What Is Calculus?

By this point, you should be familiar with using functions and solving equations (and systems of equations) involving real numbers using the techniques you learned in Algebra.  With Calculus we are allowed to do things that we are not allowed to do in Algebra using two mathematical constructs in particular: infinity, which is larger than every number, and the infinitesimal, which is smaller than every number.

Calculus I, AP Calculus AB

You remember from Algebra that the slope of a line can be determined using the change in y divided by the change in x.  The slope is simply one number that tells you the direction the line is going.  However, other functions do not have a definite slope.  You can tell if a function is increasing or decreasing by looking at a graph, and if you were to pick two values of x, say a and b, you could find what’s known as the average rate of change between a and b just by finding the change in y and dividing it by the change in x.

Sometimes, however, we want to find something called the instantaneous rate of change, or the direction in which a function is going at one value of x.  We cannot use the average rate of change formula for one point, because the changes in y and in x are both 0, and we are not allowed to divide by 0.  What we can do, however, is look at the average rate of change from a to some variable b, and see what happens to the average rate of change as we move b closer to a.  As b moves closer to a, we see that both the changes in  y and in x are approaching 0, and we can see what happens to that rate the closer they get.  This process is called differentiating a function, or taking the derivative of a function.

Another thing you will learn to do in Calculus I is to find the area bound by a function.  You know how to find the area of a rectangle by multiplying it’s length by width, but if you want to find the area bound between a curve and the x-axis, you will learn to use a process called Riemann integration.

Calculus II, AP Calculus BC

Calculus II continues what you’ve learned in Calc I, and in most cases it will cover three topics:

• Techniques of integration, where you will learn how to integrate more complicated functions, and how to choose what technique to use by looking at a function.
• Intro to differential equations, where you learn to make mathematical models using a differential equation, or an equation with a derivative in it.
• Infinite sequences and series.  An infinite sequence is an ordered list of numbers, and here you will learn how to tell whether or not a sequence converges.  An infinite series is the sum of all the terms in a sequence, and you will also learn how to tell whether or not a series will converge.  You will also learn how to represent some functions, such as sin(x), as a special kind of infinite series called a power series.

Calculus III

Calc III, often called Multivariable Calculus or Vector Calculus, involves applying the techniques learned in Calc I and II to functions of more than one variable.  You will learn how to take a partial derivative, which differentiates a multivariable function with respect to only one variable, and you will learn multiple integration, which involves integrating a multivariable function with