Feeding Hummingbirds

It's that time of the year when I get the most hummingbirds at my feeder. It's hot, and the fields are brown, but the little whirring birds make summer fun the whole season long.

B17: The Killer Vitamin

What is a vitamin? I don't mean the different types of vitamins or the shops where they can be bought. I mean, what makes something a vitamin? Perhaps a vitamin is only a useless chemical that serves no purpose, and is therefore called a vitamin to make it seem useful when it really isn't. I prefer to think of a vitamin, however, as an essential chemical that our body needs to grow and survive.

Did You Know...

I recently added a "Did You Know" app to the sidebar of my blog. The app randomly picks a math/science/technology fact from a list, and displays it. I had to write the app myself (with JavaScript and HTML), because Blogger doesn't have an app like that built-in, and even if it did, I like to have control over my things.

To get technical: the data is stored on Firebase. The app reads the data from Firebase, and adds it all to an array. There is no way to tell how many items there are without reading all of them from Firebase. The app then looks at how many elements are in the array (call that number a) and then randomly chooses a number b from 1 to a. It then takes the bth element from the array, and displays it. (I know, I didn't have to store them all in an array, but it's easier that way.)

Because of the data storage method, I can add new facts very easily: I just have to go to Firebase and add the new fact to the database. Of course, my app updates as soon as I add the new data. I can also set up the app to read data from a different Firebase location, so I can easily have multiple apps with different themes. Here's a screenshot of a Minecraft-themed app, for example:

If anybody is interested in putting a version of the app on their blog, let me know! I'd be happy to set up a version of my app with another theme, such as astronomy, animals, or sports. I can also change the style to make the font pink or green, or give it a purple background, before I send you the code. The possibilities are endless.

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Fourier Series Grapher

Here's a simple JavaScript + HTML program I wrote. It will graph a given number of terms of a Fourier series. To graph a Fourier series, type the appropriate expressions into the text fields and click "Update." The text fields support JavaScript, so you can write a whole function in there if you need to.

I also added some functions.

1. even(x) - returns true if x is even, false otherwise.
2. odd(x) - returns true if x is odd, false otherwise.
3. power(x, a) - returns x^a.

Some examples of input to try:

• odd(k)?(2/k):0
• even(k)?(4/k/PI):0
• odd(k)?(4/(k+PI)/k):0

The default Fourier series is equal to sin(2.5x) on (-π, π).

Enjoy!




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Euler Spiral

I was reading a calculus textbook when I noticed it said that cos(x²) doesn't have an elementary antiderivative. Elementary antiderivative? Clearly, they were hiding something. They didn't say it didn't have an antiderivative; they said it didn't have an elementary antiderivative. Of course, I wanted to know what the antiderivative was. If it wasn't elementary, it had to be really awesome.

I looked up the integral of sin(x²). Turns out, the integral cannot be expressed as anything other than itself. It's known as the Fresnel S integral, is written as S(x), and is defined as the integral of sin(x²). There's another Fresnel integral known as the Fresnel C integral which is written as C(x) and defined as the integral of cos(x²).

I also saw some graphs of the integrals. One really cool graph involved the parametric equations x = C(t) and y = S(t), and was called the "Euler spiral." It had a cool spirally shape, and I immediately knew that I had to graph it myself. I ended up writing an interactive JavaScript program to graph the parametric equations. Here it is; enjoy!

x = C(t)
y = S(t)

Max t:
t step size: * .001

Zoom X: %
Zoom Y: %

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