Nice slopes

Slopes define rates of change.  Rates of change define most everything else.  It makes sense that if we want to be the masters of our universe (or at least it's librarians) we need to define slope.

The slope of a line is easy. Commonly represented as and understood as or even also known as

Lots of ways to represent slope, but they all mean the same thing.  We can use them to define a line given an equations in slope intercept form: where is what's known as the intercept.  For example, given the equation , we get the graph:

.

The slope here is easy to spot as 2 because we know where sits in our equation.  Without our inherent knowledge of slope intercept form though, we could calculate it easy enough using .  Taking two values of , we can plug them into the equation an get two corresponding values of . Using these 4 numbers, we can calculate the rate of change as a ratio of and .

So, here we go:

•  
• Which means:
• 
• Which means:
• So now, we have values for , , , and  
• as such we can find = = = =
• Resulting in

Given the slope, we can find our way on the line from any other place on the line by simply moving up 2 arbitrary units of measurement, and right 1.  This is great for lines.  It's not so great for curves.  So how do we react when given something that looks like this and we want to find the slope?

The answer is to find the derivative which I'll talk about next post.