Finding the tangents and normals of a mathematical function or relation is one of the most common exercises in any calculus course. In this post, I’ll show you the newest functionality in Wolfram|Alpha for discovering and investigating them.

The simplest example of a tangent is the “tangent line” to a one-dimensional curve in the plane. Graphically, the tangent line is a line that “just touches” the curve at some point, so that if it were moved just slightly, this one point of contact would become two.

If you ask Wolfram|Alpha for the tangent line of a specific function and point, it gives it in both graphical and algebraic/numerical form:

What do your alarm clock, thermostat, coffeemaker, car radio, anti-lock brakes—and almost every other electrical and mechanical device you encounter in your daily life—all have in common? They are all examples of “control systems,” one of the most ubiquitous yet unseen modern technologies. A control system is any system or device that controls or regulates the behavior of another system. Using various kinds of sensors and actuators, these systems automatically control most common appliances, industrial processes, and even your body’s own biological processes!

Take your home’s humble thermostat. The temperature of your home depends on many factors, especially how long and how recently the home’s furnace was on. With a thermostat installed, the reverse is also true: the state of the furnace depends on the temperature of the house (it comes on if the temperature is too low, and turns off if the temperature is too high). There is a closed loop of causation formed between the home’s temperature and the state of the furnace. By design, the thermostat creates a kind of closed loop called a “negative feedback loop,” which tends to stabilize the temperature around a desired value. Most control systems are like this: sensors feed information back into the system, which is then used to decide on an action. More »

The hyperlink has been one of the most powerful tools of the information age. Links make it easier to navigate the complex web of information online by combining the information itself with the method for retrieving it. Clicking a link means “tell me more about this thing,” which naturally lends itself to “surfing.”

At Wolfram|Alpha, we strive to integrate and leverage technologies to create the most powerful computational capabilities and user experiences possible. In Wolfram|Alpha, the output comes in the form of a “report.” If you want to know more about something in the output of an Wolfram|Alpha query, clicking it as a link will generate another such report. Though we’ve had links in Wolfram|Alpha for a while, we’ve recently taken them to the next (computable) level: Wolfram|Alpha now computes links dynamically based on the output generated by your query.

Clicking a link basically feeds the plaintext of that link back into Wolfram|Alpha, creating new output with new links. Thus the navigational ability of the world wide web and the computational ability of Wolfram|Alpha are now intertwined and can feed off each other. You can now surf Wolfram|Alpha like you can surf the Internet. More »

One fall evening in 1843, a man walked past the Brougham Bridge along the Royal Canal in Dublin, Ireland. Suddenly, he felt a flash of insight so strong he was compelled to etch his thoughts into the rock on the side of the bridge. This is what he wrote:

i^{2}=j^{2}=k^{2}=ijk= -1

The man was mathematician William Rowan Hamilton, and the insight was of a number system that could represent forces and motions in three-dimensional space. Hamilton called his numbers “quaternions”, because each has four parts: a real number part, and three other parts labeled with *i*, *j*, and *k*, each of which is also a real number. For example, 2 + 3*i* + 0.342*j* – 2*k* is a quaternion. More »