A journey into: Single Variable Calculus

[arkain101]

Hello and welcome to the journey into Single Variable Calculus.

 

If you are new to calculus that is okay! Because so am I. That is exactly what this topics purpose is. That is, to expose and learn the elementry basics of Calculus for the interests "not so familiar with math" crowd. :eek:

 

So for those wondering, curious, facinated or all of the above read on. ;)

 

Recently I came accross a textbook titled "Single Variable Calculus" by James Stewart of Mcmaster University. I decided that I would give it a shot to read the textbook front to back and learn myself a little calculus. :phones:

 

I realised this was going to be a long term process and I decided to make the most of this journey and share it with others, so that we may learn together and aquire support along the way.

 

As I work my way through the book I will be sharing sections of each chapter in a way that I think is most interesting.

 

What I plan to share besides just the calculus is the logic behind the numbers through whatever form I feel will work best in the most summerized way I can accomplish. Also, I found the history blips in this book also interesting so maybe I will add them along as I go. :)

 

I think this will be quite difficult due to my inexperience of mathematics so I will also be asking you "the members" a lot of questions. :phones:

 

So this will be a bit of an artisticly creative engineering process of learning mathematics, of the Calculus kind. :phones:

 

 

For your information and for future references: Chapters of the textbook

 

1)Functions and Graphs

2)Limits and Rates of Change

3)Derivatives

4)The Mean Value Theorem and Its Applications

5)Integrals

6)Inverse Functions: Exponential, Logarithmic, and Inverse Trigonometric Functions

7)Techniques of Integration

8)Futher Applications of Integration

9)Parametric Equations and Polar Coordinates

10)Infinite Sequences and Series

 

 

I think I will begin with a review on some basic mathematics. This is something I realised I had to go through before I was even able to comprehend the first paragraph of the text book.

 

The textbook reviews:

 

Algebra: arithmetic operations, fractions, factoring, quadratic formula, binomial theorem, radicals, exponents, and mathematical induction.

 

Trigonometry: angles, trigonometric functions, trigonometric identities, graphs and trig function.

[arkain101]

Introduction:

 

After reading through unfamiliar territory I found it interesting how much more clear and understandable mathematics is now at the age of 25 then it was when I was in high school 17-18. Since highschool I have done alot of "SWIM" Studying. An abreviation I just made up that stands for "Studying what interests me." :phones: This is something I noticed alot of people (that havn't persued further education) do once they become familar with the internet and become activly online. It is like, one who jumps from subject to subject reading, searching, studying whatever seems satisfying at the time. A kind of 'laid back any subject goes class.' In many cases including my own it can be a much better environment for learning (more relaxed and at your own pace). No longer does a math book read like its in a different language as I recall when I was a student, and I want to help others see that is really is not another language, its just a matter of understanding whats going on under or behind the numbers and rules and why things are arranged like they are. (lets see if I eat my own words down the road on this journey. :phones: )

 

So far from what I have learned I have noticed that mathematics is a bit like a sport. In order to perform the sport well you have to do what works best for you. I find mathematics is similar , because it 'could' be done countless ways. Infact, you could begin to invent your very own mathematics today if you wanted to but the problem is you would probably be 'reinventing the wheel' as it is known. That is because, as I have experienced, mathematics is often the effort to make complex problems as simple as possible.

ie.

Like trying to find the area of a circle. We could follow a method similar to how we measure 1 kilogram, as 1 litre of water.

 

We could fill a flat bottomed circle dish with water that has verticle sides of a height of 1mm and a diameter of 1meter and say, a 1mm by 1meter circle has an area and volume of 1litre of water (assuming it does). Then keep trying to fill a square box with exactly 1litre of water that has 1mm high walls untill it fits. Then we could measure that box's dimensions to find information about the circle.

 

1)We could add up all the sides of the box and find the perimeter of the box which gives us the circumference of the circle.

2)We could multiply the length and width of the box and find the area, which gives us the area of the circle, calling it "close enough"

3)we could multiply the length, width, and height of the box and find the volume, which gives us the volume of the circle.

 

And this relationship could be like an international standard area for a 1meter diameter circle of which to compare to for finding the area of other sized circles. It may work (even to find pi) but it would not be that 'simple' or accurate. So instead people in the past discovered from this same type of exhaustive methods an approximate number that is, pi = 3.14159...which is more accurate and easier to use.

 

A litre for example is not a cosmic volume found by the grace of god. Its a quantity or unit people in the past made up, just as you could have done, and from that accepted unit we base alot of measurements on it as a standard.

 

 

So when we realise that math is not a secret language, and that it is just an effort to try and explain something simply and work from that we can learn to connect to it more clearly and find it more friendly and down to earth.

[freeztar]

This sounds like a great idea. I'm game! ;)

[arkain101]

(let me know if it is against the rules to copy a textbook onto the forum)

 

 

Let's review. It is good to look back and make sure you know the basics.

 

 

 

 

 

Review

 

 

 

 

Algebra

 

 

Poor pictures. They will get better. See attachments.

[arkain101]

Review

 

 

 

 

Trigonometry

 

 

Take a peak at the pictures at the bottom if you like.

