Thread: Learning Calculus

Hi,

I'm extremely interested in learning about algebra and advanced calculus but I don't really know where to start. The internet seems littered with books and different tutorials but I'd like the members of the Science forum to enlighten me with their help.

What is Calculus? And how is it applied in the real world? Who invented it?
What book is the best and most suitable for a begginer?

My education in mathematics is a simple high school degree with a few courses in college.

Calculus is a very vast topic in mathematics, but in general it is the study of change. It composes of many different subtopics such as studying the limits of functions, functions, infinite and finite series. integrals and differentiation. Its divided in to integral and differential calculus which are two very different things. It has many real world applications especially in physics, engineering, economic modelling etc....

About the book choices, i dont think i can help you with that myself as i myself am only 17 and there are definitely more experienced members here who would recommend great books for you to study.

I could suggest that you just look for a college text book that someone is trying ot get rid of. They usually provide the material in a manner that will aid you in your learning experience. They are generaly a little pricy though... But most any one would do well.

Isaac Newton is generally attributed with inventing calculus though you'd barely recognize the notation he used today.
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There are lots of text books about the broad subject but many suffer from showing only one approach of many to solving a problem. If you don't understand that approach you really need a person to explain it or are up the creek. Though it's a long time ago I used Sailvanus Thompson's book on calculus to get that alternative approach to better understand how to solve some problems. At some point you realize calculus is pretty tedious because it's mostly learning what "trick" to apply to get a function into a usable and solvable forms. The real fun comes beyond calculus when you put it all together.

Quite a few online websites as well to look at. You might start at CALCULUS.ORG

Originally Posted by Shaderwolf

I could suggest that you just look for a college text book that someone is trying ot get rid of. They usually provide the material in a manner that will aid you in your learning experience. They are generaly a little pricy though... But most any one would do well.

Price-wise it helps to get a text that is a couple of editions old. Generally textbooks don't change that much from edition to edition so you could get an older edition used for under $20 where you'd pay $100-150 (or more) for the latest ...

I am currently [re]teaching myself calculus using the 9th edition of the well-known text by Thomas and Finney. It's a pretty good one, as far as textbooks go ...

For an amusing and instructive overview of some calculus basics I like How to Ace Calculus: The Streetwise Guide by Colin Adams, et al. I've read this thing a few times and I am always surprised at how well it gets the basics across. Plus, it's fairly entertaining. There's a followup book on multivariable calculus and so on, too.

Isaac Newton is generally attributed with inventing calculus though you'd barely recognize the notation he used today.

I would rather attribute it to Gottfried Leibniz instead.

Calculus, as mentioned in this thread, is the study of functional change. I would disagree with the earlier post saying that differential and integral calculus are worlds apart; in fact, the fundamental theorem of calculus shows that they are in fact intrinsically linked.

Put simply, differentiation is a fancy way of taking a rate at a single instant in time. As you probably know, a rate is some quantity -- say for instance distance -- divided by some finite amount of time. Differentiation is a way of analyzing what happens when that finite amount of time gets closer and closer to zero.

Integration on the other hand is a fancy way of adding things up. Let's say you're trying to find the area of a circle without the handy-dandy (pi)(r^2) formula. How would you do it? Well, you'd cut the bugger up into a bazillion different rectangles and find their individual areas, then add them all up, right? Well, integration is the summation of those bazillion rectangles ... only the trick to integration is that those bazillion rectangles are essentially infinitely thin.

[QUOTE=Lynx_Fox;276045]Isaac Newton is generally attributed with inventing calculus though you'd barely recognize the notation he used today.

- that is because the notation we use today was formalised by Leibnitz whom I think you'll find is equally credited as having independantly developed the discipline. Newton was fully aware of this but as was never one to let facts get in the way of the credit, he had a similar confrontation with Hooke over the inverse square law.

Originally Posted by MadRoach

What is Calculus? And how is it applied in the real world?

