It's great to hear that things are going well for you.

I'll illustrate what's happening with an example.

I've chosen the function

When we differentiate we get

and second derivative

Here are the graphs of (black) the function; (red) the first derivative; (green) the second derivative.

I've put the three graphs one after the other, lined up in the x direction but one on top of the other in the y direction. For this reason don't worry about the numbering on the axes; you don't need it. But the changes in gradient for the function are aligned correctly with the first and second derivatives.

The function is a cubic. When you differentiate it you get a quadratic. The quadratic has two zeros and these line up with the turning points on the cubic. When you differentiate again you are finding the gradient function of dy/dx, and you get a linear function.

On the (black) function graph you can see the two turning points; the first a maximum; the second a minimum; but let's say we don't know this yet. Just to the left of a maximum the function has positive gradient; just to the right it is negative. At the turning point the gradient is zero of course.

You can see this on the red graph. Just to the left of the function's turning point the red graph is above the axis so it's values are positive. The red graph crosses its axis at the turning point and after that it has negative values. In a similar way the next turning point (a minimum) has gradient negative to the left of the turning point; zero at the turning point; then positive.

So you can tell if a turning point is a maximum by looking to see if the first derivative goes from positive, through zero, to negative. And a minimum will have first derivative that goes from negative, through zero, to positive.

One way to investigate this is to draw the graphs; another is to calculate gradients just left and right of the turning point. But the second derivative gives a quick way to find out without needing to do any graph drawing. Here's how:

Function has a maximum. Gradient function goes from positive, through zero to negative. So its gradient function must be negative at that point as dy/dx has a reducing gradient.

Function has a minimum. Gradient function goes from negative, through zero, to positive. So its gradient function must be positive at that point as dy/dx has an increasing gradient.

In my example

so the turning points are at x = - √ (1/3) and + √ (1/3) At x = - √ (1/3) the second derivative is negative => this turning point is a minimum.At x = + √ (1/3) the second derivative is postive => this turning point is a maximum.

Note: I don't even have to do the full calculation here; I just need to know if the second derivative is positive or negative at each turning point; so it's a quick way to tell.

Bob

]]>Our Sun (a star) and all the planets around it are part of a galaxy known as the Milky Way Galaxy. A galaxy is a large group of stars, gas, and dust bound together by gravity. They come in a variety of shapes and sizes. The Milky Way is a large barred spiral galaxy. All the stars we see in the night sky are in our own Milky Way Galaxy. Our galaxy is called the Milky Way because it appears as a milky band of light in the sky when you see it in a really dark area.

It is very difficult to count the number of stars in the Milky Way from our position inside the galaxy. Our best estimates tell us that the Milky Way is made up of approximately 100 billion stars. These stars form a large disk whose diameter is about 100,000 light years. Our Solar System is about 25,000 light years away from the center of our galaxy – we live in the suburbs of our galaxy. Just as the Earth goes around the Sun, the Sun goes around the center of the Milky Way. It takes 250 million years for our Sun and the solar system to go all the way around the center of the Milky Way.

We can only take pictures of the Milky Way from inside the galaxy, which means we don't have an image of the Milky Way as a whole. Why do we think it is a barred spiral galaxy, then? There are several clues.

The first clue to the shape of the Milky Way comes from the bright band of stars that stretches across the sky (and, as mentioned above, is how the Milky Way got its name). This band of stars can be seen with the unaided eye in places with dark night skies. That band comes from seeing the disk of stars that forms the Milky Way from inside the disk, and tells us that our galaxy is basically flat.

Several different telescopes, both on the ground and in space, have taken images of the disk of the Milky Way by taking a series of pictures in different directions – a bit like taking a panoramic picture with your camera or phone. The concentration of stars in a band adds to the evidence that the Milky Way is a spiral galaxy. If we lived in an elliptical galaxy, we would see the stars of our galaxy spread out all around the sky, not in a single band.

Another clue comes when astronomers map young, bright stars and clouds of ionized hydrogen in the Milky Way's disk. These clouds, called HII regions, are ionized by young, hot stars and are basically free protons and electrons. These are both important marker of spiral arms in other spiral galaxies we see, so mapping them in our own galaxy can give a clue about the spiral nature of the Milky Way. There are bright enough that we can see them through the disk of our galaxy, except where the region at the center of our galaxy gets in the way.

There has been some debate over the years as to whether the Milky Way has two spiral arms or four.

Additional clues to the spiral nature of the Milky Way come from a variety of other properties. Astronomers measure the amount of dust in the Milky Way and the dominant colors of the light we see, and they match those we find in other typical spiral galaxies. All of this adds up to give us a picture of the Milky Way, even though we can't get outside to see the whole thing.

There are billions of other galaxies in the Universe. Only three galaxies outside our own Milky Way Galaxy can be seen without a telescope, and appear as fuzzy patches in the sky with the unaided eye. The closest galaxies that we can see without a telescope are the Large and Small Magellanic Clouds. These satellite galaxies of the Milky Way can be seen from the southern hemisphere. Even they are about 160,000 light years from us. The Andromeda Galaxy is a larger galaxy that can be seen from the northern hemisphere (with good eyesight and a very dark sky). It is about 2.5 million light years away from us, but its getting closer, and researchers predict that in about 4 billion years it will collide with the Milky Way. , i.e., it takes light 2.5 million years to reach us from one of our "nearby" galaxies. The other galaxies are even further away from us and can only be seen through telescopes.

