Welcome to "Introduction to Calculus"
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[edit] Welcome!
- Welcome to "Introduction to Calculus!" I will be your professor for this course. You may call me "Prof. Googly Eyes." Feel free to change my name. Please?
- Many people think calculus is hard. This is a matter of opinion. I think it is not very dificult; then again, I probably wouldn't be writing this if I didn't really like the topic. However, this course should be easy to understand by both people who find it easy and people who find it hard. If you have any questions, just write them at the top of the discussion page, and someone should quickly add the information. Hopefully, with this, the course will not be too difficult to understand.
- This course consists of many lessons. They include this one, the next one on limits, and the third one on using derivatives.
[edit] Why is Calculus necessary?
How fast can you run?
It seems a simple question but lets think:
So if you have run 100 metres in 10 seconds then your speed was 10 metres per second. Simple.
But that assumes you were running at the same speed all the time. Maybe you started off slowly then sped up and then slowed down slightly at the end. Its likely that your speed wasn't constant so all we are calculating is your average speed over the whole distance. The really interesting question is:
How fast am I going now?
Good question. If we are very good at measuring we might be able to find out that you traveled 11 metres in one second. But that's still only your average speed over 1 second. You might actually be slowing down: so your speed when we started to measure was actually slightly faster than 11 metres per second.
You see the problem. By using the formula above we cannot actually tell you how fast you are going now, we can only give you an average.
That's where calculus comes in. Using calculus, we can tell exactly how fast something is traveling right now.
How steep is this slope?
Again the equation that comes to mind is:
This is great if the slope is the same all the way up. If you are climbing a mountain or driving up a hill in the real world then you run into the same problem. We can make an average measurement but the formula can't tell us exactly how steep the slope is right here.
[edit] Finding the speed of a falling rock from the distance
Gravity is a very important feature of our everyday world. It also provides a very good basis for an introduction to the first section of calculus.
Say that, for whatever reason, my friend has dropped a rock over a cliff. The rock takes ten seconds to reach the lake below, at which point it makes a very large plop. Now, I wonder; how fast was the rock traveling when it hit the water?
Well, I know a few things about physics. I know that on the surface of the Earth, a rock will fall 16t2 feet or 9.81t2 meters in t seconds. I also know that the average speed over an interval of time equals:
.
From this, you can say that at t seconds, the distance the rock will have traveled will be 16t2 feet, and that at t1 seconds, the distance the rock will have traveled will be 
. Next, we say that the length of the interval of time between t and t1 will be t1 − t, and that the total distance traveled during this interval will be 
. Make sense? Now, for the grand finale. Let's use our equation for average speed, our distance traveled in an interval, and the length of the interval to create an average speed equation for any point during the rock's fall. The expression for this is:
- Try-This-Yourself (TTYS) 1
- TTYS 1 will show up a lot during this part of this lesson. This is so you can follow what we are doing, by doing it yourself. First step;
create the average speed formula of a rocket in which the rocket, t seconds after blast off, will be 4t3 feet off the ground. Use the interval of time between X and Y (Or t and t1; it really doesn't matter what variables you use; the people who write calculus textbooks prefer the t1 format.)
