Imagine you are building a roller coaster. You need to create a curved shape for the track, which will be designed on a computer before being built out of metal. You want a curve that is fun to travel along, which means you want a lot of sharp curves—but not too sharp, unless you want your amusement park guests to get sick or pass out.

A similar consideration is faced by engineers building a railroad or a highway: you want the path of the road to have curves that are not too sharp—in this case, to prevent the cars or trains from having to slow down (reducing efficiency) or even to wreck. The reason, of course, is that when you travel along a strongly curved section of track or road, you feel an acceleration. The higher the curvature, the stronger the acceleration, all other things being equal. However, the acceleration you feel depends on how fast you are going along the track (the faster you go, the greater the acceleration), while curvature is a property intrinsic to the track itself.

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Today is the summer solstice—when the Sun is at its more northern point—which marks the first day of summer, as well as the longest day in 2012. It’s a great day to go outside and take advantage of all the extra sunlight, and also a good time to take a look at all of the computations Wolfram|Alpha can do revolving around the Sun.

Here in Champaign, Illinois, the sun rose at 5:25am this morning, with sunset happening at 8:27pm. This equals just over 15 hours of sunlight:

It’s usually in precalculus class that students are first exposed to the more exotic and subtle aspects of functions on the real line. This first exposure comes through studying limits and discontinuities.

Most functions that we see every day, from the parabolic arc of a thrown ball to the exponential growth of money in a bank account, are “continuous.” That is, they don’t change their value suddenly. Thought of another way, a continuous function is one you can graph without having to lift your pen up from the paper.

Conversely, a discontinuity of a function is a point where the value of the function experiences a sudden change. In technical language, a discontinuity of a function from reals to reals is a point where either the left- or right-hand limit does not exist, or where these limits exist but aren’t both equal to the value of the function at this point.

Wolfram|Alpha now has the ability to find and analyze the discontinuities of most functions of real numbers. When it comes to such functions, there are three main kinds of discontinuities.

An “infinite” discontinuity is a point where the function increases to infinity and/or decreases to negative infinity (i.e., where it has a vertical asymptote). 1/x is the standard example:

Wolfram|Alpha is different from most of the tools out there on the web that you might use to get answers. Rather than inundate you with lists of links to web pages that may or may not be useful, Wolfram|Alpha works to understand your query.

What really sets these different approaches apart is how they deal with complexity in queries. Whether there are many concurrent factors to your question or you have a unique math computation with an answer that simply does not exist on some web page, Wolfram|Alpha is your best bet for a web service that actually understands what you are asking.

One of the ways that complexity can appear in queries is in depth, when there are multiple steps to a question. To understand what we mean by “depth,” think of the beautiful Matryoshka dolls that all fit inside of each other.

To answer a query like “elevation of Steven Spielberg’s birthplace“, the first step is to recognize that “Steven Spielberg” and “birthplace” can be combined to form a location.

(elevation) of ((Steven Spielberg’s) (birthplace))

In our first post on American Community Survey estimates in Wolfram|Alpha, we showed you how Wolfram|Alpha could answer questions about the age and sex of the population in practically any town or region in the United States. But that’s only a small fraction of what we can do with this wealth of detailed demographic data. Over the next few weeks, we’ll also share some examples of how Wolfram|Alpha can help you find and analyze information about education, income, and more.

But first, let’s take a look at two of the most frequently asked for demographic topics in Wolfram|Alpha: race and Hispanic origin. If you’ve never done so before, it’s worth taking a moment to brush up on the difference between these two concepts, in Census terminology. Although people often lump the two concepts together, race and Hispanic origin are two completely separate attributes in Census data: a person can be of any race and also be of Hispanic or non-Hispanic origin. Even with the basic data we’ve had in Wolfram|Alpha since its launch, people have regularly complained that our numbers “don’t add up”—and it’s always because they’ve added Hispanic population estimates to figures for the population by race and ended up with a figure larger than the country’s total population.

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In astronomy, one of the most prized pieces of data is to determine the distance to an astronomical body. Prior to the sixteenth century, the distance between planets and the Sun was an educated guess, but an accurate value had not been determined. Without the ability to pace off the distance or use a physical measuring stick, there was no direct way to determine this. How far was the Earth from the Sun? The distance between the Earth and Sun, known as an astronomical unit, was a key piece of missing data. In 1761, one of the first international scientific endeavors was carried out. Ships carrying scientists from numerous countries were dispatched to various observation locations to observe a relatively rare event. The planet Venus was going to pass between the Earth and the Sun, and, from our point of view, would move across the solar disk.