Lesson 6: Let’s Learn Some Calculus

IMPORTANT:  Before reading this email set the intent in your mind that you are going to learn this material, and there is absolutely no reason whatsoever that you can’t.    Determine that nothing is going to stop you from mastering this class.  It’s not even really going to be that hard.

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Let’s Learn Some Calculus!

Okay. Get ready. We will start with a bang!
 
I am going to teach you calculus.
 
You can do this. You will be amazed. You will probably freak yourself out.
 
I will teach you how to use things like rats and bunnies and slicky slides to learn calculus.
 
The first thing to do is answer the question: “Just what is calculus?”
 
The funny thing is if you ask most high school or college students taking calculus that question, they won’t know the answer. I know I didn’t. They can tell you how to do the problems, but they don’t see the significance of what they are doing.
 
There are two main ideas in calculus: The first is differentiation, and the second is integration.
 
We will deal with differentiation.
 
Differentiation is finding a derivative, and a derivative is a rate.
 
Consider driving in your car. How fast are you driving? The miles per hour is the rate at which you are driving. The rate you are driving is the derivative. Not too complicated a concept, is it?
 
Here is another: When you were a kid, and you slid down a slicky slide. The derivative will tell you how fast you were going down the slide.
 
Okay, repeat this out loud: “Finding a derivative is deriving a rate.”
 
Now picture a colossal rat wearing a bright red toboggan with the letter “E” on it driving your car down a road. Why? Because “rate” is “rat” with an “E.” And when you take a derivative, you are deriving (sounds almost like driving) a rate.
 
So, the obvious next question is, “Just what does this derivative thing look like, and how do you use it to find a rate?”
 
To understand that, you first need to know what a function is.

 
Consider a function an input/output machine.  You put a number into the function, replace x with a number, and you get an output. 
 
This is what a function looks like mathematically.
 
f(x) = x+1
 
You replace x in the function with a number to solve the function.
 
For example, if you replace x with the number 1 then
 
f(1) = 1+1
 
f(1) = 2

 
Get it?  Pretty basic math, isn’t it?
 
So what is the answer if you replace x with 2 in the function f(x)= x+1
 
f(2) = 2+1
 
f(2) = 3
 
Let’s try this function.
 
f(x) = 2x + 3
 
Let’s get bold and place 4 in this function.  Replace the x with 4.
 
f(4) = 2(4) + 3
 
This function has a two in front of the x, so that means you multiply whatever you put into x by two.  So, 2 times 4 is 8, and then add 3 to get 11 as your answer.
 
f(4) = 8 + 3
 
f(4) = 11
 
Functions can get more complicated, but now you know what one is, and now I will show you how to use functions.
 
But before we go on, let’s remind ourselves we are learning about functions so that we can find a derivative. We know a derivative is a rate of change, like driving in a car at a certain mile per hour or sliding down a slicky slide at so many feet per second.
 
Don’t make this hard. Just picture yourself in your car, driving down the road, or sliding down that slide. If you can picture that, then you are doing fine. 
 
Okay, repeat this out loud: “Taking a derivative is finding a rate.”
 
Now let’s see how to put functions to use. 
 
To do that, you need a Cartesian coordinate system. 
 
Don’t freak out—that is just a piece of graph paper with a big plus sign on it.

 
The horizontal line is called the x-axis.  The vertical line is the y or f(x) axis.
 
Does the f(x) look familiar?
 
The cartesian coordinate system is used for plotting points using a function.
 
Let’s do that with the function f(x) = x+2
 
Let’s start with a 1 for x, so…
 
f(1) = 1+2
 
f(1) = 3
 
Now we look at the graph on the x line at 1 since we put in a 1 for x and we see that f(1) or y = 3, so we go to the point where x is 1 and y is 3, which also can be written as (x,y) or in this case (1,3) and we put a point on the graph.
 

 
Let’s do it again with x = 2 for f(x) = x + 2
 
f(2) = 2 + 2
 
f(2) = 4
 
So x is equal to 2 and y or f(2) = 4, also written as (2,4)
 

 
One more time with x = 3.
 
f(3) = 3 + 2
 
f(3) = 5
 
So x =3 and y or f(3) = 5 also written as (3,5)
 

Notice how you can draw a line through the points.  This is a graph of the function f(x) = x+1
 

 
What does it mean? It could mean many things, but this time let’s let x = seconds and y or f(x) = feet. 
 
