Before we start the blog post you can watch a really helpful and short introductory video here:
To
those of you who actually watched the video: BUWAHAHA! Now hopefully
you’ll go through the pain and suffering we did while working on this
project as you read our blog post and every time you read “hyperbola”
all of a sudden that cursed, catchy song will pop into your head and
you’ll hear, “hyper-BOLA! hyper-BOLA! hyper-BOLA!” Soon enough the curse
of this song will spread and no one will be able to speak of hyperbolas
the same, just as happened to us poor souls two years ago when that
dreaded song came on the radio and now I can no longer feel joy when
someone celebratorily says, “it’s Friday!” because the horror sprouts in
my mind and I feel the urge to say, “it’s friday, friday, gotta get
down on friday!” For those of you who didn’t watch the video, damn you
you smart son of a gun!
Now
onto actually introducing hyperbolas (hyper-BOLA hyper-BOLA) and math
and such. Now hyperbolic functions can be naturally occurring, they are
found in the way gravity naturally pulls down a string of beads or a
power line cable and the natural motion of falling bodies with air
resistance. This makes it kind of hard to say who really discovered or
developed or invented them, more who saw the pattern, found the
equation, and put a name to it. Hyperbolic functions were developed in
the mid-18th century by this Italian mathematician dude named Vincenzo
Riccati. This guy, Riccati founds the standard addition formula for
hyperbolic equations, as well as their derivatives and even found the
relationships between the hyperbolic functions and the exponential
function. Despite all this hard work, some people still credit
hyperbolic functions introduction into the mathematical world to a
French mathematician named Johnann Heinrich Lambert. Twas in the late
1760s that Lambert published his work concerning hyperbolic functions.
Lambert became associated with hyperbolic functions the same way Leonard
Euler is associated with circular functions: Lambert popularized the
new hyperbolic trigonometry as it is used in modern science. Basically,
he created the names we still use today, while Riccati did all the hard
work and found the equations and even the derivatives, and those are:
sinh pronounced: cinch
cosh pronounced: kosh
tanh pronounced: tanch
coth pronounced: cotch (coach)
sech pronounced: setch
csch pronounced: cosetch
These names were created by Lambert simply by taking the Latin of . . . let’s use sinh(x) as an example and getting sinus hyperbolus.
Now he used all the trig names because once we get into derivatives and
such you’ll see hyperbolic functions follow the same pattern of
trigonometric functions.
Now let’s get physical, physical, physical:
Hyperbolic
cosine curves occur all around us. When they occur naturally, they are
known as catenaries. The classic example of a catenary is the hanging
chain between two rods. It minimizes the gravitational potential energy
of the hanging object. They are also seen in the graphs of the motion of
falling bodies with air resistance.
The
Gateway Arch to the West in St. Louis is one of the most well known
applications of a catenary. The arch was engineered to be the most
sturdy arch to this day. Their goal is to have it last as long as the
pyramids have thus far. They did this by inverting a catenary. To get a
better understanding, watch this super cool video! https://www.youtube.com/watch?v=vqfVKsBkB1s
Basic Hyperbolas:
A
hyperbola is a curve where the distance of any point from the focus (a
fixed point) and the directrix (a fixed straight line) are always in the
same ration. A hyperbola is made up of two separate open curves that
are mirror images of each other. For each curve there is a directrix and
focus and going through both curves focuses is the axis of symmetry.
These are all shown and labeled in the picture below. (the picture is found at the bottom of the "Basic Hyperbolas" section)
The
equation for basic hyperbolas is
Through using the constants found in this equation one can easily find the vertices and asymptotes of this hyperbola. The vertices can be found at the points: (a,0) and (-a,0). The asymptotes are the lines created by the equations
Eccentricity:
The
eccentricity shows how curvy or uncurvy the hyperbola. The eccentricity of a hyperbola is always greater than 1. This is shown by
the ration of (a point on the curve)(focus)/(point on the curve)(point
on the directrix) or by the equation:
Trig-based Hyperbolas:
There’s
a set of trig-based hyperbolas that are all named after the trig
functions. As you could most likely guess, there are many similarities
between the typical trig functions and these hyperbolas. Although their
graphs are entirely different, the relationships between the hyperbolic
functions are just like those that we already know. By this, I mean that
tanh=sinh/cosh, etc. The trigonometric identities are similar to the
hyperbolic identities, but there are differences in sign in a few of
them.
Derivatives:
Taking
the derivatives of these functions is actually quite simple. They are
all e^x type functions, which we have worked with quite a bit this
semester. Or, once you know the equations for each trig hyperbola, then
you can use the trig-deriv relationships. For example, the derivative of
sinh is cosh. Again, they aren’t exactly like the trig functions, but
there are clear similarities. It’s also pretty neat that the inverses of
these functions are in the same sort of format as the trig inverses.
Here's the link to the quiz! Good luck!
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