The general Leibniz rule, follows our post on the binomial theorem, because of their resemblance. This rule gives you a way to obtain the nth derivative of the product of 2 functions f and g which must both be n times differentiable. The proof for this result works just like the binomial theorem, it is shown by induction. Note that here, n is a natural number and that for n = 0, (fg)^0 is by definition fg. See if you can find the nth derivative of a product using this rule !
The binomial theorem can be traced back all the way to the 10th century, in Baghdad where the Persian mathematician Abu Bakr showed this result. It was later formalized by Newton, with the introduction of the binomial coefficients and with the help of Pascal's triangle ( which we posted about already ). The formula is used to calculate binomial expansions, the calculation through which the power of a sum is expressed as a sum of powers of the added values. It can be proven by induction. Using this result, could you calculate the polynomial expansion of (1+x)^3 ? Give it a try and post your result
Known as the binomial coefficients, they're found everywhere. The formula holds for 0 <= k <=n and equals 0 otherwise (k and n are positive integers). Their most basic use is to calculate polynomial expansions. It usually reads "n choose k" because there are "n choose k" ways to choose an (unordered) subset of k elements from a fixed set of n elements, this also means that all the binomial coefficients are integers. The binomial coefficients can also be arranged in Pascal's triangle (picture 2), where the first row are the values of the binomial coefficients for n =0, the second row are the values for n =1 etc... Pascal's triangle has many hidden relations in it, the most well-known is that any number of the triangle is the sum of the 2 numbers above it. Which formula that uses binomial coefficients do you prefer ? Suggested by @keremudemir
What's in a name? Little lion cub, baby bear--common names are the bane of some mycologists' existences, while being the joy of others, mostly us amateurs. Latin binomials are important so one can communicate across cultures, across languages and regions...and be taken seriously by the serious. So one can place things accurately within the web of life. But sometimes I think it's more important to get to know a mushroom--where it likes to hang out, what it likes to do, and tastes like, and feels like--than it is to categorize dozens in a dead language. This cute little #hericiumamericanum has many common names, spanning a few related species, and probably six continents. Whether it's called monkey's head or bear's head, lion's mane or pom pom, you know it is a delicious #saprophyte found on wood, and that it may contain #neurotrophic properties and digestive benefits. You know its friendly, sometimes pink, and in this case much too tiny and cute to even consider taking home to tincture. Sure, it appreciates its big-kid adult name, but I think it also appreciates a little adoration, and a nickname or two.
This #Mongolian#Cat Is The Most #Expressive Cat In #TheWorld#FrownyCat#Frowny#Fangs
The #PallassCat (#Otocolobusmanul ), also called #Manul , is a small #Wildcat with a broad but fragmented distribution in the grasslands and montane steppes of #CentralAsia .
German naturalist Peter Simon Pallas first described the cat in 1776 under the #Binomial#Felismanul .
They inhabit elevations of up to 5,050 m (16,570 ft) in the Tibetan Plateau. They inhabit Mongolia, Tajikistan, Kyrgyzstan, Pakistan, Kazakhstan, Kashmir, and occur across much of western China. They also are found in the Transbaikal regions of Russia, and less frequently, in the Altai, Tyva, and Buryatia Republics. In 1997, they were reported for the first time as being present in the eastern Sayan Mountains.
Meet Luna, a Chaco Golden Knee Tarantula from the @clarkpest Bug Zoo.
Fun with bugs? That’s right! Clark Pest Control has developed a program for grades three through six to share fun facts and get a chance to interact with some of the coolest bugs on the planet. Some of the topics covered are:
*Insect and spider anatomy.
*Where our Bug Zoo pets live.
*How they defend themselves naturally.
*What they eat.
*Each species' technical (binomial) name.
The display includes:
*Tarantulas: Versi-color, Chaco Golden Knee, and Rose Hair.
*Scorpions: African Flat Rock Scorpion.
*Cave Spider (Tail-less Whip Scorpion)
*Vinegaroon (Whip Scorpion)
*Others: (non-bug related) reptiles and more …
This program is a free service provided by Clark Pest Control and is offered to private and public schools. #clarkpestcontrolventura#clarkpestcontrol#clarkpest