Hey, it’s been a while, I’ve been off the radar, doing you know … boring stuff … But! Lately I’ve been doing cool stuff … I’ve now converted three of my e-books to paperbacks, so now you can buy them even if you prefer the old-fashioned (yet still appealing) paper format. Here they are and more to come:
There’s more! I’ve been getting complaints from readers than on some devices the e-books from the Math Shorts series are not properly displayed. This has been fixed and the changes should be live within 24 h. You will see whether it is the updated version from the description.
Today I need to party a bit, I encourage you to do the same, but as soon as I’m sober again I’ll continue converting dem berks for your analog reading pleasure 🙂
You are made up of atoms, you occupy space and you have mass.
You know what that means?
Exponential functions have the general form:
with two constants a and b (called base). It’s quite common to use Euler’s number e = 2.7182… as the base and the exponential function expressed as such:
with two constants a and c. Converting from one form to the other is not that difficult, just use ec = b or c = ln(b). Here’s how it works:
As for the plot, you should keep two special cases in mind. For b > 1 (which corresponds to c > 0 in case of base e), the function goes through the point P(0,a) and goes to infinity as x goes to infinity.
(Exponential function with b > 1 or c > 0. For example: f(x) = 8·3x)
This is exponential growth. When 0 < b < 1 (or c < 0) this turns into exponential decline. The function again goes through the point P(0,a), but approaches zero as x goes to infinity.
(Exponential function with 0 < b < 1 or c < 0. For example: f(x) = 0.5x)
Here’s how the differentiation of exponential functions works. Given the function:
The first derivative is:
For the case of base e:
You should remember both formulas. Note that the exponential functions have the unique property that their first derivative (slope) is proportional to the function value (height above x-axis). So the higher the curve, the sharper it rises. This is why exponential growth is so explosive.
If exponential functions are combined with power or polynomial functions, just use the sum rule.
(This was an excerpt from the FREE ebook “Math Shorts – Derivatives”)
Every year about one car sized asteroid will hit the Earth and leave a formidable crater. Surely this would be a spectacular and very frightening event to witness. Let’s do the math: how likely it is that one of these asteroids will impact within 0.5 kilometers of your house?
The Earth has a radius of about 6400 kilometers. With the respective formula for a sphere, we can use this value to calculate the surface area of Earth:
S = 4 · π · (6400 km)² = 515 million km²
This is the total area available for the asteroids to hit. A region of radius 0.5 km around your house has the area:
A = π · (0.5 km)² = 0.79 km²
Assuming our car sized asteroid falls randomly onto the surface of Earth, which certainly is a legitimate assumption to make, the chance of the asteroid hitting within half a kilometer of your house (or your current location for that matter) is:
p(hit) = A / S = 1 in 655 million
So it’s amazingly small. But since on average one asteroid of this size will hit per year, the above number only covers your chance over one year. What about the chance of such a hit over the course of a life? We approach this the same way as in the snack “Homicide She Wrote”. We calculate the chance of not being hit 70 times in a row and from that deduce the chance of a hit over a life span:
p(hit 70 years) = 1 in 9 million
You think that’s a low probability? It is, the chance of being struck by lightning is much higher, but tell that to the about 800 people of the 7000 million alive today who, according to the calculated odds, are expected to actually witness such a close asteroid strike at some point in their lives.
(This was an excerpt from my introductory statistics and probability ebook Statistical Snacks)