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

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.

Euler's Method

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

Elazar and Connor | Final Exam Presentation

WORK AND FORCE

Force:
A push or pull which acts on an object. This doesn't necessarily mean the object will move, but the force is existent.

Imagine a water bottle on a table, if you push the bottle lightly it might not move. This is an example of a force acting on a force and being counteracted by friction. This leads to the ideas of force diagrams.

Think of all the forces acting on you now. What keeps you on Earth? What keeps you from falling through the ground? Are these things acting on you?

Before we take a look at work, recall what we learned about displacement. It is important to know that displacement is the difference between the final position and the initial position. Where the particle goes in between is not important. So displacement=X(final)-X(initial)


Work:

A force acting on a body which results in a displacement.

Work is related to force through calculus. Just like position is the antiderivative of velocity, work is the antiderivative of force. The only difference is the variable. With velocity and position the change is the change in time, while with work the change is with respect to the axis of displacement.

Constant force: something which doesn't change over time, gravity is a great example of this.
Equation for work when the force is constant:

Variable force: a force which varies over time, given as an equation.
Equation for work when the force is variable:

QUIZ