So now, let’s say you have a pet rabbit, and this graph shows how your rabbit is hopping.
 
At 1 second, your rabbit is 3 feet away from you. After 2 seconds, your rabbit is 4 feet away from you. After 3 seconds, your rabbit is 5 feet away from you.
 
It is easy to see from the graph that after each second passes, the rabbit is another foot away from you. The rabbit is hopping at a rate of 1 foot per second.
 
What is a derivative?
 
Taking a derivative is finding a rate.
 
So did you just find a derivative?
 
Yes. A derivative is just finding a rate. You haven’t had to do calculus yet, but we are getting there.

 

Look back at the 4 colored lines.

The black line rises 1 y for every 3 x.  Its slope is 1/3.

The red line rises 1 y for every 2 x.  Its slope is 1/2.

The blue line rises 1 y for every 1 x.  Its slope is 1/1, which is 1.

The green line rises 2 y for every 1 x.  Its slope is 2/1, which is 2.

The green line has the largest slope of 2, and the black line has the smallest slope of 1/3.
  
The thing to notice right now is look at the graph and see all the lines are straight lines.
 
Now, look at the graph below.  How is it different than figure 1?
 

 
The graph is curved. It is not a straight line.
 
So what is the slope of the curved line?
 
You can’t tell because it is constantly changing. The rise over run is different on different parts of the curve. Notice the straight-line rabbit graph. The change is the same from every point to point. The 4 line is always up 1 over 3. The 1 line is always up 2 over 1. You can’t just have one rise and run with a curve. There are many, and it depends on the part of the curve you are referencing.

Here is where we use calculus, specifically differentiation. With straight lines, we use algebra, but when graphs curve, we use calculus. Algebra just isn’t good enough anymore. Algebra can’t handle curves.
 
How do we start?

Since the curve is always changing, you decide at what point on the curve you want to find the slope—how fast the curve is changing. After you pick that point, we are now in full-blown calculus mode.
 
Let’s start.
 
Do this: Pick a point on the curve, then shrink yourself down until you are so small that the curve looks like a straight line to you or flat. Think of it as the Earth. We all know the Earth is curved, but it seems relatively flat to us because we are so small compared to the Earth. 
 
Go ahead and shrink yourself down so small that the curve you are looking at is as big as the Earth is to you right now. That way, the curve will look flat, and you can use algebra to find the slope or rate of change of the curve.
 
Okay. Well, since you can’t do that, we have to do the next best thing and use calculus to find the derivative of the function of the curve at a point we choose and then determine the slope/rate of change at that point.
 
Look at the graph below, and let’s pretend the left side of the curve is a graph of some crazy slicky slide. As you can see, it is over 36 feet high and very steep. We will start down the slide at 36 feet in the air. That is like three stories. Picture yourself standing at the top of this slide, getting ready to go down it. Can you feel yourself getting nervous? The strong wind is blowing your hair. Are you even going to go down the slide? Let’s learn a little more about the slide to help you decide. 

 

The function of the slide is f(x)= x²– 12x + 36.

What we want to do is pick out a spot on that slide at a certain time (x) and see how high we are (y) and how fast we are going.  To find out how fast we are going, we will take the derivative of the function. 

That is a little more complicated than the ones we did earlier, but this is calculus. You still got this.  This is not hard at all.
 
First, let’s make sure you know what an exponent is.
 
Look back at the function of the slide above.  The exponent is the little 2 above the first x in the function. When you see an exponent, it tells you how many times to multiply the number by itself.  An exponent is also known as a power.  Keep that word in mind; you will see it again in a few minutes.
 
Let’s do the following example.
 
2³ is 2 x 2 x 2 = 8 – Multiply 2 by itself 3 times.
 
3² is 3 x 3 = 9 = Multiply 3 by itself 1 time.
 
Ok.  Let’s look at the slicky slide. Let’s see where we are on the slide after sliding for 2 seconds.
 
Let’s put 2 in the function above.
 
Don’t worry about how we came up with the function.  That is a little beyond the scope of this class.  I don’t want to make this a math class.  But trust me when I say it’s not that hard.
 