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#7817. A man bought a television set for $800 and spent $400 for it. Find his gain percent.

]]>#3677. What does the noun *ethnocentrism* mean?

#3678. What does the noun *ethnology* mean?

Prince Louis-Victor de Broglie (Louis Victor Pierre Raymond de Broglie, 7th duc de Broglie) of the French Academy, Permanent Secretary of the Academy of Sciences, and Professor at the Faculty of Sciences at Paris University, was born at Dieppe (Seine Inférieure) on 15th August, 1892, the son of Victor, Duc de Broglie and Pauline d’Armaillé. After studying at the Lycée Janson of Sailly, he passed his school-leaving certificate in 1909. He applied himself first to literary studies and took his degree in history in 1910. Then, as his liking for science prevailed, he studied for a science degree, which he gained in 1913. He was then conscripted for military service and posted to the wireless section of the army, where he remained for the whole of the war of 1914-1918. During this period he was stationed at the Eiffel Tower, where he devoted his spare time to the study of technical problems. At the end of the war Louis de Broglie resumed his studies of general physics. While taking an interest in the experimental work carried out by his elder brother, Maurice, and co-workers, he specialized in theoretical physics and, in particular, in the study of problems involving quanta. In 1924 at the Faculty of Sciences at Paris University he delivered a thesis Recherches sur la Théorie des Quanta (Researches on the quantum theory), which gained him his doctor’s degree. This thesis contained a series of important findings which he had obtained in the course of about two years. The ideas set out in that work, which first gave rise to astonishment owing to their novelty, were subsequently fully confirmed by the discovery of electron diffraction by crystals in 1927 by Davisson and Germer; they served as the basis for developing the general theory nowadays known by the name of wave mechanics, a theory which has utterly transformed our knowledge of physical phenomena on the atomic scale.

After the maintaining of his thesis and while continuing to publish original work on the new mechanics, Louis de Broglie took up teaching duties. On completion of two year’s free lectures at the Sorbonne he was appointed to teach theoretical physics at the Institut Henri Poincaré which had just been built in Paris. The purpose of that Institute is to teach and develop mathematical and theoretical physics. The incumbent of the chair of theoretical physics at the Faculty of Sciences at the University of Paris since 1932, Louis de Broglie runs a course on a different subject each year at the Institut Henri Poincaré, and several of these courses have been published. Many French and foreign students have come to work with him and a great deal of doctorate theses have been prepared under his guidance.

Between 1930 and 1950, Louis de Broglie’s work has been chiefly devoted to the study of the various extensions of wave mechanics: Dirac’s electron theory, the new theory of light, the general theory of spin particles, applications of wave mechanics to nuclear physics, etc. He has published numerous notes and several papers on this subject, and is the author of more than twenty-five books on the fields of his particular interests.

Since 1951, together with young colleagues, Louis de Broglie has resumed the study of an attempt which he made in 1927 under the name of the theory of the double solution to give a causal interpretation to wave mechanics in the classical terms of space and time, an attempt which he had then abandoned in the face of the almost universal adherence of physicists to the purely probabilistic interpretation of Born, Bohr, and Heisenberg. Back again in this his former field of research, he has obtained a certain number of new and encouraging results which he has published in notes to Comptes Rendus de l’Académie des Sciences and in various expositions.

After crowning Louis de Broglie’s work on two occasions, the Academie des Sciences awarded him in 1929 the Henri Poincaré medal (awarded for the first time), then in 1932, the Albert I of Monaco prize. In 1929 the Swedish Academy of Sciences conferred on him the Nobel Prize for Physics “for his discovery of the wave nature of electrons”. In 1952 the first Kalinga Prize was awarded to him by UNESCO for his efforts to explain aspects of modern physics to the layman. In 1956 he received the gold medal of the French National Scientific Research Centre. He has made major contributions to the fostering of international scientific co-operation.

Elected a member of the Academy of Sciences of the French Institute in 1933, Louis de Broglie has been its Permanent Secretary for the mathematical sciences since 1942. He has been a member of the Bureau des Longitudes since 1944. He holds the Grand Cross of the Légion d’Honneur and is an Officer of the Order of Leopold of Belgium. He is an honorary doctor of the Universities of Warsaw, Bucharest, Athens, Lausanne, Quebec, and Brussels, and a member of eighteen foreign academies in Europe, India, and the U.S.A.

Louis de Broglie died on March 19, 1987.

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I am a big fan of functional programming myself.

Have you taken a look at Haskell? I suspect that you might find this language way more interesting and elegant!

]]>#7641. The scientific name is *Vicugna pacos*. Name the animal.

#7642. What do the family *Dactyloidae* represent?

#4830. A number is mistakenly divided by 10 instead of being multiplied by 10. Find the percentage change in the result due to this mistake.

]]>#1520. What does the medical term 'Hemianopia' mean?

]]>Well done!

CG#123. ABCD is a rectangle formed by the points A (-1,-1), B (-1,4), C (5,4), and D (5,-1). P, Q, R, and S are the mid-points of AB, BC, CD, and DA respectively. Is the quadrilateral PQRS a square, a rectangle, or a rhombus? Justify your answer.

]]>This is an algorithm that test if one number is prime and if the number it´s not prime returns the biggest factor of that number.

Are you going to keep it a secret and profit off of it or are you going to enlighten our poor souls as to what it is?

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