The Mathematics of Calculus

[Ostronomos]

The importance of the derivative is two-fold: it can be interpreted as rates of change and it can be interpreted as slope. 

Suppose f(t) represents the position of a moving body at time t. So f(1) is its position at time 1, f(2) is its position at time 2, etc. (notice the sequential order). We can calculate the average velocity of a moving body at time t over an interval at t=c, and let that interval shrink to zero to determine the instantaneous velocity. The average velocity of a moving body is found by the following formula:

V avg = [f(c+h) - f©] / h

This is a familiar expression. As we let h tend to 0, this expression tends to the derivative of f at c.

[Ostronomos]

Much of our understanding of the world depends on describing how things change. The following are examples of things that change:
* The motion of a gymnast doing floor exercises

* The path of a satellite

* The fluctuation of the stock market

Algebra and geometry provide us with the tools to describe the relationship among static quantities. But to describe change we need a new mathematical operation.

(...)

In these examples we are comparing two quantities that are changing.

-From the Calculus textbook I'm reading.

I've made a note in my book that change is fundamental to existence. And that it is the single, most important aspect of physical existence.

In every applied mathematics textbook, we are called to attend to the nature of existence. This requires us to gain a deep understanding of nature, if we are to become successful in our pursuit of knowledge. Mathematics appears to be an inherent property of not only the universe, but as well our minds. This guarantees to be a promising feature of metaphysics.

[Ostronomos]

To determine the instantaneous rate of change of (for example) temperature at a point on the graph, we use a tangent line. This is a line that passes through a point on the graph and is in the same direction as the graph. We then calculate the slope of the tangent line by selecting one other point aside from x=x1. The slope is any two points on the graph, including x=x1.

[confused2]

A real world example of calculus would be dropping a lemming from a height of 32 feet .. how fast does it hit the ground. Hint.. the acceleration due to gravity is assumed to be 32 (feet per second) per second. We might like to look at trick ways of answering this while we're here.

[confused2]

Trick method ..
At a height of 32 feet a lemming with mass m has a Potential Energy [mgh] of m*32*32 somethings .. we don't even care what the somethings are because..
Having fallen from 32 feet the potential energy is converted into Kinetic Energy [(1/2)mv^2] since energy is neither created nor lost we get..
(1/2)mv^2=mgh
Cancelling the m
(1/2)v^2=gh
or
v^2=2gh
=2*32*32
=2048
so
v=sqrt(2048)=45 feet per second
or about 30 mph.
Probably a bit too fast if you want to use your lemmings again.