When objects move at a constant rate, it is pretty easy to calculate their motion into the future, distance covered = speed x time. But what about when things are accelerating or decelerating, so their speed is different from moment to moment. How do you calculate their position into the future? This was once thought to be impossible.

Newton and Leibnitz both invented "the calculus" independently, although Leibnitz came up with the better notation, so that is the one that we use. The idea is that if the acceleration is constant, then you can predict the motion into the future.

Imagine dropping a stone. The instant you let go of it, it is hanging motionless, but only for an instant. The next instant it may be dropping very slowly, say 1 inch per second. An instant later it is falling at 2 inches per second. If the speed increases at a constant rate, one (inch per second) every second (or, an inch per second per second) then there is a higher order regularity to its motion which lends itself to calculation.

The basic idea is quite simple, although very revolutionary in its time. The idea is to pretend that the stone drops constantly at one inch per second for the whole first second, then at two inches per second for the second second, and so forth, accelerating to 10 inches per second in 10 seconds. (Ok, this is pretty slow for falling, I'm trying to keep the numbers simple. Maybe we're on the moon or something)

You can break it into smaller intervals: Through the first tenth of a second it is falling at 1/10 an inch per second, the second tenth it is falling at 2/10ths of an inch per second, etc. When you slice it 10 times finer, you get a smoother acceleration curve, a more accurate model of a constant acceleration.

Newton (and Leibnitz) said, what if you slice it in infinitesimal intervals, let it fall for an infinitely tiny interval and calculate its speed, then another infinitesimal interval, and so forth, and after you have accumulate an infinite number of such infinitesimal intervals you will have fallen a real distance.

An infinite number of infinitesimal intervals? That is NONSENSE! they decried! No number of infinitesimal intervals can ever add up to anything! Infinity divided by infinity is so absurd, there is no answer to that one.

But the smaller you make the intervals, the more smoothly and accurately you calculate the acceleration, and in the limit, the tiniest conceivable interval, the acceleration is perfectly modeled. Calculus makes the assumption that it IS possible to add up an infinite number of infinitesimal intervals and come up with real numbers, and it turns out that this idea is so good at modeling real-world motions that now it is not controversial any more, it is a standard tool of science. Although the time fallen is infinitesimal, and the distance covered in that time is infinitesimal, there IS an instantaneous acceleration that remains fixed, something like one inch per second per second, and THAT constant is what determines the acceleration, not only for that instant, but all the others too.

Here is an example of calculus thinking. Imagine driving your car straight on a large flat plane, then turn your wheel 10 degrees to the right and hold it there. The car will start turning at, say, 10 degrees per second to the right, a constant rate of turn. Turn the wheel 20 degrees and hold it, and the car will turn at 20 degrees per second. Here's the calculus: The rate of turn of the car is proportional to the angle of deflection of the steering wheel.

Now, instead of holding the wheel steady at one angle, lets say you install an electric motor to turn your steering wheel 1 degree every second, at a constant rate. Press a button to start the electric motor, and 10 seconds later you will be turning at 10 degrees per second, 20 seconds later it will be 20 degrees per second. Your car will be turning steeper and steeper until it is spinning around its center! Here's the calculus: The rate of turn of your car is proportional to the angle of the wheel, and the angle of the wheel is proportional to the time since you turned on the electric motor. So the rate of turn of the car is the double derivative of pressing of the button of the electric motor.

Thats the core idea behind calculus. But it works for static problems as well. Imagine you are designing a tin can for a soft drink, and are trying to decide how long and thin, or short and fat to make the can, to hold the maximum volume with a minimum of aluminum. Here's the calculus: Imagine the can so short and fat that it is just a flat circle. Now stretch it continuously to grow in length and shrink in width until you reach the other extreme, a can like a pencil, very tall but very thin. In calculus you can express that continuum as an equation, solve it for the maximum point, and figure out the shape of the cylinder that will hold the maximum volume.