At a Glance - Basic Elements
What are the nuts and bolts of calculus? What makes it tick? You’ve come to the right place. Most of the basic elements are probably new concepts, and some seem complicated at first glance.
The first basic element is the function, which we've dealing with functions since the good ol' days of Algebra. They usually look something like this: f(x) = x. A function is a rule. When we plug a value into the function, we’ll get another (or the same) value out. In each function, there is also an independent variable (the input) and a dependent variable (the output) .
But the concept that underlies literally everything in calculus is the limit. A limit is a value that the dependent variable in a function approaches as the independent variable approaches a given value. As boats approach their docks, functions approach values. One way to find a limit of a function is to plug in values close to the desired value into the independent variable and see what the dependent variable is approaching.
The concept of a slope is an idea from algebra. It’s the (change in y) divided by the (change in x) By finding the slope between two points on a graph, we were able to determine the average rate of change between those points. A derivative takes this to next level. It's essentially a slope, but now we'll be able to find the slope at a point on a curvy line, instead of just between two points. Are limits involved here? You better believe it.
An integral is a way to find an area. Integrals can be used to find the area of a circle, a square, or an irregular wavy enclosed region (think of a fancy pool). It’s essentially the opposite operation of a derivative, and it’s another way to pull more data from a graph. Computing integrals relies heavily on limits and derivatives. In calculus, the concepts just keep building off of previous ones. Get used to it.