A basic concept in calculus that simplifies many of the later concepts is the limit. Limits hinge on the idea of approach. Instead of asking what is the value of a function when x equals 3, for example, it asks what value is the function approaching as x approaches 3. This is written as follows:
Lim x --> 3 f(x) = L
which means the limit of f of x as x approaches 3 equals L. Note that when x = 3, f(x) doesn't have to equal L. The limit is the value that f(x) is approaching at that point.
Tangents are straight lines that represent the slope of a curve at a particular point on that curve. On a parabola, for instance, the slope of the tangent line will be constantly changing, and it requires limits to calculate. The slope of a line is the change in y over the change in x, but this can be calculated algebraically only over two points. With limits, you calculate the change in y over the change in x as the two points approach one another. Thus, solving the limit will give the instantaneous slope, or the slope of the curve at any single point.
Derivatives are secondary functions describing the instantaneous slope for each point of an original function. The calculations used to find the derivative of a curve are the same as those used to find a tangent of a point on that curve. However, by using variables instead of numerals, a function is generated instead of a value. The derivative has many applications in physics. For instance, the derivative of the function describing the position of an object is a graph of that object's instantaneous speed.
The process of integration is the opposite process of derivation. The process was first developed to find the area underneath curves. By breaking up the curves into rectangles with an easily calculable area, the area underneath the curve can be estimated. Using the notion of the limit, the number of rectangles used to calculate can approach infinity, and meanwhile, the sum of the areas of those rectangles will approach the exact area underneath the curve.