Integration and Differentiation

[pizzaguy]

I'm a self-learner (so far) when it comes to Calculus. I have completed College Algebra (with a grade of B) and PreCalculus (with a grade of A) and am a later-in-life than usual student (age 50). I plan on returning to College in January of 2012 and will start on a degree in either Management or Mathematics (it's a long story why it comes down to those two very different goals).

Now, what I want to ask is this: Can anyone help me with an easy, down to Earth definition for integration and differentiation as the terms are used in Mathematics?

I have started to study limits and think I understand the limit is a snapshot of the rate of change in the value of y with respect to x at a particular instant. But beyond that... I need help.THanks!

 

[micromass]

Hmm, your interpretation of limit seems to be that of the derivative. Here are my intuitive explanations:

Limit: the value that a function should take on if it were nice (continuous)
Derivative:
- the local rate of change of the function.
- the slope of the tangent line
- the velocity of a particle at a point
Integration:
- the surface under a curve
- the inverse of the derivative

 

[Stephen Tashi]

I suppose complete understanding of calculus requires 3 different types of understanding
1) intuitive understanding of the concepts
2) mechanical understanding - how to work typical problems
3) precise logical understanding - how to understand and write proofs with statements that involve logical quantifieres like "for each epsilon" and "there exists a delta".

For passing engineering courses that assume you know calculus, you need item 2). It helps to have item 1) also. If you take advanced abstract mathematics, you need item 3).

I'll assume you are asking about item 1).

Think of the derivative as the "rate of change AT at point". The simplest example is velocity. The elementary way to calculate velocity is "Distance divided by time". That only makes sense as an average since there are situations where the velocity is changing a lot during the time interval. (For example, "distance to commute to work divided by time to get to work" is an average of many different rates of travel.) Is it possible to talk about velocity AT A POINT instead of average velocity over a time interval? Yes, that would be what a derivative does. At a point x, you look at smaller and smaller time intervals ( by tradition, they are from x to x + delta x) and you divide the distance moved by the length of these time intervals. Look at the definition of the derivative in your textbooks and notice how it is the limit of a quotient.

Another interpretation of the derivative is the slope of a curve AT A POINT. First, forget what you learned in geometry about the definition of a chord and a tangent. Let's define a "chord of a function" as a line between two points on the function's graph. My calculus teacher illustrated a tangent as follows. He drew the graph of a function on the blackboard. Then labeled two points A and B on the graph. Then he held a ruler so it went between A and B. He held it at point A so it pivoted on one of his fingers. He balanced the ruler at point B by holding the tip of a pencil under it. Then he slowly traced the pencil along the graph so from B to A. The ruler slowly changed its angle since one side of it rested on the pencil. When the pencil reached point A, he declared that the ruler defined the tangent to the curve at point A. (Notice there is nothing in the definition of "tangent to a function" that says it hits the curve at only on point or is perpendicular to some diameter etc. ) Look at the definition of the derivative and try to see how the fraction in the definition defines the slope of this tangent line. (Don't get in the habit of saying that the derivative "is the tangent to the curve", it is "the slope of the tangent to a curve".

If that explains derivative, we can do integration next.

 

[pizzaguy]

So.... after reading both responses, and before we go any farther... differentiation is the act of finding the derivative? Am I ok so far?

 

[Stephen Tashi]

differentiation is the act of finding the derivative? Am I ok so far?

Yes - so far.

 

[pizzaguy]

Ok, headed to rehearsal. Back when I can get back!

Let me read this stuff over and think for a bit...

 

[pizzaguy]

If that explains derivative, we can do integration next.

I think I am ready! :)

 

[Stephen Tashi]

In caculus, the word "Integration" is used ambiguously even by instructors. On one hand it can mean finding an area. In their careful moments, people refer to this as taking a "definite integral". On the other hand it can mean finding the anti-derivative of a function. The ambiguous use of the word "integration" isn't so harmful since the two concepts are related to each other by The Fundamental Theory of calculus.

Now, before I amuse myself by writing an essay on these things, what do you already understand about them?

 

[pizzaguy]

I've complete College Algebra and "PreCalculus". This took me right up short of differentiation.

I understand the concept of a limit, but tha'ts about as far as I have gone.

(Been away - had tons of things happening in my personal life...)

 

[Stephen Tashi]

You understand limit, but not derivative? Where have I failed?

 

[vector22]

Here is a simple example

You drive a car at 50 miles per hour for 2 hours
how far did you drive?

The velocity is d(distance)/d(time) = 50

[tex] distance = \int {50} \, dt = 50t [/tex]

t = 2 so distance = 100

 

[pizzaguy]

You understand limit, but not derivative? Where have I failed?

Is not the limit also a derivative? The limit at any point IS the instantaneous rate of change at that point, and is the limit at that point, right?

 

[Stephen Tashi]

Is not the limit also a derivative? The limit at any point IS the instantaneous rate of change at that point, and is the limit at that point, right?

Wrong. Think of saws. We use a special type of saw with an abrasive blade to cut through concrete. This does not mean that the idea of "saw" is that it is something that is used to cut through concrete. The idea of "saw" is more general. Look at the definition of derivative. It is a very special kind of limit. (Are you actually trying to read written materials about Calculus or are you just glancing at them and getting a hazy impression of what they say?) When you think of limit intuitively it is natural to think of some sort of physical motion as "x approaches a", but that is just a poetic phrase. There is not necessarily any motion involved in a limit.