An Overview of Calculus
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Calculus is the study of two different types of geometric objects and their applications. The first is differential calculus which is the study of slopes of functions and rates of change. The second is integral calculus which is the study of areas under a function and summations. What makes calculus unique is that it is able to deal with really tiny changes and amounts to add or divide.
Recall that the slope of a straight line is its “rise divided by its run.” What is meant by this is that you look how much a straight line rises (or falls) divided by the amount it took to “run.” This is the slope. But what is the slope when you have a function that is not a straight line? Differential calculus answers the question and gives techniques for finding the slope (if it exists). Because the slope is the amount of “rise” per amount of “run”, the derivative is useful to mathematically describe how quickly things change with respect to something else. In short, it describes a rate of change. For example, speed is the amount of distance per time. If you were to graph distance from home and at what time you would be able to tell your speed at each time by the derivative of this function.
Integral calculus is the study of areas under a function. If we again think about a straight line, the area under the straight line is the height times the width. However, what if we don't have a straight line? How do we determine the area under an arbitrary function? Integral calculus answers the question and gives techniques for finding the area under the curve. Because the area is a length times a width, integral calculus is useful for those situations where you have “something that constantly changes” times “some interval”. For example, the net-present-value of busines's cashflows is the cashflow times a discount factor. What if those cashflows are constantly changing? The integral is useful for multiplying a constantly changing money flow times the discount function.