[arkain101]

(Black text is directly from the textbook)

(Blue text is of my own input)

 

Overview

 

A preview of Calculus.

 

Calculus is fundamnetally different from the mathematics that you have previously studied (The typical algebra and triginometry we study(studied) in school). Calculus is less static and more dynamic. It is concerned with change and motion' it deals with quantities that approach other quantities.

 

For that reason it may be useful to have an overview of the subject before beginning its intensive study. In what follows, we give a glimpse of some of the main ideas of calculus by showing how limits arise when we attempt to solve various problems.

 

 

 

The Area Problem

 

THe origins of calculs go back at least 2500 years to the ancient Greeks, who found aeras using the "method of exhaustion." They knew how to find the area A of any polygon by dividing it into triangles as in Figure 1 and adding the areas of these triangles.

 

Image: Figure 1

 

 

It is a much more difficult problem to find the area of a curved figure. Their method of exhaustion was to inscribe polygons in the figure and circumscribe polygons about the circle and then let the number of sides of the polygons increa. Figure 2 illustrates this process for the special case of a circle with inscribed regular polygons.

 

Image: Figure 2

 

Let An be the area of the inscribed polygon with n sides. As n increases, it appears that An becomes closer and closer to the area of the circle. We say that the area of the circle is the limit of the areas of the inscribed polygons, and we write:

 

[math]A \quad = \quad {\lim_{\small{n\right{\infty}}} \quad {A_n}[/math]

(This equation will be explained and elaborated on in the coming posts)

 

The Greeks themselves did not explicitly use limits. However, by indirect-reasoning, Eudoxus (greek 5th century bc) used exhaustion to prove the familiar formula for the area of a circle:

 

[math]A \; = \; \pi \, r^2[/math]

 

We shall use a similar idea in Chapter 5 to find areas of regions of the type shown in Figure 3. We shall approximate the desired area A by areas of rectagles (as in Figure 4), let the width of the rectangles decrease, and then calculate A as the limit of these sums of areas of rectangles.

 

The area problem is the central problem in the branch of calculus called integral calculus. The techniques that we shall develop in Chapter 5 for finding areas will also enable us to compute the volume of a solid, the length of a curve, the force against a dam, the mass and center of gravity of a rod, and the work done in pumping water out of a tank.

 

Image: Figure 3

 

Image: Figure 4

[arkain101]

The Tangent Problem

 

 

 

 

Consider the problem of trying to find the equation of the tangent line t to a curve with the equation [math]y = f (x)[/math] at a given point P. (We shall give a precise definition of a tangent line in Sections 2.1 and 3.1. For now you can think of it as a line that touches the curve at P as in Figure 5.) Since we know that the point P lies on the tangent line, we can find the equation of t if we know its slope m. The problem is that we need two points to compute the slope and we know only one point, P, on [math]t[/math]. To get around this problem we first find an approximation to m by taking a nearby point Q on the curve and computing the slope [math]m_{PQ}[/math] of the secant line PQ. From figure 6 we see that:

 

 

 

[math] m_{\tiny{PQ}}\, = \, \large{\frac {f(x) - f(a)}{x - a}}[/math]

 

 

Now imagine that Q moves along the curve toward P as in Figure 7. You can see that the secant line rotates and approaches the tangent line as its limiting position. This means that the slope [math]m_{PQ}[/math] of the secant line becomes closer and closer to the slope m of the tangent line. We write:

 

 

 

 

[math]m \quad = \quad {\lim_{\tiny{Q\right P}} \quad {m_{\tiny{PQ}}[/math]

 

 

and we say that m is the limit of [math]m_{PQ}[/math] as Q approaches P along the curve. Since x approaches a as Q approaches P, we could also use Equation 1 to write:

 

 

[math]m \quad = \quad {\lim_{x\right a} \quad \frac {f(x)\; -\; f(a)}{x\; - \; a}} [/math]

 

 

 

The tangent problem has given rise to the branch of calculus called differential calculus, which was not invented until more than 2000 years after integral calculus. The main ideas behind differential calculus are due to the English mathematicians John Wallis (1616-1703), Isaac Barrow (1630-1677), and Isaac Newton (1642 -1716).

 

Two branches of calculus and their chief problems, the area problem and the tangent problem, appear to be very different, but it turns out that there is a very close connection between them. The tangent problem and the area problem are inverse problems in a sense that will be described in chapter 5.

 

 

 

 

See This Image For a text book picture of everything here.

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[Galapagos]

I begin calculus w/ analytic geometry in the coming spring semester, and I have been going over some of my textbook already. I will definitely be following this thread with you! Good post!

[Nootropic]

Perhaps one of the greatest failings of students learning calculus (particularly in AP calculus classes) is the fail to understand how things work. Teachers (and some professors) figure that most of the students (who are most likely not going to study beyond the basic calculus sequence) will learn the theory somewhere else. You need to understand why the derivative of x^n is nx^n-1, not just that "it is". It's really a shame the delta-epsilon definition is not taught in basic calculus because students could really benefit from learning exactly just what a limit is (even the most rigorous professors and teachers usually don't give this definition because it is fairly difficult to wrap your head around). All in all: Learn the theory! It will certainly benefit you in further study.