Elementary Calculus: Example 3: Inscribing a Cylinder Into a Sphere Find the shape of the cylinder of maximum volume which can be inscribed in a given sphere. The shape of a right circular cylinder can be described by the ratio of the radius of its base to its height. This ratio for the inscribed cylinder of maximum volume should be a number which does not depend on the radius of thesphere. For example, we should get the same shape whether the radius of the sphere is given in inches or centimeters. Let r be the radius of the given sphere, x the radius of the base of the cylinder, h its height, and V its volume. Figure 3.6.4 From the sketch we can read off the formulas V = πx2h, x2 + (½h)2 = r2, 0 ≤ x ≤ r. r is a constant. Solution One: Eliminating One Variable Sollution Two: Implicit Differentiation
Question Corner -- Why is e^(pi*i) = -1? Navigation Panel: (These buttons explained below) Question Corner and Discussion Area Asked by Brad Peterson, student, Roy High on January 29, 1997: I was watching an episode of The Simpsons the other day, the one where Homer gets sucked into the third dimension, and in this 3-D world, there was an equation that said . So I put it into the calculator and it worked, but I have no idea why, because e to any power isnt supposed to be a negative number, and I thought pi was in no way related to e. We'd be glad to explain; that's exactly what this area is here for. The first question to ask, though, is not "why does ", but rather, "what does even mean?" Once that question is answered, it will be much more clear why . for all x, a fact which is known as de Moivre's formula, and illustrates how closely related the exponential function is to the trigonometric functions. So now, the question is, why is the "right" thing to define what e raised to an imaginary power means? (i.e., a); in other words, . If .
AvantGame by Jane McGonigal 8 math talks to blow your mind Mathematics gets down to work in these talks, breathing life and logic into everyday problems. Prepare for math puzzlers both solved and unsolvable, and even some still waiting for solutions. Ron Eglash: The fractals at the heart of African designs When Ron Eglash first saw an aerial photo of an African village, he couldn’t rest until he knew — were the fractals in the layout of the village a coincidence, or were the forces of mathematics and culture colliding in unexpected ways? How big is infinity? Arthur Benjamin does “Mathemagic” A whole team of calculators is no match for Arthur Benjamin, as he does astounding mental math in the blink of an eye. Scott Rickard: The beautiful math behind the ugliest music What makes a piece of music beautiful? Margaret Wertheim: The beautiful math of coralThe intricate forms of a coral reef can only be expressed through hyperbolic geometry — and the only way humans can model it is by crocheting!
Geometry Help Looking for some Geometry Help? Our materials here review the basic terms and concepts in geometry and provide further lessons to help you develop your understanding of geometry and its applications to solving problems in real life. Geometry is about the shape and size of things. It is the study of points, lines, angles, shapes, their relationships, and their properties. Videos have been included in almost all the following topics to help reinforce your understanding. Angles Triangles Polygons Circles Circle Theorems Solid Geometry Geometrical Formulas Coordinate Geometry and Graphs Geometric Construction Geometry Transformation Geometry Proofs (Videos) Triangle Medians and Centroids (2D Proof) Area Circumradius Formula Proof Proof that the diagonals of a rhombus are perpendicular bisectors of each other Geometry Practice Questions Free SAT Practice Questions (with Hints & Solutions) - Geometry OML Search We welcome your feedback, comments and questions about this site or page.
Happy number A happy number is a number defined by the following process: Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number equals 1 (where it will stay), or it loops endlessly in a cycle which does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers (or sad numbers).[1] Overview[edit] More formally, given a number , define a sequence , ... where is the sum of the squares of the digits of . If a number is happy, then all members of its sequence are happy; if a number is unhappy, all members of the sequence are unhappy. For example, 19 is happy, as the associated sequence is: The 143 happy numbers up to 1,000 are: The happiness of a number is preserved by rearranging the digits, and by inserting or removing any number of zeros anywhere in the number. Sequence behavior[edit] If n is not happy, then its sequence does not go to 1. , or For and above,
public data Indicateurs de développement humain Rapport sur le développement humain 2013, Programme des Nations Unies pour le développement Les données utilisées pour calculer l'Indice de développement humain (IDH) et autres indices composites présentés dans le Rapport sur le développement humain ... Eurostat, Indicateurs démographiques Eurostat Indicateurs démographiques annuels. Chômage en Europe (données mensuelles) données sur le chômage harmonisé pour les pays européens. Salaire minimum en Europe Salaire mensuel brut minimum en euros ou parités de pouvoir d'achat, données semi-annuelles. Dette publique en Europe Statistiques sur les finances publiques des pays européens.
Learning Calculus: Overcoming Our Artificial Need for Precision Accepting that numbers can do strange, new things is one of the toughest parts of math: There’s numbers between the numbers we count with? (Yes — decimals)There’s a number for nothing at all? (Sure — zero)The number line is two dimensional? Calculus is a beautiful subject, but challenges some long-held assumptions: Numbers don’t have to be perfectly accurate? Today’s post introduces a new way to think about accuracy and infinitely small numbers. Counting Numbers vs. Not every number is the same. Our first math problems involve counting: we have 5 apples and remove 3, or buy 3 books at $10 each. We later learn about fractions and decimals, and things get weird: What’s the smallest fraction? It gets worse. We’re hit with a realization: we have limited accuracy for quantities that are measured, not counted. What do I mean? That’s cute, but you didn’t answer my question — what number is it? You may pout, open your calculator and say it’s “18.8495…”. We don’t know! Why? But I need exact numbers!
Numbers Real Number Sets Real Number Line Real Number Venn Diagram Complex Number Sets Complex Number Plane z = x + iy, i = √−1 Complex Number Venn Diagram Properties of the Number Sets Closed under addition (multiplication, subtraction, division) means the sum (product, difference, quotient) of any two numbers in the set is also in the set.Dense: Between any two numbers there is another number in the set.Continuous with no gaps. The complex numbers are the algebraic completion of the real numbers. Infinity, ∞ The integers, rational numbers, and algebraic numbers are countably infinite, meaning there is a one-to-one correspondence with the counting numbers. PDF format for printing PDF | Terms of Use | Buy Poster
Babylon Berlin Cet article ou cette section contient des informations sur une série télévisée en cours de production, programmée ou prévue. Le texte est susceptible de contenir des informations spéculatives et son contenu peut être nettement modifié au fur et à mesure de l’avancement de la série et des informations disponibles s’y rapportant.La dernière modification de cette page a été faite le 6 septembre 2023 à 11:15. Babylon Berlin est une série télévisée allemande dramatique et policière créée par Tom Tykwer, Achim von Borries et Hendrik Handloegten, diffusée depuis le 13 octobre 2017 sur Sky 1, une chaîne de Sky Deutschland, simultanément en Allemagne et en Autriche. Elle est tirée des romans policiers de Volker Kutscher[1], Le premier ouvrage de la série écrite par Volker Kutscher s'intitule Der Nasse Fisch (« le poisson mouillé »). En France, la série est diffusée à partir du 27 août 2018 sur Canal+ en France, et à partir du 5 janvier 2020 sur RTS 1 en Suisse. Musique[modifier | modifier le code]