equations involving partial derivatives explained in intuitive manner. Turn on CC because the audio is really low. https://youtu.be/VHelUNnq5To?si=iZ7-XRIYOWRDbNZs
James Bond - Agent 7? and other problems with leading zeroes. Good quick read, especially for programmers and coders, from John D. Cook.
Grab some strips of paper as you work through this article from Quanta Mag. #MobiusStrip
"In 1977, two mathematicians conjectured that to be twisted in a Möbius strip, a rectangle of width 1 must be longer than √3, as in the strip on the lower right. In August 2023, Schwartz proved that they were correct: Any closer to a square than that, and there’s no way to twist the rectangle into a Möbius strip."
https://www.quantamagazine.org/mathematicians-identify-the-best-versions-of-iconic-shapes-20240105/
Maths in Real Life: Fighting Big Ag Pollution with Maps and Math
"While geostatistics involves very complex math, illustrating spatial data and analytical findings on a map can help communicate the science by rooting the information in a personal sense of place, like showing how a factory-polluted river empties into a lake where people swim, boat, and fish."
"As a spatial statistician working on environmental policy, I use math to make sense of our environment and draw maps to communicate my research findings to policymakers and the general public."
When is f(g(x)) ≤ g(f(x))? What conditions are necessary for f(g(x)) ≤ g(f(x))? Are there real-life applications for this inequality? Worthy questions to consider from intermediate algebra through calculus.
These questions are posed by John D. Cook. I look forward to comments being posted there.
https://www.johndcook.com/blog/2023/12/06/fgx-versus-gfx/
SPOILER ALERT! Theorem below.
![Let f be continuous with domain 0 ≤ x < 1 or 0 ≤ x ≤ 1, f(0) = 0, f(1) > 1 (including the possibility that f(1) = +∞); let g be continuous with domain the range of f, and g(1) ≤ 1. Let f(x)/x and g(x)/x be strictly increasing on their domains. Finally let f(x) ≠ x for 0 < x < 1. Then f(g(x)) ≤ g(f(x)) for 0 < x < 1.
[1] Ralph P. Boas. Inequalities for a Collection. Mathematics Magazine, January 1979, Vol. 52,No. 1, pp. 28-31](https://media.mathstodon.xyz/media_attachments/files/111/552/040/149/017/180/original/038666dbc4a32526.gif "Let f be continuous with domain 0 ≤ x < 1 or 0 ≤ x ≤ 1, f(0) = 0, f(1) > 1 (including the possibility that f(1) = +∞); let g be continuous with domain the range of f, and g(1) ≤ 1. Let f(x)/x and g(x)/x be strictly increasing on their domains. Finally let f(x) ≠ x for 0 < x < 1. Then f(g(x)) ≤ g(f(x)) for 0 < x < 1.
[1] Ralph P. Boas. Inequalities for a Collection. Mathematics Magazine, January 1979, Vol. 52,No. 1, pp. 28-31")
Interesting piece of the Aperiodical's Carnival 222 (https://aperiodical.com/carnival-of-mathematics/) hosted this month by John D. Cook (https://www.johndcook.com/blog/2023/12/02/222nd-carnival-of-mathematics/) is a conversation with ChatGPT about what happens when you ask ChatGPT to solve Elchanan Mossel’s dice problem.
"You throw a die until you get 6. What is the expected number of throws (including the throw giving 6) conditioned on the event that all throws gave even numbers?"
Try answering the problem on your own first before reading on. Careful, there is a common misconception important to avoid.
https://chat.openai.com/share/37432585-8712-4d5b-be37-a730047c9887