Here we go.  The function below tells us where we are on the slide at a certain time.  So let’s see where we are at 2 seconds by plugging a two into the function.

f(x)= x²– 12x + 36

f(2) = (2)² -12(2) + 36
 
f(2) = 4 – 24 + 36
 
f(2) = 16
 
After 2 seconds, we are 16 feet high on the slide.  Remember, we started at about 36 feet.
 
Now we want to see how fast we are going at that point, and to do that, we finally take the derivative.
 

From the picture above, you can see that there are many notations that tell you to take a derivative.  We will use a few of them interchangeably in this class. Just understand they all mean the same thing. 

The symbol dy/dx is one of the symbols for taking a derivative so let’s take the derivative of our function.
 
Mathematically it looks like this.
 
dy/dx = x² -12x +36
 
This says, “take the derivative of y with respect to x.” To take this derivative, we are going to use the power rule. I told you that you would see the word power again. This is so simple.  Remember, my ten-year-old niece did this.

The Power Rule

 f'(x) = x² -12x +36

Notice the new notation.  This tells you to find “f prime of x.”  Which also means take the derivative.

To take the derivative, start with the x².  You take the exponent, which is two, and multiply it by the number in front of the x.
 
But wait a minute, there is no number in front of the x. 
 
Yes, there is.  If you don’t see a number, it means the number 1 is there.  You could write 1x.
 
We take the 1 and multiply it by the number 2 (the exponent), then decrease the exponent, which is 2 by 1. Now the exponent is 1.  The first part of the derivative is now 2x¹, which is 2x.  Don’t forget you don’t have to write ones.

Now, look at the -12x.  Numbers with just an x in front of them lose the x. So the second part of the derivative is just -12.

 
Finally, look at 36.  Numbers by themselves with no letter just disappear.  So, 36 is gone.
 
We finally did it.

f(x) = x² -12x +36 
f'(x) = 2x – 12 

The derivative of the function x² -12x +36 is 2x -12.
 
So now let’s see how fast we are going on the slide after 3 seconds.  To find this out, all we do is replace x with 3 into the derivative.

f'(x) = 2x – 12 
f'(3) = 2(3) – 12 
f'(3) = 6 – 12
f'(3) = -6  
 
So, after 3 seconds, you are going down the slide at a rate of -6 feet per second.
 
What? How can you slide negative feet? 
 
All that means in calculus is you are going down or backward.  If the number is positive, it means you are going up or forwards.  Obviously, you are going down the slide, so the number is negative.
 
Okay, let’s do one more real quick.
 
This time let’s see how high and fast we are going at 5 seconds on the slide.
 
First, find out how high you are by going back to the original function.
 
f(x) = x² – 12x + 36
f(5) = (5)² – 12(5) +36
 
f(5) = 25 – 60 + 36
 
f(5) = 1
 
At 5 seconds, you are now only 1 foot high on the slide. 
 
How fast are you going?
 
We already took the derivative.  So now all we do is plug 5 into it.
 
Remember, the derivative is 2x -12.

f'(x) = 2x – 12 
f'(5) = 2(5) – 12 
f'(5) = 10 – 12
f'(5) = -2
 
Remember, the answer is negative because you are going down.
 
At 5 seconds, you have slowed down to -2 feet per second.

You can see that it makes sense because the slope is getting smaller.

So now that you have all your information, do you want to go down the slide?  What do you think?
 
You just did Calculus.  You get it, don’t you?  Wasn’t too hard, was it? 
 
You just took on the math of maths. Calculus!  Calculus makes Algebra look like a wimp.
 
Most things are so much easier than you think they are.  In fact, many things which many think are hard are actually very easy if you will just take a little time.
 
Congratulations, you now know more Calculus than probably 90%, if not more, of the people on the planet.  That has to make you feel good about yourself.
 
Okay.  Yes, we did do some very easy Calculus.  The math in Calculus can get so hard that only computers can do it.  But be assured you now know enough Calculus for the purposes of this class. You will use your newfound skill for taking derivatives in future classes.  It will come in handy when we discuss Quantum Theory.

Okay, now it is time to draw a picture that represents what you have just learned and teach someone else.  Again, please do this.  It is very important.  You will understand why as we get into later lessons. Next lesson, we need to start discussing light.