A brief introduction to advanced mathematics

Calculus is a scary word. As a high school senior, I dropped out of AP Calculus after the first semester. However, I later took Calculus I, II, III, and IV in College. I will begin by listing some prerequisites to Calculus I, followed by a brief introduction to Calculus I (called Differential Calculus).


Prerequisites:
At least one year of algebra is required
One year of stats is helpful but not required
One year of physics is helpful, but if you have not taken physics, then this will help you there too!





Calculus is often broken into 3 major sections: Differential Calculus, Integral Calculus, and Differential Equations (a branch of Calculus, used most often in applied Calculus). To illustrate Calculus I, I would like you to get a sheet of paper, and draw an axis on it, using x and y axis; +. Draw the following line: y=2x+0.
This should be a straight line, starting in the lower left, and moving to the upper right. Now, what is the slope? what is slope?

formal def: Slope is the change in y over change in x
informal def: slope is how steep the line is.

in our case, slope is 2, because for every unit on the x axis we move, we move 2 units on the y axis. There now, that wasn't so hard, right?

Next: draw the following line on your paper: y=x^2 (read as x squared). This should be a parabola (think a half-pipe). Now I ask you, what is the slope, or how steep is the line? If you answered i don't know, than your on the right track! the slope varies, in the middle, the slope is very flat, but as we get higher and higher, the slope gets steeper and steeper. Calculus I is all about finding the slope on equations like this one, where the slope changes.

Lets start with some approximations. try to approximate the slope at each unit of interval (0,1,2,3,4 etc). write it out on a table like this:

 x   |   slope
0    |   0 (flat)
1    |   2 (about the same as our first line)
2    |   4 (about double the steepness of 2)
3    |   6 (starting to notice a pattern?)


if we create a new line, using our slope approximations as the y value, we should be a line of y=2x. This is a representation of how the slope of y=x^2 changes as it gets steeper. That's differential calculus! Calculus I is all about finding the slope of non-linear lines (aka, lines that are not straight). Now, what if we don't want to draw out all the lines, and we want to solve this all algebraically?

The formal equation for determining the slope at a given point is:

f '(x) = lim (delta x->0)   f(x+delta x)-f(x) 
                                           delta x

from here, we substitute in our equation:


f '(x) = lim (delta x -> 0) f(x+delta x)^2 - f(x)^2
                                               delta x


f '(x) = lim (delta x -> -) x^2 + 2x*delta x + delta x^2 - x^2
                                                  delta x

f '(x) = lim (delta x -> 0) 2x*delta x + delta x^2
                                                   delta x

f '(x) = lim (delta x-> 0 2x+ delta x

now at this point, remember that delta x is approaching zero, so we can cancel out delta x

we are left with f '(x) = 2x

simple right? :) not really? oh well, there is a trick for exponents. This will be much easier, believe me.

y=x^2

first, we take the exponent (2) and multiply it by our x variable.

y=2x^2

next, we take our exponent, and subtract 1 from the total

y=2x^1

which is the same as saying

y=2x

huh. that's odd, how did that work? well, there is a proof on why this works, but for now lets just test it on a few equations until we believe it. I will let you research the proof on your own.

Lets try this one:     y=2x^2

go ahead and try it using all three methods weve tried above.

I got y=4x, what did you get?

Okay, that's enough math for today.

woof!