Integration by Trig Substitution

Blog Post on google docs because I am not a fan of the Blogger's formatting. 

https://docs.google.com/document/d/1E5TqLG3DPgaYTAACS_1_mfq2iioFC9CYd3c6V6NFvpc/edit?usp=sharing

Arc Length of a Curve!

The arc length of a curve is a fairly simple concept. We've already learned how to find the arc length of circles, but what about functions like cos(x+5) . . . Well first let's define arc length:

The length of a curve if were to be “rectified”.

"Rectified" is simply straightening out a curve from it's "curvy" shape to a linear line that can be measured. Now, you could do this with string, but you can also do it with calculus. Before we get into the calculus, here is a link to visualize a rectified curve: click here

So here is the calculus–again, nothing you don't already know how to do. Since, we're only going to go over how to find the arc length rectangularly

The first step is having a function that you would like to have the , and the formula.

For specifically rectangular form, ds is replaced with a formula showed below:

Since you already know how to find derivatives, the process is fairly simple from here on out. You can either use your graphing calculator by using "fnINT" of the rectangular formula above, or you can use Geogebra! Below is a 10-minute video I made to show how to take a real life object, and find the arc length of it. Not surprisingly, solving with calculus is much easier than measuring a curve with string.

If you would like to check out some examples of how to solve polar functions or parabolic functions, check out this website: click here. The formulas are also fairly simple, and easy to solve based on the calculus you know. Adios amigos.

Eliot

Mathematics of the Natural World

Patterns in nature are visual regularities of form that occur in the natural world. These patterns can be modeled with mathematics and physics. Natural patterns can include symmetries, fractals, spirals, meanders, waves and dunes, foams and bubbles, arrays, cracks, and stripes (some examples are shown below):

Romanesco Broccoli in fractal form

radial symmetry

bilateral symmetry in a zebra's stripes

radial symmetry in a kiwi

logarithmic spiral of a Nautilus

dune meanders

tessellation array of scales

tessellation array of scales

sand dunes at equivalent angles

crack patterns

inelastic crack patterns

meanders

phyllotaxis Fibonacci spirals

phyllotaxis arrangement

Philosophers, mathematicians, and physicists have applied their skills to the natural world across the ages. Early Greek philosophers Plato, Pythagoras, and Empedocles often studied natural form, hoping to explain the ordered occurrence of patterns. In the 19th century, Joseph Plateau developed the theory of minimal surface area as shown in soap bubble films, and was able to mathematically model the concept. Ernst Haeckel painted thousands of Radiolaria (small marine organisms) to show their symmetry in detail. D'Arcy Thompson extensively studied and modeled the growth patterns of flora and fauna, applying mathematics to spiral growth. Alan Turing established methods for predicting morphogenesis in embryos that would eventually become spots and stripes. Benoit Mandelbrot and Aristid Lindenmayer developed the concept of fractals that can be used to approximate plant growth patterns.

While the models are not always spot on, the conceptual process of predicting the patterns of the natural world has broadened our horizons and increased our appreciation of the beauty of nature.

Partial Differentiation

Partial Differentiation

Earlier in the year we learned how to differentiate a function.Today we took that a step further by talking about PARTIAL DIFFERENTIATION!!! Basically, partial differentiation is the process of finding the derivative in regards to a single term–either x or y. This process can be used when a function has more than one variable. Before we can take the partial derivative of a function, we need to learn about some new notation. denotes that a function is in terms of both the x and y denotes that a partial derivative needs to be taken in terms of x denotes that a partial derivative needs to be taken in terms of y

Likewise, the partial derivative notations above can be written as  and  or as  and .

For the majority of first and second semester we used limit definitions to understand how and why derivatives work. Even though limits are not the most efficient way to find derivatives, they’re important when understanding them fully. Remember that derivatives have a limit definition of . The definition of a partial derivative using limits can be split into both an x and a y component, but both follow the basic structure of the original limit definition. However, each variable is approaching a separate number. As h approaches zero, x approaches a and y approaches b. This is written as:

 and

Notice, that in the numerator, h is placed with a, since that is the value x is approaching. Likewise, when taking the partial derivative of y, h is placed with b, since that is the value that y is approaching. Basically what the limit definition explains is that partial differentiation requires that you only take the derivative with regards to one variable at a time and leave the other as a constant. That really is the most important thing to keep in mind when finding partial derivatives. ALWAYS treat the variable that you’re NOT taking the partial derivative of as a constant. For example...

Before getting into higher order partial derivatives, it’s also important to understand the geometric representation of a partial derivatives. In this case we’re looking at a three-dimensional surface, in the x, y, z axes. Just as the derivative of a function represents the slope of a tangent line to  at ,  and  are the slopes of tangents lines. However, there’s a difference. They are slopes of traces of surfaces, which are curves that represent the intersection of the surface and the plane given by  or . That is, when taking the derivative in terms of y, it is the slope of the tangent plane in the y direction. Likewise, the derivative in terms of x represents the slope of the tangent plane in the x direction. Later, as we talk about higher order partial derivatives, you’ll see that when you take a derivative in terms of both x and y, it’s geometric representation can be described as the rate of change of the slope in the x-direction as one moves in the y-direction. In order to understand these concepts more clearly, this applet will be helpful.

Taking higher order partial derivatives is relatively easy. You simply take the partial derivative in terms of either x or y, and then take the partial derivative in terms of x or y again, depending on the problem. Here’s an example:

Just like there were differentials for functions with one variable, there are also differentials for functions with multiple variables. We learned this year that differentials were:

With multiple variables, this becomes written as:

Those two differential equations above are synonymous. What this is saying is that the differential of a function with two variables is just equal to the sum of the partial derivative of x and y multiplied by a change in x and change in y respectively. This can likewise be expanded if you had a function with more than two variables, such as :

If you’re still feeling a bit stuck, here are two PatrickJMT videos to refer to!

Partial Differentiation and Higher Order Partial Differentiation

Euler's Method

https://docs.google.com/document/d/1mLQORMBumLDS1R9fpaJsSz_jtnfEkz4T5Uplkga1snc/edit

Hyper-BOLA (Lexi and Autumn)

Before we start the blog post you can watch a really helpful and short introductory video here:

http://www.youtube.com/watch?v=E3IGKxRFlfM

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! 

QUIZ